tag:blogger.com,1999:blog-70722046726770436412024-03-13T10:34:32.966-07:00Welcome to Mathmatics Universe: For Class V To XIIthis blog is all about the mathematics of class V to XII maths,and detail of the solution ,all chapter contain in the there class.I am provide the sol of maths to all class.Anonymoushttp://www.blogger.com/profile/12500986417656529636noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-7072204672677043641.post-30816178296748039192012-12-05T10:00:00.000-08:002012-12-06T05:09:53.410-08:00Basic arithmetics math formulas<div dir="ltr" style="text-align: left;" trbidi="on">
<b>TIME AND DISTANCE </b><br />
<b><br /></b>
<br />
<div class="bix-div-introduction" style="background-color: white; font-family: Verdana; font-size: 12px;">
<ol type="1">
<li><div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<i class="ga-fhead" style="color: #555555; font-family: Verdana, Arial, Helvetica, sans-serif; font-style: normal; font-weight: bold;">Speed, Time and Distance:</i></div>
<div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<table cellpadding="0" cellspacing="0" class="ga-tbl-answer" style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px;"><tbody>
<tr align="center" class="ga-tr-divident"><td class="ga-td-line-rpad" rowspan="2" style="padding-right: 7px; vertical-align: middle;">Speed =</td><td class="ga-td-line" rowspan="2" style="height: auto; padding: 0px; vertical-align: middle; white-space: nowrap;"><img src="http://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif" style="border: 0px; vertical-align: middle;" /></td><td class="ga-td-divident" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; padding-bottom: 2px; vertical-align: bottom;">Distance</td><td class="ga-td-line" rowspan="2" style="height: auto; padding: 0px; vertical-align: middle; white-space: nowrap;"><img src="http://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif" style="border: 0px; vertical-align: middle;" /></td><td class="ga-td-line" rowspan="2" style="height: auto; padding: 0px; vertical-align: middle; white-space: nowrap;">,</td><td class="ga-td-line-lrpad" rowspan="2" style="padding-left: 7px; padding-right: 7px; vertical-align: middle;">Time =</td><td class="ga-td-line" rowspan="2" style="height: auto; padding: 0px; vertical-align: middle; white-space: nowrap;"><img src="http://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif" style="border: 0px; vertical-align: middle;" /></td><td class="ga-td-divident" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; padding-bottom: 2px; vertical-align: bottom;">Distance</td><td class="ga-td-line" rowspan="2" style="height: auto; padding: 0px; vertical-align: middle; white-space: nowrap;"><img src="http://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif" style="border: 0px; vertical-align: middle;" /></td><td class="ga-td-line" rowspan="2" style="height: auto; padding: 0px; vertical-align: middle; white-space: nowrap;">,</td><td class="ga-td-line-lrpad" rowspan="2" style="padding-left: 7px; padding-right: 7px; vertical-align: middle;">Distance = (Speed x Time).</td></tr>
<tr align="center" class="ga-tr-divisor"><td class="ga-td-divisor" style="padding-top: 2px; vertical-align: top;">Time</td><td class="ga-td-divisor" style="padding-top: 2px; vertical-align: top;">Speed</td></tr>
</tbody></table>
</div>
</li>
<li><div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<i class="ga-fhead" style="color: #555555; font-family: Verdana, Arial, Helvetica, sans-serif; font-style: normal; font-weight: bold;">km/hr to m/sec conversion:</i></div>
<div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<i class="ga-fhead" style="color: #555555; font-family: Verdana, Arial, Helvetica, sans-serif; font-style: normal; font-weight: bold;"><br /></i></div>
<div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<table cellpadding="0" cellspacing="0" class="ga-tbl-answer" style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px;"><tbody>
<tr align="center" class="ga-tr-divident"><td class="ga-td-line-rpad" rowspan="2" style="padding-right: 7px; vertical-align: middle;"><i class="ga-var" style="margin-left: 1px; margin-right: 1px;">x</i> km/hr =</td><td class="ga-td-line" rowspan="2" style="height: auto; padding: 0px; vertical-align: middle; white-space: nowrap;"><img src="http://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif" style="border: 0px; vertical-align: middle;" /></td><td class="ga-td-line-rpad" rowspan="2" style="padding-right: 7px; vertical-align: middle;"><i class="ga-var" style="margin-left: 1px; margin-right: 1px;">x</i> x</td><td class="ga-td-divident" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; padding-bottom: 2px; vertical-align: bottom;">5</td><td class="ga-td-line" rowspan="2" style="height: auto; padding: 0px; vertical-align: middle; white-space: nowrap;"><img src="http://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif" style="border: 0px; vertical-align: middle;" /></td><td class="ga-td-line-lpad" rowspan="2" style="padding-left: 7px; vertical-align: middle;">m/sec.</td></tr>
<tr align="center" class="ga-tr-divisor"><td class="ga-td-divisor" style="padding-top: 2px; vertical-align: top;">18</td></tr>
</tbody></table>
</div>
</li>
<li><div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<i class="ga-fhead" style="color: #555555; font-family: Verdana, Arial, Helvetica, sans-serif; font-style: normal; font-weight: bold;">m/sec to km/hr conversion:</i></div>
<div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<i class="ga-fhead" style="color: #555555; font-family: Verdana, Arial, Helvetica, sans-serif; font-style: normal; font-weight: bold;"><br /></i></div>
<div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<i class="ga-fhead" style="color: #555555; font-family: Verdana, Arial, Helvetica, sans-serif; font-style: normal; font-weight: bold;"><br /></i></div>
<div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<table cellpadding="0" cellspacing="0" class="ga-tbl-answer" style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px;"><tbody>
<tr align="center" class="ga-tr-divident"><td class="ga-td-line-rpad" rowspan="2" style="padding-right: 7px; vertical-align: middle;"><i class="ga-var" style="margin-left: 1px; margin-right: 1px;">x</i> m/sec =</td><td class="ga-td-line" rowspan="2" style="height: auto; padding: 0px; vertical-align: middle; white-space: nowrap;"><img src="http://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif" style="border: 0px; vertical-align: middle;" /></td><td class="ga-td-line-rpad" rowspan="2" style="padding-right: 7px; vertical-align: middle;"><i class="ga-var" style="margin-left: 1px; margin-right: 1px;">x</i> x</td><td class="ga-td-divident" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; padding-bottom: 2px; vertical-align: bottom;">18</td><td class="ga-td-line" rowspan="2" style="height: auto; padding: 0px; vertical-align: middle; white-space: nowrap;"><img src="http://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif" style="border: 0px; vertical-align: middle;" /></td><td class="ga-td-line-lpad" rowspan="2" style="padding-left: 7px; vertical-align: middle;">km/hr.</td></tr>
<tr align="center" class="ga-tr-divisor"><td class="ga-td-divisor" style="padding-top: 2px; vertical-align: top;">5</td></tr>
</tbody></table>
</div>
</li>
<li><div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
If the ratio of the speeds of A and B is <i class="ga-var" style="font-family: Verdana, Arial, Helvetica, sans-serif; margin-left: 1px; margin-right: 1px;">a</i> : <i class="ga-var" style="font-family: Verdana, Arial, Helvetica, sans-serif; margin-left: 1px; margin-right: 1px;">b</i>, then the ratio of the</div>
<div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<br /></div>
<div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<table cellpadding="0" cellspacing="0" class="ga-tbl-answer" style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px;"><tbody>
<tr align="center" class="ga-tr-divident"><td class="ga-td-line-rpad" rowspan="2" style="padding-right: 7px; vertical-align: middle;">the times taken by then to cover the same distance is</td><td class="ga-td-divident" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; padding-bottom: 2px; vertical-align: bottom;">1</td><td class="ga-td-line-lrpad" rowspan="2" style="padding-left: 7px; padding-right: 7px; vertical-align: middle;">:</td><td class="ga-td-divident" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; padding-bottom: 2px; vertical-align: bottom;">1</td><td class="ga-td-line-lrpad" rowspan="2" style="padding-left: 7px; padding-right: 7px; vertical-align: middle;">or <i class="ga-var" style="margin-left: 1px; margin-right: 1px;">b</i> : <i class="ga-var" style="margin-left: 1px; margin-right: 1px;">a</i>.</td></tr>
<tr align="center" class="ga-tr-divisor"><td class="ga-td-divisor" style="padding-top: 2px; vertical-align: top;"><i class="ga-var" style="margin-left: 1px; margin-right: 1px;">a</i></td><td class="ga-td-divisor" style="padding-top: 2px; vertical-align: top;"><i class="ga-var" style="margin-left: 1px; margin-right: 1px;">b</i></td></tr>
</tbody></table>
</div>
<div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<br /></div>
</li>
<li><div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
Suppose a man covers a certain distance at <i class="ga-var" style="font-family: Verdana, Arial, Helvetica, sans-serif; margin-left: 1px; margin-right: 1px;">x</i> km/hr and an equal distance at <i class="ga-var" style="font-family: Verdana, Arial, Helvetica, sans-serif; margin-left: 1px; margin-right: 1px;">y</i>km/hr. Then,</div>
<div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<br /></div>
<div style="font-family: Verdana, Tahoma, Arial, sans-serif; line-height: 17px;">
<table cellpadding="0" cellspacing="0" class="ga-tbl-answer" style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px;"><tbody>
<tr align="center" class="ga-tr-divident"><td class="ga-td-line-rpad" rowspan="2" style="padding-right: 7px; vertical-align: middle;">the average speed during the whole journey is</td><td class="ga-td-line" rowspan="2" style="height: auto; padding: 0px; vertical-align: middle; white-space: nowrap;"><img src="http://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif" style="border: 0px; vertical-align: middle;" /></td><td class="ga-td-divident" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; padding-bottom: 2px; vertical-align: bottom;">2<i class="ga-var" style="margin-left: 1px; margin-right: 1px;">xy</i></td><td class="ga-td-line" rowspan="2" style="height: auto; padding: 0px; vertical-align: middle; white-space: nowrap;"><img src="http://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif" style="border: 0px; vertical-align: middle;" /></td><td class="ga-td-line-lpad" rowspan="2" style="padding-left: 7px; vertical-align: middle;">km/hr.</td></tr>
<tr align="center" class="ga-tr-divisor"><td class="ga-td-divisor" style="padding-top: 2px; vertical-align: top;"><i class="ga-var" style="margin-left: 1px; margin-right: 1px;">x</i> + <i class="ga-var" style="margin-left: 1px; margin-right: 1px;">y</i></td></tr>
</tbody></table>
</div>
</li>
</ol>
</div>
<hr style="background-color: #d4d4d4; border-width: 0px; clear: both; color: #d4d4d4; font-family: Verdana, Tahoma, Arial, sans-serif; height: 1px;" />
<br /></div>
Anonymoushttp://www.blogger.com/profile/12500986417656529636noreply@blogger.com2tag:blogger.com,1999:blog-7072204672677043641.post-35183625318331682222012-12-02T23:09:00.003-08:002012-12-04T22:09:05.449-08:00Introduction to Coordinate Geometry<div dir="ltr" style="text-align: left;" trbidi="on">
<br />
<h2 style="background-color: white; border-color: rgb(192, 192, 192); border-style: solid; border-width: 0px 0px 1px; clear: both; color: #447799; font-family: sans-serif; font-size: 17px; margin: 25px 0px 0px;">
What are coordinates</h2>
<div>
<div>
In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on axioms and theorems to derive truth. Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry.</div>
<div>
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (3 dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. The numerical output, however, might also be a vector or a shape. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.</div>
</div>
<div>
<br /></div>
<div>
<a class="image" href="http://en.wikipedia.org/wiki/File:Punktkoordinaten.svg" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12px; line-height: 19.200000762939453px; text-align: center; text-decoration: initial;"><img alt="" class="thumbimage" height="292" src="http://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Punktkoordinaten.svg/350px-Punktkoordinaten.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Punktkoordinaten.svg/525px-Punktkoordinaten.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Punktkoordinaten.svg/700px-Punktkoordinaten.svg.png 2x" style="background-color: white; border: 1px solid rgb(204, 204, 204); vertical-align: middle;" width="350" /></a></div>
<div>
<br /></div>
<div>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Coordinates">Coordinates</span></h3>
</div>
<div>
<br /></div>
<div>
<div>
In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position. These are typically written as an ordered pair (x, y). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates (x, y, z).</div>
<div>
Other coordinate systems are possible. On the plane the most common alternative is polar coordinates, where every point is represented by its radius r from the origin and its angle θ. In three dimensions, common alternative coordinate systems include cylindrical coordinates and spherical coordinates</div>
<div>
<br /></div>
<div>
<a class="image" href="http://en.wikipedia.org/wiki/File:Cartesian-coordinate-system.svg" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12px; line-height: 19.200000762939453px; text-align: center;"><img alt="" class="thumbimage" height="250" src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/250px-Cartesian-coordinate-system.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/375px-Cartesian-coordinate-system.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/500px-Cartesian-coordinate-system.svg.png 2x" style="background-color: white; border: 1px solid rgb(204, 204, 204); vertical-align: middle;" width="250" /></a></div>
<div>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
</h3>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
Equations of curves</h3>
</div>
<div>
<br /></div>
<div>
In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation. For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal. These points form a line, and y = x is said to be the equation for this line. In general, linear equations involving x and y specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.</div>
</div>
<div>
<div>
Usually, a single equation corresponds to a curve on the plane. This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x2 + y2 = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces (see below), or as a system of parametric equations. The equation x2 + y2 = r2 is the equation for any circle with a radius of r.</div>
</div>
<div>
<br /></div>
<div>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Distance_and_angle">Distance and angle</span></h3>
</div>
<div>
<span class="mw-headline"></span><br />
<div>
<span class="mw-headline">n analytic geometry, geometric notions such as distance and angle measure are defined using formulas. These definitions are designed to be consistent with the underlying Euclidean geometry. For example, using Cartesian coordinates on the plane, the distance between two points (x1, y1) and (x2, y2) is defined by the formula</span></div>
<span class="mw-headline">
</span>
<br />
<div>
<span class="mw-headline"> <img alt="d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2},\!" class="tex" src="http://upload.wikimedia.org/math/0/1/2/012f0f054e332b767a2b646220da5684.png" style="background-color: white; border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /> </span></div>
<span class="mw-headline">
</span>
<br />
<div>
<span class="mw-headline"><br /></span></div>
<span class="mw-headline">
</span>
<br />
<div>
<span class="mw-headline">which can be viewed as a version of the Pythagorean theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula</span></div>
<span class="mw-headline">
</span>
<br />
<div>
<span class="mw-headline"> <img alt="\theta = \arctan(m)\!" class="tex" src="http://upload.wikimedia.org/math/4/4/6/44674d3480ad4ec9419187194e3ca272.png" style="background-color: white; border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /></span></div>
<span class="mw-headline">
</span>
<div>
<span class="mw-headline">where m is the slope of the line.</span></div>
<span class="mw-headline">
<div>
<br /></div>
<div>
<a class="image" href="http://en.wikipedia.org/wiki/File:Distance_Formula.svg" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12px; line-height: 19.200000762939453px; text-align: center; text-decoration: initial;"><img alt="" class="thumbimage" height="200" src="http://upload.wikimedia.org/wikipedia/commons/thumb/6/67/Distance_Formula.svg/250px-Distance_Formula.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Distance_Formula.svg/375px-Distance_Formula.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Distance_Formula.svg/500px-Distance_Formula.svg.png 2x" style="background-color: white; border: 1px solid rgb(204, 204, 204); vertical-align: middle;" width="250" /></a><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 12px; line-height: 19.200000762939453px; text-align: center;"></span><br />
<div class="thumbcaption" style="border: none; font-family: sans-serif; font-size: 11px; line-height: 1.4em; padding: 3px !important;">
<div class="magnify" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; border: none !important; float: right;">
<a class="internal" href="http://en.wikipedia.org/wiki/File:Distance_Formula.svg" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; border: none !important; color: #0b0080; display: block; text-decoration: initial;" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/static-1.21wmf4/skins/common/images/magnify-clip.png" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; border: none !important; display: block; vertical-align: middle;" width="15" /></a></div>
The distance formula on the plane follows from the Pythagorean theorem.</div>
</div>
</span></div>
<div>
<span class="mw-headline"><br /></span></div>
<div>
<span class="mw-headline"></span><br />
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline">
<span class="mw-headline" id="Section_of_a_line">Section of a line</span></span></h3>
<span class="mw-headline">
</span>
<br />
<div>
<span class="mw-headline"><span class="mw-headline"></span><br /></span>
<br />
<div style="background-color: white; margin-bottom: 0.5em; margin-top: 0.4em;">
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;"></span></span></span></div>
<span class="mw-headline"><span class="mw-headline">
</span></span>
<br />
<div style="font-size: small; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">In Analytical Geometry a section of a line can be given by the formula where (c,d)&(e,f) are the endpoints of the line & m:n is the ratio of division</span></span></span></div>
<span class="mw-headline"><span class="mw-headline">
</span></span>
<br />
<div style="font-size: small; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">S(a,b)=(nc+me/m+n, nd+mf/m+n)</span></span></span></div>
<span class="mw-headline"><span class="mw-headline">
</span></span>
<br />
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
</div>
<span class="mw-headline"><span class="mw-headline">
</span></span>
<br />
<h3 style="background-image: none; border-bottom-style: none; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline"><span class="mw-headline">
<span style="font-family: sans-serif;"><span class="mw-headline" id="Transformations">Transformations</span></span></span></span></h3>
<span class="mw-headline"><span class="mw-headline">
</span></span>
<div style="font-size: small; line-height: 19.200000762939453px;">
<div>
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">Transformations are applied to parent functions to turn it into a new function with similar characteristics. For example, the parent function y=1/x has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote,and occupies either the 1st and 3rd or 2nd and 4th quadrant. In general, if y = f(x), then it can be transformed into y = af(b(x − k)) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis. The b value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like a, reflects the function in the y-axis when it is negative. The k and h values introduce translations, h, vertical, and k horizontal. Positive h and k values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.</span></span></span></div>
<div>
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">Transformations can be applied to any geometric equation whether or not the equation represents a function. Transformations can be considered as individual transactions or in combinations.</span></span></span></div>
<div>
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">Suppose that R(x,y) is a relation in the xy plane. For example</span></span></span></div>
<div>
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">x2 + y2 -1= 0</span></span></span></div>
<div>
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">is the relation that describes the unit circle. The graph of R(x,y) is changed by standard transformations as follows:</span></span></span></div>
<div>
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">Changing x to x-h moves the graph to the right h units.</span></span></span></div>
<div>
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">Changing y to y-k moves the graph up k units.</span></span></span></div>
<div>
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">Changing x to x/b stretches the graph horizontally by a factor of b. (think of the x as being dilated)</span></span></span></div>
<div>
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">Changing y to y/a stretches the graph vertically.</span></span></span></div>
<div>
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">Changing x to xcosA+ ysinA and changing y to -xsinA + ycosA rotates the graph by an angle A.</span></span></span></div>
<div>
<span class="mw-headline"><span class="mw-headline"><span style="font-family: sans-serif;">There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered. For more information, consult the Wikipedia article on affine transformations.</span></span></span></div>
</div>
<span class="mw-headline"><span class="mw-headline">
<div style="font-size: small; line-height: 19.200000762939453px;">
<span style="font-family: sans-serif;"><span style="font-size: 17px;"><br /></span></span></div>
<div style="font-size: small; line-height: 19.200000762939453px;">
<span style="font-family: sans-serif;"><span style="font-size: 17px;"><b>Intersections</b></span></span></div>
<div style="font-size: small; line-height: 19.200000762939453px;">
<span style="font-family: sans-serif;"><span style="font-size: 17px;"><b><br /></b></span></span></div>
<div>
<span style="font-family: sans-serif;"><span style="font-size: x-small;"><span style="line-height: 19.200000762939453px;">While this discussion is limited to the xy-plane, it can easily be extended to higher dimensions. For two geometric objects P and Q represented by the relations P(x,y) and Q(x,y) the intersection is the collection of all points (x,y) which are in both relations. For example, P might be the circle with radius 1 and center (0,0): P = {(x,y) | x2+y2=1} and Q might be the circle with radius 1 and center (1,0): Q = {(x,y) | (x-1)2+y2=1}. The intersection of these two circles is the collection of points which make both equations true. Does the point (0,0) make both equations true? Using (0,0) for (x,y), the equation for Q becomes (0-1)2+02=1 or (-1)2=1 which is true, so (0,0) is in the relation Q. On the other hand, still using (0,0) for (x,y) the equation for P becomes (0)2+02=1 or 0=1 which is false. (0,0) is not in P so it is not in the intersection.</span></span></span></div>
<div>
<span style="font-family: sans-serif;"><span style="font-size: x-small;"><span style="line-height: 19.200000762939453px;"><br /></span></span></span></div>
<div>
<span style="font-family: sans-serif;"><span style="font-size: 13px; line-height: 19.200000762939453px;">The intersection of P and Q can be found by solving the simultaneous equations:</span></span></div>
<div>
<span style="font-family: sans-serif;"><span style="font-size: 13px; line-height: 19.200000762939453px;"><br /></span></span></div>
<div>
<span style="font-family: sans-serif;"><span style="font-size: 13px; line-height: 19.200000762939453px;"> <b> x</b></span><b><sup style="line-height: 1em;">2</sup><span style="font-size: 13px; line-height: 19.200000762939453px;">+y</span><sup style="line-height: 1em;">2</sup><span style="font-size: 13px; line-height: 19.200000762939453px;"> = 1</span></b></span></div>
<div>
<span style="font-family: sans-serif;"><span style="font-size: 13px; line-height: 19.200000762939453px;"><b><br /></b></span></span></div>
<div>
<span style="font-family: sans-serif;"><b><span style="font-size: 13px; line-height: 19.200000762939453px;"> (x-1)</span><sup style="line-height: 1em;">2</sup><span style="font-size: 13px; line-height: 19.200000762939453px;">+y</span><sup style="line-height: 1em;">2</sup><span style="font-size: 13px; line-height: 19.200000762939453px;"> = 1</span></b></span></div>
<h3 class="style5" style="font-family: Arial, Helvetica, sans-serif; font-size: 20px;">
Explanation for the coordinate geometry of a straight line--</h3>
<div>
<br /></div>
</span></span></div>
<span class="mw-headline">
</span></div>
<div>
<span style="line-height: 19.200000762939453px;"><span class="mw-headline" style="font-family: sans-serif; font-size: x-small;"></span></span><br />
<div style="background-color: white; margin-bottom: 0.5em; margin-top: 0.4em;">
<span class="mw-headline" style="font-family: sans-serif; font-size: x-small; line-height: 19.200000762939453px;"></span><br />
<span class="mw-headline" style="font-family: sans-serif; font-size: x-small; line-height: 19.200000762939453px;">Every straight line can be represented algebraically in the form y = mx + c , where</span><br />
<span class="mw-headline" style="font-family: sans-serif; font-size: x-small; line-height: 19.200000762939453px;"><br /></span>
<span class="mw-headline" style="font-family: sans-serif; font-size: x-small; line-height: 19.200000762939453px;"> m = represents the gradient of a line (its slope, steepness)</span><br />
<span class="mw-headline" style="font-family: sans-serif; font-size: x-small; line-height: 19.200000762939453px;"><br /></span>
<span class="mw-headline" style="font-family: sans-serif; font-size: x-small; line-height: 19.200000762939453px;"> c = represents the y -intercept (a point where the line crosses the y axis)</span><br />
<span class="mw-headline" style="font-family: sans-serif; font-size: x-small; line-height: 19.200000762939453px;"><br /></span>
<span class="mw-headline" style="font-family: sans-serif; font-size: x-small; line-height: 19.200000762939453px;">Furthermore, there are several ways in which you can describe a straight line algebraically</span><br />
<span class="mw-headline" style="font-family: sans-serif; font-size: x-small; line-height: 19.200000762939453px;"><br /></span>
<span class="mw-headline" style="font-family: sans-serif; font-size: x-small; line-height: 19.200000762939453px;"><b style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 16px; line-height: normal;">Equation of a line=</b></span><br />
<br />
<table cellpadding="0" cellspacing="0" style="color: black; font-family: 'Times New Roman'; text-align: start;"><tbody>
<tr><td valign="top" width="85"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<img height="23" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate1.doc" width="90" /></div>
</td><td valign="top" width="358"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
Line gradient <i>m </i>, the intercept on <i>y </i>-axis is c</div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<br /></div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<table cellpadding="0" cellspacing="0" style="color: black; font-family: 'Times New Roman'; text-align: start;"><tbody>
<tr><td valign="top" width="85"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<img height="21" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate2.doc" width="61" />,</div>
</td><td valign="top" width="358"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
Line gradient <i>m </i>, passes through the origin</div>
</td></tr>
</tbody></table>
</div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<br /></div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<table cellpadding="0" cellspacing="0" style="color: black; font-family: 'Times New Roman'; text-align: start;"><tbody>
<tr><td valign="top" width="85"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<img height="23" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate3.doc" width="75" />,</div>
</td><td valign="top" width="358"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
Line gradient 1, makes an angle of 45 0 with the <i>x </i>-axis, and the intercept on the <i>y </i>axis is <i>c</i></div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<i><br /></i></div>
</td></tr>
<tr><td valign="top" width="85"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<img height="26" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate4.doc" width="46" />,</div>
</td><td valign="top" width="358"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
Line is parallel to the <i>x </i>axis, through (0, <i>k)</i></div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<i><br /></i></div>
</td></tr>
<tr><td valign="top" width="85"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<img height="25" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate5.doc" width="46" /></div>
</td><td valign="top" width="358"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
Line is the <i>x </i>axis</div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<br /></div>
</td></tr>
<tr><td valign="top" width="85"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<img height="21" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate6.doc" width="48" />.</div>
</td><td valign="top" width="358"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
Line is parallel to the <i>y </i>axis, through ( <i>k </i>, 0)</div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<br /></div>
</td></tr>
<tr><td valign="top" width="85"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<img height="20" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate7.doc" width="48" />,</div>
</td><td valign="top" width="358"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
Line is the <i>y </i>axis</div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<br /></div>
</td></tr>
<tr><td valign="top" width="85"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<img height="26" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate8.doc" width="123" /></div>
</td><td valign="top" width="358"><div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
, General form of the equation of straight line</div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<br /></div>
</td></tr>
</tbody></table>
</div>
<div class="style1">
The gradient measures the steepness of the line.</div>
<div class="style1">
<br /></div>
<div class="style1">
It is defined as</div>
<div class="style1">
<img height="51" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate9.doc" width="98" />, or <img height="56" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate91.doc" width="98" /></div>
<div class="style1">
<br /></div>
<div class="style1">
When the gradient is 1, the line makes a 45 0 angle with either axes. If the gradient is 0, the line is parallel to the <i>x </i>axis.</div>
</td></tr>
</tbody></table>
<span style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 16px; font-weight: bold;"><br /></span>
<span style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 16px; font-weight: bold;">Equation of a straight line given the gradient and a point=</span><br />
<br />
<div style="text-align: left;">
On the coordinate plane, the slant of a line is called the slope. Slope is the ratio of the change in the y-value over the change in the x-value, also called rise over run.<br />
Given any two points on a line, you can calculate the slope of the line by using this formula:<br />
</div>
<div style="text-align: left;">
<i> slope = </i><img align="absmiddle" alt="change in y/change in x" height="45" src="http://www.onlinemathlearning.com/image-files/coord-geo-slope-1.gif" width="121" /></div>
<br />
<span style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 16px; font-weight: bold;"><br /></span>
<br />
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px; text-align: left;">
f the point is given by its coordinates <img height="26" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate92.doc" width="56" />, and the gradient of a line is given as <i>m </i>, you can deduce the equation of that line.</div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px; text-align: left;">
You are using the formula for gradient, <img height="56" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate93.doc" width="98" />, to derive a formula for the line itself. The coordinates of the point can be substituted, while the <i>y 2 </i>and <i>x 2 </i>need to remain (without the superscript numbers).</div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px; text-align: left;">
Then simply substitute the given values into</div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px; text-align: left;">
<img height="26" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate94.doc" width="146" /></div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<br /></div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<span style="font-size: 16px; font-weight: bold;">The equation of a line given two points=</span></div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<span style="font-size: 16px; font-weight: bold;"><br /></span></div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
</div>
<div class="style1">
When you have this kind of problem, you take that, as both points belong to the same line, the gradients at both points will be the same.</div>
<div class="style1">
It makes sense therefore to say that <img height="56" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate95.doc" width="138" /></div>
<div class="style1">
All you need to do in this case will be to substitute coordinates you have for the given points<img height="26" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate96.doc" width="71" /> and <img height="26" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate97.doc" width="75" />.</div>
<div class="style1">
<br /></div>
<div class="style1">
<span style="font-size: 16px; font-weight: bold;">Parallel and perpendicular lines-</span></div>
<div class="style1">
<span style="font-size: 16px; font-weight: bold;"><br /></span></div>
<div class="style1">
</div>
<div class="style1">
When two lines are parallel, their gradient is the same: <img height="26" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate98.doc" width="70" /><br />
<br /></div>
<div class="style1">
When two lines are perpendicular, their product equals -1: <img height="26" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate99.doc" width="90" />.</div>
<div class="style1">
<br /></div>
<div class="style1">
<span style="font-size: 16px; font-weight: bold;"><br /></span></div>
<div class="style1">
<span style="font-size: 16px; font-weight: bold;">The line length</span></div>
<div class="style1">
<br /></div>
<div class="style1">
</div>
<div class="style1">
The length of the line segment joining two points will relate to their coordinates. Have a good look at the diagram</div>
<div align="center" class="style1">
<img height="215" src="http://www.mathsisgoodforyou.com/images/geometricaldiagrams/lenghtofline.jpg" width="200" /></div>
<div class="style1">
The length joining the point A and C can be found by using Pythagoras' Theorem:</div>
<div align="center" class="style1">
<img height="70" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate991.doc" width="363" /></div>
<div align="center" class="style1">
<br /></div>
<div align="center" class="style1">
<br /></div>
<div align="center" class="style1">
</div>
<div class="style6" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-weight: bold; text-align: start;">
Mid-point of a line</div>
<div class="style6" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-weight: bold; text-align: start;">
<br /></div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px; text-align: start;">
Mid-point of the line can be found by using the same principle</div>
<div align="center" class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<img height="190" src="http://www.mathsisgoodforyou.com/images/geometricaldiagrams/midpoint.jpg" width="200" /></div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px; text-align: start;">
So the point between A and C will have the coordinates</div>
<div align="center" class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px;">
<img height="51" src="http://www.mathsisgoodforyou.com/images/mathsequ/coordinate992.doc" width="140" /></div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px; text-align: start;">
If you know the midpoint, you can easily find the perpendicular bisector of a given line. This new line will go through the midpoint of the given line, and it will be perpendicular to it.</div>
<div class="style1" style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 14px; text-align: start;">
<br /></div>
<h1 title="ref-138">
Pairs of Straight Lines</h1>
<h4 style="text-align: left;" title="ref-138">
Any two lines through the Origin may be written as <img alt="y = mx" height="12" src="http://www.codecogs.com/images/eqns/13f9e90e75735c078c7fae17683cf0a9.gif" style="margin-bottom: -4px;" width="57" /> and <img alt="y = tx" height="16" src="http://www.codecogs.com/images/eqns/8ac30682c105fb72935f8604534e130d.gif" style="margin-bottom: -4px;" width="48" /> where <img alt="m" height="9" src="http://www.codecogs.com/images/eqns/bba3dd9e37c86269b4bc9880d7f0e995.gif" style="margin-bottom: -1px;" width="16" /> and <img alt="t" height="13" src="http://www.codecogs.com/images/eqns/11d247c39f98d9ed91065925e37532f9.gif" style="margin-bottom: -1px;" width="8" /> are their gradients. So <img alt="(y - mx)(y - tx) = 0" height="18" src="http://www.codecogs.com/images/eqns/b4811c67345e96eee29ebb4acc63c340.gif" style="margin-bottom: -5px;" width="155" /> giving <img alt="y - mx" height="12" src="http://www.codecogs.com/images/eqns/ffb94d1c15c36e099c7b0367e63f854c.gif" style="margin-bottom: -4px;" width="55" /> or <img alt="y - tx = 0" height="16" src="http://www.codecogs.com/images/eqns/43d7828d5b67896b0cad464d120b9a9c.gif" style="margin-bottom: -4px;" width="77" /> must represent the pair. </h4>
<h4 style="text-align: left;" title="ref-138">
The general form of this equation is given by: </h4>
<div class="formulaDsp">
</div>
<div class="formulaDsp">
<img alt="ax^2+2hxy+by^2=0" height="21" src="http://www.codecogs.com/images/eqns/9455375fb49316fae11b3277243da2f4.gif" width="169" /> </div>
<div class="formulaDsp">
<br /></div>
<div class="formulaDsp">
This equation must represent a pair of straight lines, real or imaginary, through the origin. These can be written as</div>
<div class="formulaDsp">
</div>
<div class="formulaDsp">
<div class="formulaDsp">
<img alt="b\left(\frac{y}{x} \right)^2 + 2h\left(\frac{y}{x} \right) + a = 0" height="39" src="http://www.codecogs.com/images/eqns/5872ed1e3812dbccab9371dc17bb983b.gif" width="200" /> </div>
<div class="formulaDsp">
<br /></div>
<div class="formulaDsp">
Since <img alt="\displaystyle \frac{y}{x}" height="31" src="http://www.codecogs.com/cache/eqns/f110e4ac3fd6ec0a44fa13598b79ce1f.gif" style="margin-bottom: -11px;" width="10" /> is the gradient of a line through the origin <br />
the roots of this equation must be the gradients of the lines <img alt="m" height="9" src="http://www.codecogs.com/images/eqns/bba3dd9e37c86269b4bc9880d7f0e995.gif" style="margin-bottom: -1px;" width="16" /> and <img alt="t" height="13" src="http://www.codecogs.com/images/eqns/11d247c39f98d9ed91065925e37532f9.gif" style="margin-bottom: -1px;" width="8" />.
<br />
Therefore <img alt="\displystyle m + t= - \frac{2h}{t}\f" height="22" src="http://www.codecogs.com/cache/eqns/c5aec2289b65713dd4eabdb4bdc54a1b.gif" style="margin-bottom: -6px;" width="92" />
and <img alt="\displaystyle m t = \frac{a}{b}" height="31" src="http://www.codecogs.com/cache/eqns/11d691ca4c760dd57c5b6093661f0584.gif" style="margin-bottom: -11px;" width="54" /> </div>
<div class="formulaDsp">
<br /></div>
<div class="formulaDsp">
<h2>
Angles Between Lines</h2>
<h4 style="text-align: left;">
Suppose that the lines <img alt="y = mx" height="12" src="http://www.codecogs.com/images/eqns/13f9e90e75735c078c7fae17683cf0a9.gif" style="margin-bottom: -4px;" width="57" /> and <img alt="y = tx" height="16" src="http://www.codecogs.com/images/eqns/8ac30682c105fb72935f8604534e130d.gif" style="margin-bottom: -4px;" width="48" /> are represented by the following equation:
</h4>
<div class="formulaDsp">
<img alt="ax^2 + 2hxy + by^2 = 0" height="21" src="http://www.codecogs.com/images/eqns/559d1c511c5d53365db47b64fb3db7d2.gif" width="169" /></div>
If the angle between them is <img alt="\theta" height="14" src="http://www.codecogs.com/images/eqns/fa3af604b5504f4e45ded5b524b78187.gif" style="margin-bottom: -1px;" width="10" /> then:
<br />
<div class="formulaDsp">
<img alt="\tan \theta = \frac{m - t}{1 - mt}= \frac{\sqrt{(m + t)^2 - 4mt}}{1 + mt}" height="43" src="http://www.codecogs.com/cache/eqns/b688193c77f11b0c9decc8d53208ab53.gif" width="282" /></div>
Hence
<br />
<div class="formulaDsp">
<img alt="tan\;\theta = \frac{\sqrt{4h^2/b^2 - 4a/b}}{1 + a/b}" height="48" src="http://www.codecogs.com/images/eqns/67f758d07dad5f9588c9b2861bc68210.gif" width="195" /></div>
therefore
<br />
<div class="formulaDsp">
<img alt="\tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b}" height="44" src="http://www.codecogs.com/images/eqns/f52db6febbfb7dde422e4cf801cc88eb.gif" width="148" /></div>
<strong>N.B.</strong> The lines will be parallel if the values of this fraction become infinite. i.e. <img alt="a + b = 0" height="14" src="http://www.codecogs.com/images/eqns/8630f4a5b79296ed551e54fa1181018c.gif" style="margin-bottom: -2px;" width="69" /><br />
<h2>
To Find The Equation Of The Angle Bisectors</h2>
<h4 style="text-align: left;">
An angle <strong>bisector</strong> divides the angle into two angles with equal measures. An angle only has one bisector. Each point of an angle bisector is <strong>equidistant</strong> from the sides of the angle. </h4>
<h4 style="text-align: left;">
As before suppose that the lines <img alt="y = mx" height="12" src="http://www.codecogs.com/images/eqns/13f9e90e75735c078c7fae17683cf0a9.gif" style="margin-bottom: -4px;" width="57" /> and <img alt="y = tx" height="16" src="http://www.codecogs.com/images/eqns/8ac30682c105fb72935f8604534e130d.gif" style="margin-bottom: -4px;" width="48" /> are represented by:
</h4>
<div class="formulaDsp">
<img alt="ax^2 + 2hxy + by^2 = 0" height="21" src="http://www.codecogs.com/images/eqns/559d1c511c5d53365db47b64fb3db7d2.gif" width="169" /></div>
The equation of the angle bisectors will be:
<br />
<div class="formulaDsp">
<img alt="\frac{y - mx}{\sqrt{1 + m^2}} = \pm \frac{y - tx}{\sqrt{1 + t^2}}" height="44" src="http://www.codecogs.com/images/eqns/dc8910a8a0b7bc19ba24f44ec5f7b189.gif" width="173" /></div>
<div class="formulaDsp">
<img alt="\therefore\;\;\;\;\;(1 + t^2)(y - mx)^2 = (1 + m^2)(y - tx)^2" height="21" src="http://www.codecogs.com/images/eqns/e154b7565d776b328597aae9bcbbed7f.gif" width="341" /></div>
or
<br />
<div class="formulaDsp">
<img alt="x^2(m^2 - t^2) - 2xy(m + mt^2\;-t\;-tm^2) + y^2(t^2 - m^2) = 0" height="21" src="http://www.codecogs.com/images/eqns/9c4fdf19f8f9be32e64cdbbd17f2b5ad.gif" width="459" /></div>
Since <img alt="m" height="9" src="http://www.codecogs.com/images/eqns/bba3dd9e37c86269b4bc9880d7f0e995.gif" style="margin-bottom: -1px;" width="16" /> is not equal to <img alt="t" height="13" src="http://www.codecogs.com/images/eqns/11d247c39f98d9ed91065925e37532f9.gif" style="margin-bottom: -1px;" width="8" />, divide the above equation by <img alt="(m - t)" height="18" src="http://www.codecogs.com/images/eqns/9c3dd4c029e761af27755beb46eff832.gif" style="margin-bottom: -5px;" width="54" />
<br />
<div class="formulaDsp">
<img alt="x^2(m + t) - 2xy(1 - mt) - y^2(m + t) = 0" height="21" src="http://www.codecogs.com/images/eqns/948b27f5e043031930d801fe6b4bee96.gif" width="326" /></div>
Substituting for <img alt="(m+t)" height="18" src="http://www.codecogs.com/images/eqns/6b4058b0f44ce2dbbd1f6e6c722524c1.gif" style="margin-bottom: -5px;" width="54" /> and <img alt="mt" height="13" src="http://www.codecogs.com/images/eqns/beda159081becb6187f820ac0e98f715.gif" style="margin-bottom: -1px;" width="23" />:
<br />
<div class="formulaDsp">
<img alt="x^2(- \frac{2h}{b}) - 2xy(1 - \frac{a}{b}) - y^2(-\frac{2h}{b}) = 0" height="40" src="http://www.codecogs.com/images/eqns/74012922abc3a0cb6df8db351f01dcca.gif" width="303" /></div>
or
<br />
<div class="formulaDsp">
<img alt="(x^2 - y^2)(- 2h) = 2xy(b - a)" height="21" src="http://www.codecogs.com/images/eqns/0b2a8a7d77a7720f5131ab2f760cd912.gif" width="223" /></div>
<div class="greybox">
Therefore the required equation is
<br />
<div class="formulaDsp">
</div>
<div class="formulaDsp">
<img alt="\frac{x^2 - y^2}{xy} = \frac{a - b}{h}" height="46" src="http://www.codecogs.com/images/eqns/f9a488d5735745946d752202e4cb7c98.gif" width="126" /> </div>
<div class="formulaDsp">
<h2>
To Find The Equation Of The Pair Of Lines Joining The Points Of Intersection Of The Following Two Lines, To The Origin:</h2>
<div class="formulaDsp">
<img alt="ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0" height="21" src="http://www.codecogs.com/images/eqns/4dd5a1910637f347e5088b673ab50568.gif" width="300" /></div>
</p> <p><br />
<div class="formulaDsp">
<img alt="lx + my + n = 0" height="19" src="http://www.codecogs.com/images/eqns/d2c58aa4769070fb253e19023bb188a0.gif" width="131" /></div>
From the linear equation express 1 as a linear function of <img alt="x" height="9" src="http://www.codecogs.com/images/eqns/cb5dc0936e0c5c4e5594438d118d384d.gif" style="margin-bottom: -1px;" width="11" /> and <img alt="y" height="12" src="http://www.codecogs.com/images/eqns/4e8ce9e3a6f5c0619114a4b2f94db712.gif" style="margin-bottom: -4px;" width="10" />. i.e.:
<br />
<div class="formulaDsp">
<img alt="1\;= - \frac{(lx + my)}{n}" height="41" src="http://www.codecogs.com/images/eqns/3784a278acdd78e7ba85cd48b95d8295.gif" width="132" /></div>
Use this to build up every term of the quadratic equation to the second degree and we get:
<br />
<div class="formulaDsp">
<img alt="ax^2 + 2hxy + by^2 + (2gx + 2fy)\left(- \frac{lx + my}{n} \right) + c\left(- \frac{lx + my}{n} \right)^2 = 0" height="49" src="http://www.codecogs.com/images/eqns/14a3c44e792e5c9878521892e3f50dc8.gif" width="543" /></div>
Every term here is of the second degree and since any point which satisfies both:
<br />
<div class="formulaDsp">
<img alt="- \frac{(lx + my)}{n} = 1" height="41" src="http://www.codecogs.com/images/eqns/f4241ec6bc68e2a145ad3f7e710f1743.gif" width="128" /></div>
and
<br />
<div class="formulaDsp">
<img alt="2hxy + by^2 + 2gx + 2fy + c = 0" height="21" src="http://www.codecogs.com/images/eqns/ae5072e9a00f7de207bf3191441c057f.gif" width="251" /></div>
must also satisfy this new equation, it must represent the required pair of lines. </div>
<div class="formulaDsp">
</div>
<div class="formulaDsp">
<h2>
To Find The Condition That The General Equation Of The Second Degree Should Represent A Pair Of Straight Lines..</h2>
<h2>
So far we have considered only pairs of straight lines through the origin. </h2>
<h4 style="text-align: left;">
<span style="font-weight: normal;">The equation of the pair of lines <img alt="ax + by + c = 0" height="16" src="http://www.codecogs.com/images/eqns/2afcdd868cf18d2747936df47922a64a.gif" style="margin-bottom: -4px;" width="115" /> and <img alt="lx + my + n = 0" height="16" src="http://www.codecogs.com/images/eqns/670603f6027c2ded3132959b114dd93f.gif" style="margin-bottom: -4px;" width="121" /> is obviously given by the equation:
</span></h4>
<div class="formulaDsp">
<img alt="(ax + by + c)(lx + my + n) = 0" height="20" src="http://www.codecogs.com/images/eqns/2ad57e35d8cca00825592c2263c4a45d.gif" width="246" /></div>
And it is worth noting that the equation:
<br />
<div class="formulaDsp">
<img alt="a(x - \alpha )^2 + 2h(x - \alpha )(y - \beta ) + b(y - \beta )^2 = 0" height="21" src="http://www.codecogs.com/images/eqns/614dee6d1b6fd1911002e22c3aae6df5.gif" width="359" /></div>
represents a pair of straight lines through the point <img alt="(\alpha, \beta )" height="18" src="http://www.codecogs.com/images/eqns/087627a28ee4124284e6c74a68b7f86a.gif" style="margin-bottom: -5px;" width="41" /> and parallel to the pair given by:
<br />
<div class="formulaDsp">
<img alt="ax^2 + 2hxy + by^2 = 0" height="21" src="http://www.codecogs.com/images/eqns/559d1c511c5d53365db47b64fb3db7d2.gif" width="169" /></div>
The general equation in the second degree:
<br />
<div class="formulaDsp">
<img alt="ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0" height="21" src="http://www.codecogs.com/images/eqns/4dd5a1910637f347e5088b673ab50568.gif" width="300" /></div>
will represent a pair of straight lines <strong>if it factorizes</strong>. Expanding the equation as a quadratic in x we get:
<br />
<div class="formulaDsp">
<img alt="ax^2 + 2x(hy + g) + (by^2 + 2fy\;+c) = 0" height="21" src="http://www.codecogs.com/images/eqns/d0336ac7c28641adb0a34fff15436bf2.gif" width="314" /></div>
When we solve for <img alt="x" height="9" src="http://www.codecogs.com/images/eqns/cb5dc0936e0c5c4e5594438d118d384d.gif" style="margin-bottom: -1px;" width="11" /> we will get an expression containing a square root. If the equation represents a pair of lines <img alt="x" height="9" src="http://www.codecogs.com/images/eqns/cb5dc0936e0c5c4e5594438d118d384d.gif" style="margin-bottom: -1px;" width="11" /> must be expressible as one or other of two linear expressions in <img alt="x" height="9" src="http://www.codecogs.com/images/eqns/cb5dc0936e0c5c4e5594438d118d384d.gif" style="margin-bottom: -1px;" width="11" /> and <img alt="y" height="12" src="http://www.codecogs.com/images/eqns/4e8ce9e3a6f5c0619114a4b2f94db712.gif" style="margin-bottom: -4px;" width="10" /> and so this square root must be rational. <img alt="(hy + g)^2 - a(by^2 + 2fy+c)" height="20" src="http://www.codecogs.com/images/eqns/42e83a77f9758fe8cfe48d6df0bd0f40.gif" style="margin-bottom: -5px;" width="207" /> must be a perfect square.
<br />
<div class="formulaDsp">
The condition for this is given by: <img alt="(hy - af)^2 = (h^2 - ab)(g^2 - ac)" height="21" src="http://www.codecogs.com/images/eqns/c385d4ba9cad9d671ec65a6ceab5ab83.gif" width="248" /></div>
<div class="greybox">
Which simplifies to become:
<br />
<div class="formulaDsp">
<img alt="af^2 + bg^2 + ch^2 = 2fgh + abc" height="21" src="http://www.codecogs.com/images/eqns/d98683990bb52f42ebb40f048287297e.gif" width="233" /></div>
</div>
<h2>
<b style="font-family: Verdana, Arial, Helvetica, sans-serif; text-align: -webkit-center;">The Equation Of A Circle</b></h2>
<div>
(x - a)<sup>2</sup> + (y - b)<sup>2</sup> = r<sup>2</sup><br />
where<br />
a is the x co-ordinate of the centre of the circle<br />
b is the y co-ordinate of the centre of the circle<br />
r is the radius of the circle</div>
<div>
<br /></div>
<div>
<b>In terms of showing method and avoiding error</b><br />
especially with <a href="http://www.maths.com/numbers.directed.htm" style="color: blue; font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 11px; font-weight: bold; text-decoration: initial;">directed numbers</a><br />
<b>the following format can be very useful<br />[x - (a)]<sup>2</sup> + [y - (b)]<sup>2</sup> = (r)<sup>2</sup></b></div>
<div>
<b><sup><br /></sup></b></div>
<div>
<b><sup><br /></sup></b></div>
<div>
<div>
<span style="color: red;">(x + 2)<sup>2</sup> + (y - 3)<sup>2</sup> = 5<sup>2</sup></span></div>
<br />
<div>
<span style="color: #007700;">[x + 2]<sup>2</sup> + [y - 3]<sup>2</sup> = 5<sup>2</sup></span></div>
<br />
<div>
<span style="color: #007700;">[x - (-2)]<sup>2</sup> + [y - (3)]<sup>2</sup> = (5)<sup>2</sup></span></div>
<br />
<div>
<span style="color: red;">a = (-2)</span></div>
<div>
-2 is the x co-ordinate<br />
of the centre of the circle</div>
<div>
<span style="color: red;">b = (3)</span></div>
<div>
3 is the y co-ordinate<br />
of the centre of the circle</div>
<div>
<span style="color: red;">r = (5)</span></div>
<div>
5 is the radius<br />
of the circle</div>
</div>
</div>
<div class="formulaDsp">
</div>
<div class="formulaDsp">
</div>
<div class="formulaDsp">
</div>
<div class="formulaDsp">
</div>
<div class="formulaDsp">
</div>
<div class="formulaDsp">
<br />
<h2>
</h2>
</div>
</div>
<h2>
</h2>
<h2>
</h2>
</div>
</div>
<h1 title="ref-138">
</h1>
<div align="center" class="style1">
<br /></div>
<div align="center" class="style1">
<br /></div>
<div align="center" class="style1">
<br /></div>
<div align="center" class="style1">
<br /></div>
<div align="center" class="style1">
<br /></div>
<div align="center" class="style1">
<br /></div>
<div align="center" class="style1">
<br /></div>
<div align="center" class="style1">
<br /></div>
<div align="center" class="style1">
<br /></div>
<div align="center" class="style1">
<br /></div>
<div style="color: black; font-family: 'Times New Roman'; font-size: medium;">
<br /></div>
<br />
<br />
<br />
<br />
<span style="color: #000033; font-family: Arial, Helvetica, sans-serif; font-size: 16px; font-weight: bold;"><br /></span>
</div>
<span class="mw-headline">
</span></div>
</div>
Anonymoushttp://www.blogger.com/profile/12500986417656529636noreply@blogger.com1Naraina, New Delhi, Delhi, India28.6302834 77.139887828.6163464 77.1201468 28.6442204 77.1596288tag:blogger.com,1999:blog-7072204672677043641.post-64887694424166643762012-12-01T22:20:00.000-08:002012-12-04T22:03:07.640-08:00CIRCLE FORMULA AND THEORY DETAIL<div dir="ltr" style="text-align: left;" trbidi="on">
<br />
<div style="background-color: white; margin-bottom: 0.5em; margin-top: 0.4em;">
</div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;">A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius.</span></span></div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;">A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.</span></span></div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;">A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant.</span></span></div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;">A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to the axis of the cone.</span></span><img alt="CIRCLE 1.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/1d/CIRCLE_1.svg/220px-CIRCLE_1.svg.png" /></div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"><br /></span></span></div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
</div>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; line-height: 19.200000762939453px; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Terminology"><span style="color: red;">Terminology</span></span></h2>
<div>
<span class="mw-headline"></span><br />
<div>
<ol style="text-align: left;"><span class="mw-headline">
<li><b>Chord: a line segment whose endpoints lie on the circle.</b></li>
<li><b>Diameter: the longest chord, a line segment whose endpoints lie on the circle and which passes through the centre; or the length of such a segment, which is the largest distance between any two points on the circle.</b></li>
<li><b>Radius: a line segment joining the center of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.</b></li>
<li><b>Circumference: the length of one circuit along the circle itself.</b></li>
<li><b>Tangent: a straight line that touches the circle at a single point.</b></li>
<li><b>Secant: an extended chord, a straight line cutting the circle at two points.</b></li>
<li><b>Arc: any connected part of the circle's circumference.</b></li>
<li><b>Sector: a region bounded by two radii and an arc lying between the radii.</b></li>
<li><b>Segment: a region bounded by a chord and an arc lying between the chord's endpoints.</b></li>
</span></ol>
<div>
<table cellpadding="0" cellspacing="0" style="background-color: white; color: black; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; text-align: start;"><tbody>
<tr><td><div class="thumb tright" style="background-color: transparent; clear: right; float: right; margin: 0.5em 0px 1.3em 1.4em; width: auto;">
<div class="thumbinner" style="background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); font-size: 12px; min-width: 100px; overflow: hidden; padding: 3px !important; text-align: center; width: 222px;">
<a class="image" href="http://en.wikipedia.org/wiki/File:CIRCLE_LINES.svg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;"><img alt="" class="thumbimage" height="222" src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/06/CIRCLE_LINES.svg/220px-CIRCLE_LINES.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/CIRCLE_LINES.svg/330px-CIRCLE_LINES.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/CIRCLE_LINES.svg/440px-CIRCLE_LINES.svg.png 2x" style="background-color: white; border: 1px solid rgb(204, 204, 204); vertical-align: middle;" width="220" /></a><br />
<div class="thumbcaption" style="border: none; font-size: 11px; line-height: 1.4em; padding: 3px !important; text-align: left;">
<div class="magnify" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; border: none !important; float: right;">
<a class="internal" href="http://en.wikipedia.org/wiki/File:CIRCLE_LINES.svg" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; border: none !important; color: #0b0080; display: block; text-decoration: initial;" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/static-1.21wmf4/skins/common/images/magnify-clip.png" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; border: none !important; display: block; vertical-align: middle;" width="15" /></a></div>
Chord, secant, tangent, radius, and diameter</div>
</div>
</div>
</td><td><div class="thumb tright" style="background-color: transparent; clear: right; float: right; margin: 0.5em 0px 1.3em 1.4em; width: auto;">
<div class="thumbinner" style="background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); font-size: 12px; min-width: 100px; overflow: hidden; padding: 3px !important; text-align: center; width: 222px;">
<a class="image" href="http://en.wikipedia.org/wiki/File:Circle_slices.svg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;"><img alt="" class="thumbimage" height="220" src="http://upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Circle_slices.svg/220px-Circle_slices.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Circle_slices.svg/330px-Circle_slices.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Circle_slices.svg/440px-Circle_slices.svg.png 2x" style="background-color: white; border: 1px solid rgb(204, 204, 204); vertical-align: middle;" width="220" /></a><br />
<div class="thumbcaption" style="border: none; font-size: 11px; line-height: 1.4em; padding: 3px !important; text-align: left;">
<div class="magnify" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; border: none !important; float: right;">
<a class="internal" href="http://en.wikipedia.org/wiki/File:Circle_slices.svg" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; border: none !important; color: #0b0080; display: block; text-decoration: initial;" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/static-1.21wmf4/skins/common/images/magnify-clip.png" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; border: none !important; display: block; vertical-align: middle;" width="15" /></a></div>
Arc, sector, and segment</div>
</div>
</div>
</td></tr>
</tbody></table>
<h2 style="background-color: white; background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; line-height: 19.200000762939453px; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline">
<span class="mw-headline" id="Analytic_results"><span style="color: red;">Analytic results</span></span></span></h2>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline">
<span class="mw-headline" id="Length_of_circumference" style="font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px;">Length of circumference-</span><span style="background-color: transparent; line-height: 19.200000762939453px;"><span style="font-family: sans-serif; font-size: x-small;">The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:</span></span></span></h3>
</div>
</div>
<span class="mw-headline">
</span>
<br />
<div>
<span class="mw-headline"><span style="color: yellow;"><img alt="C = 2\pi r = \pi d.\," class="tex" src="http://upload.wikimedia.org/math/8/e/a/8ea94f2f4f25c73df88369317e602d94.png" style="background-color: white; border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /></span></span></div>
<span class="mw-headline">
</span>
<div>
<span class="mw-headline"><span style="color: yellow;"><br /></span></span></div>
<span class="mw-headline">
<div>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Area_enclosed"><span style="color: red;">Area enclosed-</span></span></h3>
</div>
<div>
As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius,[3] which comes to π multiplied by the radius squared:</div>
<div>
<img src="http://upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Circle_Area.svg/220px-Circle_Area.svg.png" /><img alt="\mathrm{Area} = \pi r^2.\," class="tex" src="http://upload.wikimedia.org/math/0/1/f/01fcc4892814dea3c4385c7b9187db0c.png" style="background-color: white; border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /></div>
<div>
<br /></div>
<div>
<span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">Equivalently, denoting diameter by </span><i style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">d</i></div>
<div>
<i style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"><br /></i></div>
<div>
<img alt="\mathrm{Area} = \frac{\pi d^2}{4} \approx 0{.}7854d^2," class="tex" height="68" src="http://upload.wikimedia.org/math/f/8/5/f8596418b1db3586760016310e5f49bb.png" style="background-color: white; border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" width="320" /></div>
<div>
<br /></div>
<div>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Equations"><span style="color: red;">Equations---</span></span></h3>
</div>
<div>
<span style="background-color: white; font-family: sans-serif; font-size: 15px; line-height: 19.200000762939453px;"><b>Cartesian coordinates--</b></span><span style="font-family: sans-serif; font-size: 15px; line-height: 19.200000762939453px;">In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that</span></div>
<div>
<span style="font-family: sans-serif; font-size: 15px; line-height: 19.200000762939453px;"><br /></span></div>
<div>
<img alt="\left(x - a \right)^2 + \left( y - b \right)^2=r^2." class="tex" src="http://upload.wikimedia.org/math/e/9/3/e9326e126151d2fb2e0573e8b5f57310.png" style="background-color: white; border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /><br />
<br />
<br />
<br />
<img src="http://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Circle_center_a_b_radius_r.svg/220px-Circle_center_a_b_radius_r.svg.png" style="background-color: white; font-family: sans-serif; font-size: small; line-height: 19.200000762939453px;" /></div>
<div>
<br /></div>
<div>
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;">This equation, also known as Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x − a and y − b. If the circle is centred at the origin (0, 0), then the equation simplifies to</span></span></div>
<div>
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"> </span></span></div>
<div>
<img alt="x^2 + y^2 = r^2.\!\ " class="tex" src="http://upload.wikimedia.org/math/2/5/0/250a578d869e4ac1918e793c149f9196.png" style="background-color: white; border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /></div>
<div>
<br /></div>
<div>
The equation can be written in parametric form using the trigonometric functions sine and cosine as</div>
<div>
<br /></div>
<div>
<br />
<br />
<br />
<dd style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"> <img alt="x = a+r\,\cos t,\," class="tex" src="http://upload.wikimedia.org/math/4/1/2/4124b2be94e347e27e7e936d9b134375.png" style="border: none; vertical-align: middle;" /></dd><dd style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"> <img alt="y = b+r\,\sin t\," class="tex" src="http://upload.wikimedia.org/math/7/7/c/77c3eb62b4580e2fc8fff315dd2c4643.png" style="border: none; vertical-align: middle;" /></dd><dd style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-size: x-small;"><br /></span></dd><dd style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-size: x-small;"><br /></span></dd><dd style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-size: x-small;">In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the circle onto the line passing through the centre parallel to the x-axis.</span><span style="font-size: x-small;">In homogeneous coordinates each conic section with equation of a circle is of the form</span></dd><dd style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><br /></dd><dd style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"> <img alt="ax^2+ay^2+2b_1xz+2b_2yz+cz^2 = 0.\," class="tex" src="http://upload.wikimedia.org/math/5/7/f/57f11a8fc9a36ab4b9b8c8fa4eb5b105.png" style="border: none; vertical-align: middle;" /></dd><dd style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-size: 15px;"><br /></span></dd><dd style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-size: 15px;"><b><span style="color: red;">Polar coordinates-</span></b></span></dd><dd style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-size: 15px;"><b><span style="color: red;"><br /></span></b></span></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"><b>In polar coordinates the equation of a circle is:</b></span></span></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"><b><br /></b></span></span></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"> <img alt="r^2 - 2 r r_0 \cos(\theta - \phi) + r_0^2 = a^2\," class="tex" src="http://upload.wikimedia.org/math/5/5/1/5519c99adba7f0fa41a882dbea4ed887.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">where a is the radius of the circle, is the polar coordinate of a generic point on the circle, and is the polar coordinate of the centre of the circle (i.e., r0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle). For a circle centred at the origin, i.e. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies on the circle, the equation becomes</dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><br /></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"> <img alt="r = 2 a\cos(\theta - \phi).\," class="tex" src="http://upload.wikimedia.org/math/1/9/1/191a9ad893211888cf7658ba3cabf815.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><br /></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">In the general case, the equation can be solved for </span><i style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">r</i><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">, giving</span></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"><br /></span></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"> <img alt="r = r_0 \cos(\theta - \phi) + \sqrt{a^2 - r_0^2 \sin^2(\theta - \phi)}," class="tex" src="http://upload.wikimedia.org/math/6/1/0/610dcbaf1efc668802d5017f1e36e352.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"> the solution with a minus sign in front of the square root giving the same curve.</span></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="color: red; font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"><b><br /></b></span></span></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"><b style="color: red;">Complex plane-</b><span style="color: red;">-</span>In the complex plane, a circle with a centre at c and radius (r) has the equation . In parametric form this can be written .</span></span></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;">The slightly generalised equation for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with , since . Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.</span></span></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="color: red; font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"><b><br /></b></span></span></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"> <img alt="|z-c|^2 = r^2\," src="http://upload.wikimedia.org/math/f/5/9/f594d237b498070e39255bfcb46afee0.png" />; </dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"> <img alt="z = re^{it}+c" class="tex" src="http://upload.wikimedia.org/math/6/9/0/69019590943f4779caae2be2ce4b30b4.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin: 0px; vertical-align: middle;" /><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">.</span></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="color: red; font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"><b><br /></b></span></span></dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; line-height: 19.200000762939453px; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Properties"><span style="color: red;">Properties</span><span style="font-weight: normal;">-</span></span></h2>
</dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><div>
<span class="mw-headline"><span style="font-weight: normal;"></span></span><br />
<div>
<ol style="text-align: left;"><span class="mw-headline"><span style="font-weight: normal;">
<li>The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.)</li>
<li>The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T.</li>
<li>All circles are similar.</li>
<li>A circle's circumference and radius are proportional.</li>
<li>The area enclosed and the square of its radius are proportional.</li>
<li>The constants of proportionality are 2π and π, respectively.</li>
<li>The circle which is centred at the origin with radius 1 is called the unit circle.</li>
<li>Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.</li>
<li>Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.</li>
</span></span></ol>
</div>
<span class="mw-headline"><span style="font-weight: normal;">
</span></span></div>
</dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><h3 style="background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Chord"><span style="color: red;">Chord</span></span></h3>
</dd><dd style="background-color: white; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mw-headline"></span><br />
<ol style="text-align: left;"><span class="mw-headline">
<li>Chords are equidistant from the centre of a circle if and only if they are equal in length.</li>
<li>The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector:</li>
<li>A perpendicular line from the centre of a circle bisects the chord.</li>
<li>The line segment (circular segment) through the centre bisecting a chord is perpendicular to the chord.</li>
<li>If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.</li>
<li>If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.</li>
<li>If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplemental.</li>
<li>For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.</li>
<li>An inscribed angle subtended by a diameter is a right angle (see Thales' theorem).</li>
<li>The diameter is the longest chord of the circle.</li>
<li>If the intersection of any two chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then ab = cd.</li>
<li>If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then a2 + b2 + c2 + d2 equals the square of the diameter.</li>
<li>The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two chords intersecting at the same point, and is given by 8r 2 – 4p 2 (where r is the circle's radius and p is the distance from the center point to the point of intersection).</li>
<li>The disance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.</li>
</span></ol>
<span class="mw-headline"><span style="color: red;">
</span></span>
<br />
<div>
<h3 style="background-image: none; border-bottom-style: none; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline">
</span></h3>
<h3 style="background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline">
<span class="mw-headline" id="Tangent"><span style="color: red;">Tangent--</span></span></span></h3>
<div>
<span class="mw-headline"><span class="mw-headline" style="font-weight: normal; line-height: 19.200000762939453px;"></span></span><br />
<ul style="list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;"><span class="mw-headline"><span class="mw-headline" style="font-weight: normal; line-height: 19.200000762939453px;">
<li style="margin-bottom: 0.1em;"><span style="font-family: sans-serif; font-size: x-small;"></span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: sans-serif; font-size: x-small;">The line perpendicular drawn to a radius through the end point of the radius is a tangent to the circle.</span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: sans-serif; font-size: x-small;">A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle.</span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: sans-serif; font-size: x-small;">Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.</span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: sans-serif; font-size: x-small;">If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the angles ∠BOA and ∠BPA are supplementary.</span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: sans-serif; font-size: x-small;">If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = 1⁄2arc(AQ).</span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: sans-serif; font-size: x-small;"><br /></span></li>
<li style="margin-bottom: 0.1em;"><span style="font-family: sans-serif; font-size: x-small;"> </span></li>
</span></span></ul>
<span class="mw-headline"><span class="mw-headline" style="font-weight: normal; line-height: 19.200000762939453px;">
</span></span></div>
</div>
<span class="mw-headline">
</span></dd></div>
</span></div>
<br /></div>
Anonymoushttp://www.blogger.com/profile/12500986417656529636noreply@blogger.com0tag:blogger.com,1999:blog-7072204672677043641.post-58199947077820697632012-12-01T21:24:00.000-08:002012-12-04T22:01:22.707-08:00List of trigonometric identities<div dir="ltr" style="text-align: left;" trbidi="on">
<span style="font-family: sans-serif; font-size: x-small; line-height: 19.200000762939453px;">In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle. Only the former are covered in this article.</span><br />
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;">These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.</span></span></div>
<br />
<h3 style="background-color: white; color: #333333; font-family: Verdana; font-size: 12px; line-height: 18px; margin: 1.5em 0px; padding: 0px; text-decoration: inherit;">
<span style="font-family: Verdana, Arial, Helvetica, sans-serif;"><span style="font-size: x-small;">Useful trig formulas for learning trigonometric concepts.</span></span></h3>
<table border="0" cellpadding="0" cellspacing="0" style="background-color: white; border-spacing: 0px; color: #333333; empty-cells: show; font-family: Verdana; font-size: 12px; height: 115px; line-height: 18px; margin: 0px; padding: 0px; text-align: left; text-decoration: inherit; width: 53%px;"><tbody>
<tr><td height="190" style="vertical-align: top;" valign="top" width="43%"><br /><img height="99" src="http://z.about.com/d/math/1/0/C/1/trigform.gif" width="199" /></td><td height="190" style="vertical-align: top;" valign="top" width="57%"><div style="font-family: inherit; font-style: inherit; margin-bottom: 1.5em; margin-top: 1.5em; padding: 0px; text-decoration: inherit;">
<img height="185" src="http://0.tqn.com/d/math/1/0/F/1/trigtri.gif" width="208" /></div>
</td></tr>
<tr><td colspan="2" style="vertical-align: top;" valign="top"><div style="font-family: inherit; font-style: inherit; margin-bottom: 1.5em; margin-top: 1.5em; padding: 0px; text-decoration: inherit;">
<span style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: x-small;"><br />Triangle ABC is any triangle with side lengths <i>a,b,c</i></span></div>
<blockquote style="font-family: inherit; font-style: inherit; margin: 1.5em 0px 1.5em 3em; padding: 0px; quotes: ''; text-decoration: inherit;">
<blockquote style="font-family: inherit; font-style: inherit; margin: 1.5em 0px 1.5em 3em; padding: 0px; quotes: ''; text-decoration: inherit;">
<div style="font-family: inherit; font-style: inherit; margin-bottom: 1.5em; margin-top: 1.5em; padding: 0px; text-decoration: inherit;">
<span style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: x-small;">Law of Cosines<br /><br /><img height="14" src="http://0.tqn.com/d/math/1/0/E/1/cosines.gif" width="197" /></span></div>
<div style="font-family: inherit; font-style: inherit; margin-bottom: 1.5em; margin-top: 1.5em; padding: 0px; text-decoration: inherit;">
<span style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: x-small;">Law of Sines<br /><br /><img height="26" src="http://0.tqn.com/d/math/1/0/D/1/sines.gif" width="168" /></span></div>
</blockquote>
</blockquote>
</td></tr>
<tr><td style="vertical-align: top;" width="43%"> </td><td style="vertical-align: top;" width="57%"><br /></td></tr>
</tbody></table>
<br />
<br />
<h2 style="background-color: white; background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; line-height: 19.200000762939453px; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Pythagorean_identity"><span style="color: red;">Pythagorean identity</span></span></h2>
<div>
The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity<span style="color: red;">:</span></div>
<br />
<img alt="\cos^2\theta + \sin^2\theta = 1\!" class="tex" src="http://upload.wikimedia.org/math/e/4/1/e41af8919b1aa560e52032ef6e2c4415.png" style="background-color: white; border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /><br />
<span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">where </span><span class="texhtml" style="background-color: white; font-family: 'Times New Roman', 'Nimbus Roman No9 L', Times, serif; font-size: 15px; line-height: 19.200000762939453px; white-space: nowrap;">cos<sup style="line-height: 1em;">2</sup> <i>θ</i></span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"> means </span><span class="texhtml" style="background-color: white; font-family: 'Times New Roman', 'Nimbus Roman No9 L', Times, serif; font-size: 15px; line-height: 19.200000762939453px; white-space: nowrap;">(cos(<i>θ</i>))<sup style="line-height: 1em;">2</sup></span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"> and </span><span class="texhtml" style="background-color: white; font-family: 'Times New Roman', 'Nimbus Roman No9 L', Times, serif; font-size: 15px; line-height: 19.200000762939453px; white-space: nowrap;">sin<sup style="line-height: 1em;">2</sup> <i>θ</i></span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"> means </span><span class="texhtml" style="background-color: white; font-family: 'Times New Roman', 'Nimbus Roman No9 L', Times, serif; font-size: 15px; line-height: 19.200000762939453px; white-space: nowrap;">(sin(<i>θ</i>))<sup style="line-height: 1em;">2</sup></span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">.</span><br />
<span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"><br /></span>
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;">This can be viewed as a version of the Pythagorean theorem, and follows from the equation x2 + y2 = 1 for the unit circle. This equation can be solved for either the sine or the cosine:</span></span><br />
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"><br /></span></span>
<img alt="\sin\theta = \pm \sqrt{1-\cos^2\theta} \quad \text{and} \quad \cos\theta = \pm \sqrt{1 - \sin^2\theta}. \, " class="tex" src="http://upload.wikimedia.org/math/a/1/1/a113f3184dce25494b2df8ffa03f8640.png" style="background-color: white; border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /><br />
<br />
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline"><span style="color: red;"><br /></span></span></h3>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Related_identities">Related identities</span></h3>
<div>
<span class="mw-headline">Dividing the Pythagorean identity through by either cos2 θ or sin2 θ yields two other identities:</span></div>
<div>
<span class="mw-headline"><br /></span></div>
<div>
<span class="mw-headline"><img alt="1 + \tan^2\theta = \sec^2\theta\quad\text{and}\quad 1 + \cot^2\theta = \csc^2\theta.\!" class="tex" src="http://upload.wikimedia.org/math/c/9/3/c9314fabd99644575500be53eebbc8ab.png" style="background-color: white; border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /></span></div>
<div>
<span class="mw-headline"><br /></span></div>
<div>
<span class="mw-headline"><br /></span></div>
<div>
<span class="mw-headline"><b>Using these identities together with the ratio identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):</b></span></div>
<br />
<table class="wikitable" style="background-color: white; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin: 1em 0px; text-align: center;"><tbody>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;">in terms of</th><th scope="col" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \sin \theta\!" class="tex" src="http://upload.wikimedia.org/math/b/b/c/bbc9297318ee661c13fe114d28820b15.png" style="border: none; vertical-align: middle;" /></th><th scope="col" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \cos \theta\!" class="tex" src="http://upload.wikimedia.org/math/9/b/b/9bb4c5f619fb6d5f255c30d632fac8ad.png" style="border: none; vertical-align: middle;" /></th><th scope="col" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \tan \theta\!" class="tex" src="http://upload.wikimedia.org/math/3/e/6/3e64c04b4e255b35bc688310f5286ba8.png" style="border: none; vertical-align: middle;" /></th><th scope="col" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \csc \theta\!" class="tex" src="http://upload.wikimedia.org/math/3/3/0/3300ffc3669677063340e0c6e1e00f7a.png" style="border: none; vertical-align: middle;" /></th><th scope="col" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \sec \theta\!" class="tex" src="http://upload.wikimedia.org/math/5/f/2/5f213e7bb36c7b9cf85642e3e24f87db.png" style="border: none; vertical-align: middle;" /></th><th scope="col" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \cot \theta\!" class="tex" src="http://upload.wikimedia.org/math/f/f/1/ff14478418c00facf373c27122df9541.png" style="border: none; vertical-align: middle;" /></th></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \sin \theta =\!" class="tex" src="http://upload.wikimedia.org/math/3/e/e/3ee8894913e50367c7e39dea01ebb813.png" style="border: none; vertical-align: middle;" /></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \sin \theta\ " class="tex" src="http://upload.wikimedia.org/math/6/2/d/62de3aac5ab6b695b615498d05065189.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\sqrt{1 - \cos^2 \theta}\! " class="tex" src="http://upload.wikimedia.org/math/8/f/4/8f412524cf3b802dc80fb691e8ba0be2.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}}\! " class="tex" src="http://upload.wikimedia.org/math/9/1/d/91d7dd01d19591dd6c52a59e1c9efa07.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \frac{1}{\csc \theta}\! " class="tex" src="http://upload.wikimedia.org/math/d/3/a/d3ac18a12b2585c706c73858e9d3936f.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta}\! " class="tex" src="http://upload.wikimedia.org/math/8/8/a/88a6e7644bcf7056afdaa09356de037a.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{1}{\sqrt{1 + \cot^2 \theta}}\! " class="tex" src="http://upload.wikimedia.org/math/e/7/9/e792b64e6d978849e819f278bc05e979.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \cos \theta =\!" class="tex" src="http://upload.wikimedia.org/math/0/f/3/0f33c748d1b205237d4489b1703f873d.png" style="border: none; vertical-align: middle;" /></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\sqrt{1 - \sin^2\theta}\! " class="tex" src="http://upload.wikimedia.org/math/1/2/9/1296e22d95236c7d5d42f3dfccf6912c.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \cos \theta\! " class="tex" src="http://upload.wikimedia.org/math/9/b/b/9bb4c5f619fb6d5f255c30d632fac8ad.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{1}{\sqrt{1 + \tan^2 \theta}}\! " class="tex" src="http://upload.wikimedia.org/math/5/5/0/550ad529263a113e25696b48daf583aa.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{\sqrt{\csc^2 \theta - 1}}{\csc \theta}\! " class="tex" src="http://upload.wikimedia.org/math/0/b/e/0be75c99660ff0e9a64416a1fb65d46c.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \frac{1}{\sec \theta}\! " class="tex" src="http://upload.wikimedia.org/math/9/7/2/97206a2b5fb0d4bcbd229f0cb9124587.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{\cot \theta}{\sqrt{1 + \cot^2 \theta}}\! " class="tex" src="http://upload.wikimedia.org/math/b/9/4/b948b7622c4cc020ed03bb270d6cb946.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \tan \theta =\!" class="tex" src="http://upload.wikimedia.org/math/4/9/9/499b65875f7b4dbcd3205550877d2f94.png" style="border: none; vertical-align: middle;" /></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}\! " class="tex" src="http://upload.wikimedia.org/math/c/c/6/cc64fe0132469276ff430afc86a01210.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta}\! " class="tex" src="http://upload.wikimedia.org/math/0/9/3/0936b643973a8e12960c4064a844a83a.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \tan \theta\! " class="tex" src="http://upload.wikimedia.org/math/3/e/6/3e64c04b4e255b35bc688310f5286ba8.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{1}{\sqrt{\csc^2 \theta - 1}}\! " class="tex" src="http://upload.wikimedia.org/math/2/9/9/2993fc75cd4d21398557279ebfcb2095.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\sqrt{\sec^2 \theta - 1}\! " class="tex" src="http://upload.wikimedia.org/math/d/6/5/d651adef1611f66709fb0d63472ba289.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \frac{1}{\cot \theta}\! " class="tex" src="http://upload.wikimedia.org/math/7/6/2/76295459b3992dc231b7ad90ad3ebe7c.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \csc \theta =\!" class="tex" src="http://upload.wikimedia.org/math/6/9/8/698bc74df1a3a60e3e6c1886191f0ee2.png" style="border: none; vertical-align: middle;" /></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \frac{1}{\sin \theta}\! " class="tex" src="http://upload.wikimedia.org/math/5/d/7/5d709ef743564b011d09b53d9cd1925f.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{1}{\sqrt{1 - \cos^2 \theta}}\! " class="tex" src="http://upload.wikimedia.org/math/0/2/d/02db090735695af99596ad763b999eab.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{\sqrt{1 + \tan^2 \theta}}{\tan \theta}\! " class="tex" src="http://upload.wikimedia.org/math/6/6/9/66979f6706b5b22946a3415a8be7fa78.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \csc \theta\! " class="tex" src="http://upload.wikimedia.org/math/3/3/0/3300ffc3669677063340e0c6e1e00f7a.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{\sec \theta}{\sqrt{\sec^2 \theta - 1}}\! " class="tex" src="http://upload.wikimedia.org/math/2/4/e/24ed564c5a78167c378ea40a83437eca.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\sqrt{1 + \cot^2 \theta}\! " class="tex" src="http://upload.wikimedia.org/math/2/0/3/20386f63ae4897ad3930bcd45a6538ec.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \sec \theta =\!" class="tex" src="http://upload.wikimedia.org/math/1/f/5/1f5a5c0af9edef9f6099772cceacec58.png" style="border: none; vertical-align: middle;" /></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{1}{\sqrt{1 - \sin^2 \theta}}\! " class="tex" src="http://upload.wikimedia.org/math/8/c/a/8ca18591c65a03c130f8d0993358c769.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \frac{1}{\cos \theta}\! " class="tex" src="http://upload.wikimedia.org/math/e/7/3/e73ba92e37073e084a483da86aef5324.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\sqrt{1 + \tan^2 \theta}\! " class="tex" src="http://upload.wikimedia.org/math/6/c/2/6c231d387adc96b0694bb202af2a36d3.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{\csc \theta}{\sqrt{\csc^2 \theta - 1}}\! " class="tex" src="http://upload.wikimedia.org/math/e/5/f/e5f2b761a3c83502dba9ac7626dcacd5.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \sec \theta\! " class="tex" src="http://upload.wikimedia.org/math/5/f/2/5f213e7bb36c7b9cf85642e3e24f87db.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{\sqrt{1 + \cot^2 \theta}}{\cot \theta}\! " class="tex" src="http://upload.wikimedia.org/math/1/0/c/10c7a5fa83a1b732ab67dd88fb2965d4.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \cot \theta =\!" class="tex" src="http://upload.wikimedia.org/math/c/0/0/c0016a9de185411f17a12a2161698725.png" style="border: none; vertical-align: middle;" /></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}\! " class="tex" src="http://upload.wikimedia.org/math/1/c/a/1ca70aca5c212cc597a8e9a096348de4.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{\cos \theta}{\sqrt{1 - \cos^2 \theta}}\! " class="tex" src="http://upload.wikimedia.org/math/6/4/5/645e7a6681dacb6936b83d6755fd979f.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \frac{1}{\tan \theta}\! " class="tex" src="http://upload.wikimedia.org/math/3/e/9/3e9f01f5a6dccf67517c084e2b166e49.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\sqrt{\csc^2 \theta - 1}\! " class="tex" src="http://upload.wikimedia.org/math/8/9/4/894b8c6d50c0a73547835eff76649358.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pm\frac{1}{\sqrt{\sec^2 \theta - 1}}\! " class="tex" src="http://upload.wikimedia.org/math/a/f/e/afe282e97712e098a9e305320e83f016.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt=" \cot \theta\! " class="tex" src="http://upload.wikimedia.org/math/f/f/1/ff14478418c00facf373c27122df9541.png" style="border: none; vertical-align: middle;" /></td></tr>
</tbody></table>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.2em;"><h2>
<b><u><span style="color: red;">Angel</span></u></b></h2>
<dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">1 full circle = 360 degrees = 2<img alt="\pi" class="tex" src="http://upload.wikimedia.org/math/5/2/2/522359592d78569a9eac16498aa7a087.png" style="border: none; vertical-align: middle;" /> radians = 400 grads.</dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
The following table shows the conversions for some common angles:</div>
<br />
<table class="wikitable" style="background-color: white; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin: 1em 0px; text-align: center;"><tbody>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Degree_(angle)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;" title="Degree (angle)">Degrees</a></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">30°</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">60°</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">120°</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">150°</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">210°</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">240°</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">300°</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">330°</td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Radian" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;" title="Radian">Radians</a></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac\pi6\!" class="tex" src="http://upload.wikimedia.org/math/c/0/5/c05ebaa249164524dbc2dac351c3556c.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac\pi3\!" class="tex" src="http://upload.wikimedia.org/math/2/c/6/2c696a1246645871dc5e921a8f05bca3.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac{2\pi}3\!" class="tex" src="http://upload.wikimedia.org/math/e/7/8/e78006f8ffdc93d426dc942d9827adfb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac{5\pi}6\!" class="tex" src="http://upload.wikimedia.org/math/2/9/6/296ac932b5da6324e6988f17c0320c38.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac{7\pi}6\!" class="tex" src="http://upload.wikimedia.org/math/5/2/7/52769b9cb7f20d86fae447d288155145.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac{4\pi}3\!" class="tex" src="http://upload.wikimedia.org/math/1/7/4/174c255b1299c2fbecd04eca1571b38d.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac{5\pi}3\!" class="tex" src="http://upload.wikimedia.org/math/3/b/1/3b1101467bda67d947a4522b4c85c775.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac{11\pi}6\!" class="tex" src="http://upload.wikimedia.org/math/2/4/0/24035e3198adf0407d939892ac8273fb.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Grad_(angle)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;" title="Grad (angle)">Grads</a></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">33⅓ grad</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">66⅔ grad</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">133⅓ grad</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">166⅔ grad</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">233⅓ grad</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">266⅔ grad</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">333⅓ grad</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">366⅔ grad</td></tr>
<tr><td colspan="9" style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Degree_(angle)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;" title="Degree (angle)">Degrees</a></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">45°</td><td bgcolor="#F8F8FF" style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">90°</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">135°</td><td bgcolor="#F8F8FF" style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">180°</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">225°</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">270°</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">315°</td><td bgcolor="#F8F8FF" style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">360°</td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Radian" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;" title="Radian">Radians</a></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac\pi4\!" class="tex" src="http://upload.wikimedia.org/math/e/c/5/ec5266e77efc6eb985f5e126fdad1d4e.png" style="border: none; vertical-align: middle;" /></td><td bgcolor="#F8F8FF" style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac\pi2\!" class="tex" src="http://upload.wikimedia.org/math/f/8/f/f8f84325b07ed25b0555aae40aa04191.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac{3\pi}4\!" class="tex" src="http://upload.wikimedia.org/math/0/a/0/0a041dffe73b1dda3c73ee4d26e56346.png" style="border: none; vertical-align: middle;" /></td><td bgcolor="#F8F8FF" style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\pi\!" class="tex" src="http://upload.wikimedia.org/math/b/a/f/bafaac6a706b3d07c96d280f86da6d1b.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac{5\pi}4\!" class="tex" src="http://upload.wikimedia.org/math/5/4/5/5455d70a13a621da64c17cbd15171344.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac{3\pi}2\!" class="tex" src="http://upload.wikimedia.org/math/1/2/a/12a5020710d086bfcfe992fb7f42f2b5.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\frac{7\pi}4\!" class="tex" src="http://upload.wikimedia.org/math/6/a/c/6acbf666c2bff73eadc7ce11bf3a5c59.png" style="border: none; vertical-align: middle;" /></td><td bgcolor="#F8F8FF" style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="2\pi\!" class="tex" src="http://upload.wikimedia.org/math/4/b/b/4bb27c5d64417b16c651406d2e4b3ff5.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Grad_(angle)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;" title="Grad (angle)">Grads</a></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">50 grad</td><td bgcolor="#F8F8FF" style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">100 grad</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">150 grad</td><td bgcolor="#F8F8FF" style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">200 grad</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">250 grad</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">300 grad</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">350 grad</td><td bgcolor="#F8F8FF" style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">400 grad</td></tr>
</tbody></table>
<br />
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Trigonometric_functions" style="color: red;">Trigonometric functions</span></h3>
<div>
<span class="mw-headline"></span><br />
<div style="background-color: white; color: black; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<span class="mw-headline">The primary trigonometric functions are the <a href="http://en.wikipedia.org/wiki/Sine" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;" title="Sine">sine</a> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Cosine" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;" title="Cosine">cosine</a> of an angle. These are sometimes abbreviated sin(<i>θ</i>) and cos(<i>θ</i>), respectively, where <i>θ</i> is the angle, but the parentheses around the angle are often omitted, e.g., sin <i>θ</i> and cos <i>θ</i>.</span></div>
<span class="mw-headline">
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<b><span style="color: #444444;">The <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Tangent_function" style="background-image: none; text-decoration: initial;" title="Tangent function">tangent</a> (tan) of an angle is the <a href="http://en.wikipedia.org/wiki/Ratio" style="background-image: none; text-decoration: initial;" title="Ratio">ratio</a> of the sine to the cosine:</span></b></div>
<div style="background-color: white; color: black; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<img alt="\tan\theta = \frac{\sin\theta}{\cos\theta}." class="tex" src="http://upload.wikimedia.org/math/0/e/5/0e504c85d19558e8df4d9dc0e0bcb3bb.png" style="border: none; vertical-align: middle;" /></div>
<div style="background-color: white; margin-bottom: 0.5em; margin-top: 0.4em;">
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;">Finally, the reciprocal functions secant (sec), cosecant (csc), and cotangent (cot) are the reciprocals of the cosine, sine, and tangent:</span></span></div>
<div style="background-color: white; margin-bottom: 0.5em; margin-top: 0.4em;">
<img alt="\sec\theta = \frac{1}{\cos\theta},\quad\csc\theta = \frac{1}{\sin\theta},\quad\cot\theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}." class="tex" src="http://upload.wikimedia.org/math/8/2/1/82170e617d3eacae343ebbd2ca3c10b0.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /></div>
<div style="background-color: white; margin-bottom: 0.5em; margin-top: 0.4em;">
</div>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; line-height: 19.200000762939453px; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Inverse_functions"><span style="color: red;">Inverse functions</span></span></h2>
<div>
<span class="mw-headline"></span><br />
<div>
<span class="mw-headline">Main article: Inverse trigonometric functions</span></div>
<span class="mw-headline">
<div>
The inverse trigonometric functions are partial inverse functions for the trigonometric functions. For example, the inverse function for the sine, known as the inverse sine (sin−1) or arcsine (arcsin or asin), satisfies</div>
<div>
<br /></div>
<div>
<img alt="\sin(\arcsin x) = x\quad\text{for} \quad |x| \leq 1 " class="tex" src="http://upload.wikimedia.org/math/7/7/b/77bb30c6a0b3379c3c1f2d326c2882d9.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /></div>
<div>
<br /></div>
<div>
and</div>
<div>
<br /></div>
<div>
<img alt="\arcsin(\sin x) = x\quad\text{for} \quad |x| \leq \pi/2. " class="tex" src="http://upload.wikimedia.org/math/5/1/3/513f43047e7e9951339625a3fa9206fa.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /></div>
<div>
<br /></div>
<div>
<span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">This article uses the notation below for inverse trigonometric functions:</span></div>
<div>
<span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"><br /></span></div>
<div>
<table class="wikitable" style="background-color: white; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin: 1em 0px; text-align: center;"><tbody>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Function</th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">sin</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">cos</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">tan</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">sec</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">csc</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">cot</td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Inverse</th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">arcsin</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">arccos</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">arctan</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">arcsec</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">arccsc</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">arccot</td></tr>
</tbody></table>
<h2 style="background-color: white; background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; line-height: 19.200000762939453px; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Symmetry.2C_shifts.2C_and_periodicity"><span style="color: red;">Symmetry, shifts, and periodicity</span></span></h2>
</div>
<div>
<span class="mw-headline"></span><br />
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline">
<span class="mw-headline" id="Symmetry">Symmetry-</span><span style="font-size: 13px; font-weight: normal;">When the trigonometric functions are reflected from certain angles, the result is often one of the other trigonometric functions. This leads to the following identities:</span></span></h3>
<span class="mw-headline">
<div>
<table class="wikitable" style="background-color: white; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin: 1em 0px; text-align: start;"><tbody>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Reflected in <img alt="\theta=0 " class="tex" src="http://upload.wikimedia.org/math/1/9/f/19f79ae2c22ce446fc72582b4db5dfc7.png" style="border: none; vertical-align: middle;" /><sup class="reference" id="cite_ref-3" style="font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial; white-space: nowrap;">[</a></sup></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Reflected in <img alt="\theta= \pi/2" class="tex" src="http://upload.wikimedia.org/math/6/d/b/6db0d32d690a884ce45baec7768bd983.png" style="border: none; vertical-align: middle;" /><br />
(co-function identities)<sup class="reference" id="cite_ref-4" style="font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial; white-space: nowrap;">[</a></sup></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Reflected in <img alt="\theta= \pi" class="tex" src="http://upload.wikimedia.org/math/e/c/6/ec6e904af8b95767456d081daccf51ad.png" style="border: none; vertical-align: middle;" /></th></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="
\begin{align}
\sin(-\theta) &= -\sin \theta \\
\cos(-\theta) &= +\cos \theta \\
\tan(-\theta) &= -\tan \theta \\
\csc(-\theta) &= -\csc \theta \\
\sec(-\theta) &= +\sec \theta \\
\cot(-\theta) &= -\cot \theta
\end{align}
" class="tex" src="http://upload.wikimedia.org/math/e/5/a/e5a4322488a7fc6448ac229f5d0b5ff7.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="
\begin{align}
\sin(\tfrac{\pi}{2} - \theta) &= +\cos \theta \\
\cos(\tfrac{\pi}{2} - \theta) &= +\sin \theta \\
\tan(\tfrac{\pi}{2} - \theta) &= +\cot \theta \\
\csc(\tfrac{\pi}{2} - \theta) &= +\sec \theta \\
\sec(\tfrac{\pi}{2} - \theta) &= +\csc \theta \\
\cot(\tfrac{\pi}{2} - \theta) &= +\tan \theta
\end{align}
" class="tex" src="http://upload.wikimedia.org/math/3/7/1/371cb8bac9a829b73714b43e2cac4132.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="
\begin{align}
\sin(\pi - \theta) &= +\sin \theta \\
\cos(\pi - \theta) &= -\cos \theta \\
\tan(\pi - \theta) &= -\tan \theta \\
\csc(\pi - \theta) &= +\csc \theta \\
\sec(\pi - \theta) &= -\sec \theta \\
\cot(\pi - \theta) &= -\cot \theta \\
\end{align}
" class="tex" src="http://upload.wikimedia.org/math/9/8/e/98eb98a18cfaff005607b95fe15c641b.png" style="border: none; vertical-align: middle;" /></td></tr>
</tbody></table>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Shifts_and_periodicity">Shifts and periodicity--</span><span style="font-size: 13px; font-weight: normal;">By shifting the function round by certain angles, it is often possible to find different trigonometric functions that express the result more simply. Some examples of this are shown by shifting functions round by π/2, π and 2π radians. Because the periods of these functions are either π or 2π, there are cases where the new function is exactly the same as the old function without the shift.</span></h3>
</div>
<div>
<table class="wikitable" style="background-color: white; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin: 1em 0px; text-align: start;"><tbody>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Shift by π/2</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Shift by π<br />
Period for tan and cot</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Shift by 2π<br />
Period for sin, cos, csc and sec<sup class="reference" id="cite_ref-6" style="font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-6" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial; white-space: nowrap;">[</a></sup></th></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="
\begin{align}
\sin(\theta + \tfrac{\pi}{2}) &= +\cos \theta \\
\cos(\theta + \tfrac{\pi}{2}) &= -\sin \theta \\
\tan(\theta + \tfrac{\pi}{2}) &= -\cot \theta \\
\csc(\theta + \tfrac{\pi}{2}) &= +\sec \theta \\
\sec(\theta + \tfrac{\pi}{2}) &= -\csc \theta \\
\cot(\theta + \tfrac{\pi}{2}) &= -\tan \theta
\end{align}
" class="tex" src="http://upload.wikimedia.org/math/9/2/3/923afb2e4b7ae94b2f706923641b0691.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="
\begin{align}
\sin(\theta + \pi) &= -\sin \theta \\
\cos(\theta + \pi) &= -\cos \theta \\
\tan(\theta + \pi) &= +\tan \theta \\
\csc(\theta + \pi) &= -\csc \theta \\
\sec(\theta + \pi) &= -\sec \theta \\
\cot(\theta + \pi) &= +\cot \theta \\
\end{align}
" class="tex" src="http://upload.wikimedia.org/math/8/4/4/84432bb4458c6216810fc9ef6fe2bd0f.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="
\begin{align}
\sin(\theta + 2\pi) &= +\sin \theta \\
\cos(\theta + 2\pi) &= +\cos \theta \\
\tan(\theta + 2\pi) &= +\tan \theta \\
\csc(\theta + 2\pi) &= +\csc \theta \\
\sec(\theta + 2\pi) &= +\sec \theta \\
\cot(\theta + 2\pi) &= +\cot \theta
\end{align}
" class="tex" src="http://upload.wikimedia.org/math/1/3/c/13c34cc3e94395c30dd3f4212a039d90.png" style="border: none; vertical-align: middle;" /></td></tr>
</tbody></table>
</div>
</span></div>
<div>
<h2 style="background-color: white; background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Angle_sum_and_difference_identities" style="font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19.200000762939453px;"><span style="color: red;">Angle sum and difference identities-</span></span><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">These are also known as the </span><i style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">addition and subtraction theorems</i><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"> or </span><i style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">formulæ</i><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">. They were originally established by the 10th century Persian mathematician </span><a href="http://en.wikipedia.org/wiki/Ab%C5%AB_al-Waf%C4%81%27_B%C5%ABzj%C4%81n%C4%AB" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; text-decoration: initial;" title="Abū al-Wafā' Būzjānī">Abū al-Wafā' Būzjānī</a><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">. One method of proving these identities is to apply </span><a href="http://en.wikipedia.org/wiki/Euler%27s_formula" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; text-decoration: initial;" title="Euler's formula">Euler's formula</a><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">. The use of the symbols </span><img alt="\pm" class="tex" src="http://upload.wikimedia.org/math/5/7/2/5722e2f6169308b8be3542900c6d6553.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin: 0px; vertical-align: middle;" /><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"> and </span><img alt="\mp" class="tex" src="http://upload.wikimedia.org/math/0/2/f/02f4d839cf96b60132fea53480d63648.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin: 0px; vertical-align: middle;" /><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"> is described in the article </span><a href="http://en.wikipedia.org/wiki/Plus-minus_sign" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; text-decoration: initial;" title="Plus-minus sign">plus-minus sign</a><span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">.</span></h2>
</div>
<div>
<span class="mw-headline"><br /></span></div>
<div>
<span class="mw-headline"><table class="wikitable" style="background-color: white; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin: 1em 0px; text-align: start;"><tbody>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Sine</th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \!" class="tex" src="http://upload.wikimedia.org/math/e/2/1/e21be04c83072e08c84eb34ef64d52f5.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Cosine</th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta\," class="tex" src="http://upload.wikimedia.org/math/d/7/7/d777676afb5ddc2c9f403f3569b49176.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Tangent</th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="\tan(\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}" class="tex" src="http://upload.wikimedia.org/math/5/1/a/51ae1ae6b131eedcf53585e6fe0b72ed.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Arcsine</th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="\arcsin\alpha \pm \arcsin\beta = \arcsin\left(\alpha\sqrt{1-\beta^2} \pm \beta\sqrt{1-\alpha^2}\right)" class="tex" src="http://upload.wikimedia.org/math/b/7/3/b734c360c4ccd6a7a4a054dc23db8619.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Arccosine</th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="\arccos\alpha \pm \arccos\beta = \arccos\left(\alpha\beta \mp \sqrt{(1-\alpha^2)(1-\beta^2)}\right)" class="tex" src="http://upload.wikimedia.org/math/7/0/4/70430ce08442f8ab9bea75766871b59c.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Arctangent</th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="\arctan\alpha \pm \arctan\beta = \arctan\left(\frac{\alpha \pm \beta}{1 \mp \alpha\beta}\right)" class="tex" src="http://upload.wikimedia.org/math/f/5/a/f5a0f7dfc2765a6ba210d82b5bd6866a.png" style="border: none; vertical-align: middle;" /><sup class="reference" id="cite_ref-13" style="line-height: 1em;"><br /></sup></td></tr>
</tbody></table>
</span></div>
<div>
<span style="font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"><br /></span></div>
<div>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Matrix_form"><span style="color: red;">Matrix form-</span></span><span style="font-size: 13px;">The sum and difference formulae for sine and cosine can be written in </span><a href="http://en.wikipedia.org/wiki/Matrix_(mathematics)" style="background-image: none; color: #0b0080; font-size: 13px; text-decoration: initial;" title="Matrix (mathematics)">matrix</a><span style="font-size: 13px;"> form as:</span></h3>
</div>
<div>
<span style="font-size: 13px;"><br /></span></div>
<div>
<img alt="
\begin{align}
& {} \quad
\left(\begin{array}{rr}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{array}\right)
\left(\begin{array}{rr}
\cos\phi & -\sin\phi \\
\sin\phi & \cos\phi
\end{array}\right) \\[12pt]
& = \left(\begin{array}{rr}
\cos\theta\cos\phi - \sin\theta\sin\phi & -\cos\theta\sin\phi - \sin\theta\cos\phi \\
\sin\theta\cos\phi + \cos\theta\sin\phi & -\sin\theta\sin\phi + \cos\theta\cos\phi
\end{array}\right) \\[12pt]
& = \left(\begin{array}{rr}
\cos(\theta+\phi) & -\sin(\theta+\phi) \\
\sin(\theta+\phi) & \cos(\theta+\phi)
\end{array}\right)
\end{align}
" class="tex" src="http://upload.wikimedia.org/math/6/d/6/6d660d7ae58e070731b043ae52d38692.png" style="background-color: white; border: none; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; vertical-align: middle;" /></div>
<div>
<span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">This shows that these matrices form a </span><a href="http://en.wikipedia.org/wiki/Group_representation" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; text-decoration: initial;" title="Group representation">representation</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"> of the rotation group in the plane (technically, the </span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Special_orthogonal_group" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; text-decoration: initial;" title="Special orthogonal group">special orthogonal group</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"> </span><i style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">SO</i><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;">(2)), since the composition law is fulfilled: subsequent multiplications of a vector with these two matrices yields the same result as the rotation by the sum of the angles.</span><br />
<span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px;"><br /></span>
<br />
<h2 style="text-align: left;">
<b>Reciprocal identities</b></h2>
<b><br /></b>
<br />
<img align="BOTTOM" alt="displaymath161" height="107" src="http://www.sosmath.com/trig/Trig5/trig5/img1.gif" width="348" /><br />
<br />
<h2 style="text-align: left;">
<b><br /></b></h2>
<h2 style="text-align: left;">
<b>Pythagorean Identities</b></h2>
<b><br /></b>
<br />
<img align="BOTTOM" alt="displaymath162" height="35" src="http://www.sosmath.com/trig/Trig5/trig5/img2.gif" width="480" /><br />
<br />
<h2 style="text-align: left;">
<b><br /></b></h2>
<h2 style="text-align: left;">
<b>Quotient Identities</b></h2>
<div>
<b><br /></b></div>
<div>
<b><br /></b></div>
<br />
<img align="BOTTOM" alt="displaymath163" height="46" src="http://www.sosmath.com/trig/Trig5/trig5/img3.gif" width="229" /><br />
<br />
<h2 style="text-align: left;">
<b><br /></b></h2>
<h2 style="text-align: left;">
<b>Co-Function Identities</b></h2>
<div>
<b><br /></b></div>
<div>
<b><br /></b></div>
<br />
<img align="BOTTOM" alt="displaymath164" height="98" src="http://www.sosmath.com/trig/Trig5/trig5/img4.gif" width="487" /><br />
<br />
<h2 style="text-align: left;">
<b><br /></b></h2>
<h2 style="text-align: left;">
<b>Even-Odd Identities</b></h2>
<div>
<b><br /></b></div>
<div>
<b><br /></b></div>
<br />
<img align="BOTTOM" alt="displaymath165" height="42" src="http://www.sosmath.com/trig/Trig5/trig5/img5.gif" width="470" /><br />
<br />
<b><br /></b>
<b><br /></b>
<h2 style="text-align: left;">
<b>Sum-Difference Formulas</b></h2>
<br />
<img align="BOTTOM" alt="displaymath166" height="160" src="http://www.sosmath.com/trig/Trig5/trig5/img6.gif" width="656" /><br />
<br />
<h2 style="text-align: left;">
<b>Double Angle Formulas</b></h2>
<b><br /></b>
<b><br /></b>
<br />
<img align="BOTTOM" alt="align99" height="150" src="http://www.sosmath.com/trig/Trig5/trig5/img7.gif" width="201" /><br />
<br />
<b><br /></b>
<h2 style="text-align: left;">
<b>Power-Reducing/Half Angle Formulas</b></h2>
<b><br /></b>
<br />
<img align="BOTTOM" alt="displaymath167" height="179" src="http://www.sosmath.com/trig/Trig5/trig5/img8.gif" width="169" /><br />
<br />
<b><br /></b>
<h2 style="text-align: left;">
<b>Sum-to-Product Formulas</b></h2>
<b><br /></b>
<br />
<img align="BOTTOM" alt="displaymath168" height="259" src="http://www.sosmath.com/trig/Trig5/trig5/img9.gif" width="355" /><br />
<br />
<h2 style="text-align: left;">
<b><br /></b></h2>
<h2 style="text-align: left;">
<b>Product-to-Sum Formulas</b></h2>
<div>
<b><br /></b></div>
<br />
<img align="BOTTOM" alt="displaymath169" height="229" src="http://www.sosmath.com/trig/Trig5/trig5/img10.gif" width="319" /><br />
<a href="" name="addition" style="background-color: #ffffee;"></a><br />
<h2 style="font-family: Arial; font-size: small;">
<a href="" name="addition" style="font-size: small;"><h2>
<span style="font-weight: normal;">Addition Formulas</span></h2>
<div>
<span style="font-weight: normal;"><br /></span></div>
cos(X + Y) = cosX cosY - sinX sinY<br /><br />cos(X - Y) = cosX cosY + sinX sinY<br /><br />sin(X + Y) = sinX cosY + cosX sinY<br /><br />sin(X - Y) = sinX cosY - cosX sinY<br /><br />tan(X + Y) = [ tanX + tanY ] / [ 1 - tanX tanY]<br /><br />tan(X - Y) = [ tanX - tanY ] / [ 1 + tanX tanY] </a><div style="text-align: left;">
<a href="" name="addition" style="font-size: small;"><br /></a></div>
<a href="" name="addition" style="font-size: small;">cot(X + Y) = [ cotX cotY - 1 ] / [ cotX + cotY]<br /><br />cot(X - Y) = [ cotX cotY + 1 ] / [ cotX - cotY]</a></h2>
<div>
<a href="" name="addition" style="font-size: small;"><br /></a></div>
<h2 style="text-align: left;">
<a href="" name="addition"><span style="font-family: Arial; font-size: x-small;">Difference of Squares Formulas</span></a></h2>
<h2 style="text-align: left;">
<a href="" name="addition"></a><a href="" name="difference_squares" style="font-family: Arial; font-size: small;">sin<sup> 2</sup>X - sin<sup> 2</sup>Y = sin(X + Y)sin(X - Y) </a><a href="" name="addition"></a><a href="" name="difference_squares" style="font-family: Arial; font-size: small;"><br /></a><a href="" name="addition"></a><a href="" name="difference_squares" style="font-family: Arial; font-size: small;">cos<sup> 2</sup>X - cos<sup> 2</sup>Y = - sin(X + Y)sin(X - Y) </a><a href="" name="addition"></a><a href="" name="difference_squares" style="font-family: Arial; font-size: small;"><br /></a><a href="" name="addition"></a><a href="" name="difference_squares" style="font-family: Arial; font-size: small;">cos<sup> 2</sup>X - sin<sup> 2</sup>Y = cos(X + Y)cos(X - Y)</a></h2>
<div>
<a href="" name="addition"></a><a href="" name="difference_squares" style="font-family: Arial; font-size: small;"></a><a href="" name="mulitple_angle"><h2>
Multiple Angle Formulas</h2>
</a><div>
<a href="" name="mulitple_angle"></a><a href="" name="mulitple_angle" style="background-color: #ffffee; font-family: Arial; font-size: small;"><b>sin(3X) = 3sinX - 4sin<sup> 3</sup>X<br /><br />cos(3X) = 4cos<sup> 3</sup>X - 3cosX<br /><br />sin(4X) = 4sinXcosX - 8sin<sup> 3</sup>XcosX<br /><br />cos(4X) = 8cos<sup> 4</sup>X - 8cos<sup> 2</sup>X + 1</b></a></div>
<div>
<span style="color: red;"><b><br /></b></span></div>
<div>
<span style="color: red;"><b><br /></b></span></div>
<h2 style="text-align: left;">
<b>Power Reducing Formulas</b></h2>
<div>
<a href="" name="power_reducing" style="background-color: #ffffee; font-family: Arial; font-size: small;">sin<sup> 2</sup>X = 1/2 - (1/2)cos(2X))<br /><br />cos<sup> 2</sup>X = 1/2 + (1/2)cos(2X))<br /><br />sin<sup> 3</sup>X = (3/4)sinX - (1/4)sin(3X)<br /><br />cos<sup> 3</sup>X = (3/4)cosX + (1/4)cos(3X)<br /><br />sin<sup> 4</sup>X = (3/8) - (1/2)cos(2X) + (1/8)cos(4X)<br /><br />cos<sup> 4</sup>X = (3/8) + (1/2)cos(2X) + (1/8)cos(4X)<br /><br />sin<sup> 5</sup>X = (5/8)sinX - (5/16)sin(3X) + (1/16)sin(5X)<br /><br />cos<sup> 5</sup>X = (5/8)cosX + (5/16)cos(3X) + (1/16)cos(5X)<br /><br />sin<sup> 6</sup>X = 5/16 - (15/32)cos(2X) + (6/32)cos(4X) - (1/32)cos(6X)<br /><br />cos<sup> 6</sup>X = 5/16 + (15/32)cos(2X) + (6/32)cos(4X) + (1/32)cos(6X)</a></div>
<div>
<b><br /></b></div>
<div>
<b><br /></b></div>
</div>
<div style="font-family: Arial; font-size: small; text-align: left;">
<a href="" name="addition" style="font-size: small;"></a><a href="" name="periodicity" style="background-color: #ffffee;"><h2>
Trigonometric Functions Periodicity</h2>
<div>
<br /></div>
</a><div>
<a href="" name="periodicity" style="background-color: #ffffee;"></a><a href="" name="periodicity">sin (X + 2Pi) = sin X , period 2Pi<br /><br />cos (X + 2Pi) = cos X , period 2Pi<br /><br />sec (X + 2Pi) = sec X , period 2Pi<br /><br />csc (X + 2Pi) = csc X , period 2Pi<br /><br />tan (X + Pi) = tan X , period Pi<br /><br />cot (X + Pi) = cot X , period Pi</a></div>
</div>
<br />
</div>
</span></div>
</span></div>
</div>
Anonymoushttp://www.blogger.com/profile/12500986417656529636noreply@blogger.com2Naraina, New Delhi, Delhi, India28.6302834 77.139887828.6163464 77.1201468 28.6442204 77.1596288tag:blogger.com,1999:blog-7072204672677043641.post-86116241959557717992012-11-30T22:58:00.000-08:002012-12-02T23:44:51.792-08:00What is a Bodmas formula and how to use it<div dir="ltr" style="text-align: left;" trbidi="on">
BODMAS-<br />
1-The letter <b>B</b> is telling you to do the brackets first<br />
<br />
<br />
2-The letters <b>DM</b> represent division and multiplication<br />
<br />
<br />
3- The letters <b>AS</b> represent addition and subtraction<br />
<br />
<br />
<br />
<br />
<div style="background-color: #eee8dd; font-family: Georgia, 'Times New Roman', serif; font-size: 13px; line-height: 21px;">
<span style="background-color: #ffe599; font-family: Arial; font-size: medium;"><span style="background-color: white;"><strong><span style="color: #333333;"> </span><span style="color: magenta;"> 1</span></strong></span><b><span style="color: magenta;"><span style="background-color: white;">. ( ) </span><span style="background-color: yellow;"> Brackets</span></span></b></span></div>
<div style="background-color: #eee8dd; font-family: Georgia, 'Times New Roman', serif; font-size: 13px; line-height: 21px;">
</div>
<div style="background-color: #eee8dd; font-family: Georgia, 'Times New Roman', serif; font-size: 13px; line-height: 21px;">
<span style="color: magenta; font-family: Arial;"><b><span style="font-size: medium;"> 2. </span><span style="font-family: arial,sans-serif; font-size: large;">÷ <span style="font-size: medium;">and x <span style="background-color: lime;"> Left to right</span></span></span></b></span></div>
<div style="background-color: #eee8dd; font-family: Georgia, 'Times New Roman', serif; font-size: 13px; line-height: 21px;">
</div>
<div style="background-color: #eee8dd; font-family: Georgia, 'Times New Roman', serif; font-size: 13px; line-height: 21px;">
<span style="color: magenta; font-family: Arial; font-size: medium;"><b> 3. + and -- <span style="background-color: cyan;">Left to right</span> </b></span></div>
<div style="background-color: #eee8dd; color: #333333; font-family: Georgia, 'Times New Roman', serif; font-size: 13px; line-height: 21px;">
</div>
<br />
</div>
Anonymoushttp://www.blogger.com/profile/12500986417656529636noreply@blogger.com0tag:blogger.com,1999:blog-7072204672677043641.post-73550738549204719882012-11-30T22:54:00.001-08:002012-11-30T22:56:15.077-08:00Divisibility Rule<div dir="ltr" style="text-align: left;" trbidi="on">
<br />
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Divisibility_by_2">Divisibility by 2</span></h3>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
First, take any even number (for this example it will be 376) and note the last digit in the number, discarding the other digits. Then take that digit (6) while ignoring the rest of the number and determine if it is divisible by 2. If it is divisible by 2, then the original number is divisible by 2.</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<b>Example</b></div>
<ol style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; list-style-image: none; margin: 0.3em 0px 0px 3.2em; padding: 0px;">
<li style="margin-bottom: 0.1em;">376 (The original number)</li>
<li style="margin-bottom: 0.1em;"><s>37</s> <u>6</u> (Take the last digit)</li>
<li style="margin-bottom: 0.1em;">6 ÷ 2 = 3 (Check to see if the last digit is divisible by 2)</li>
<li style="margin-bottom: 0.1em;">376 ÷ 2 = 188 (If the last digit is divisible by 2, then the whole number is divisible by 2)</li>
</ol>
<div>
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"><br /></span></span></div>
<div>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Divisibility_by_3">Divisibility by 3</span></h3>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
First, take any number (for this example it will be 492) and add together each digit in the number (4 + 9 + 2 = 15). Then take that sum (15) and determine if it is divisible by 3. The original number is divisible by 3 if and only if the final number is divisible by 3.</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
If a number is a multiplication of 3 consecutive numbers then that number is always divisible by 3. This is useful for when the number takes the form of (<i>n</i> × (<i>n</i> − 1) × (<i>n</i> + 1))</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<b>Ex.</b></div>
<ol style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; list-style-image: none; margin: 0.3em 0px 0px 3.2em; padding: 0px;">
<li style="margin-bottom: 0.1em;">492 (The original number)</li>
<li style="margin-bottom: 0.1em;">4 + 9 + 2 = 15 (Add each individual digit together)</li>
<li style="margin-bottom: 0.1em;">15 is divisible by 3 at which point we can stop. Alternatively we can continue using the same method if the number is still too large:</li>
<li style="margin-bottom: 0.1em;">1 + 5 = 6 (Add each individual digit together)</li>
<li style="margin-bottom: 0.1em;">6 ÷ 3 = 2 (Check to see if the number received is divisible by 3)</li>
<li style="margin-bottom: 0.1em;">492 ÷ 3 = 164 (If the number obtained by using the rule is divisible by 3, then the whole number is divisible by 3)</li>
</ol>
<div>
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"><br /></span></span></div>
</div>
<div>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Divisibility_by_4">Divisibility by 4</span></h3>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4 this is because 100 is divisible by 4 and so adding hundreds, thousands, etc. is simply adding another number that is divisible by 4. If any number ends in a two digit number that you know is divisible by 4 (e.g. 24, 04, 08, etc.), then the whole number will be divisible by 4 regardless of what is before the last two digits.</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
Alternatively, one can simply divide the number by 2, and then check the result to find if it is divisible by 2. If it is, the original number is divisible by 4. In addition, the result of this test is the same as the original number divided by 4.</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<b>Ex.</b><br /><b>General rule</b></div>
<ol style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; list-style-image: none; margin: 0.3em 0px 0px 3.2em; padding: 0px;">
<li style="margin-bottom: 0.1em;">2092 (The original number)</li>
<li style="margin-bottom: 0.1em;"><s>20</s> <u>92</u> (Take the last two digits of the number, discarding any other digits)</li>
<li style="margin-bottom: 0.1em;">92 ÷ 4 = 23 (Check to see if the number is divisible by 4)</li>
<li style="margin-bottom: 0.1em;">2092 ÷ 4 = 523 (If the number that is obtained is divisible by 4, then the original number is divisible by 4)</li>
</ol>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<b>Alternative example</b></div>
<ol style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; list-style-image: none; margin: 0.3em 0px 0px 3.2em; padding: 0px;">
<li style="margin-bottom: 0.1em;">1720 (The original number)</li>
<li style="margin-bottom: 0.1em;">1720 ÷ 2 = 860 (Divide the original number by 2)</li>
<li style="margin-bottom: 0.1em;">860 ÷ 2 = 430 (Check to see if the result is divisible by 2)</li>
<li style="margin-bottom: 0.1em;">1720 ÷ 4 = 430 (If the result is divisible by 2, then the original number is divisible by 4)</li>
</ol>
<div>
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"><br /></span></span></div>
</div>
<div>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Divisibility_by_5">Divisibility by 5</span></h3>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
Divisibility by 5 is easily determined by checking the last digit in the number (47<b>5</b>), and seeing if it is either 0 or 5. If the last number is either 0 or 5, the entire number is divisible by 5.<sup class="reference" id="cite_ref-Pascal.27s-criterion_1-10" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Divisibility_rule#cite_note-Pascal.27s-criterion-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial; white-space: nowrap;">[1]</a></sup><sup class="reference" id="cite_ref-last-m-digits_2-7" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Divisibility_rule#cite_note-last-m-digits-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial; white-space: nowrap;">[2]</a></sup></div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
If the last digit in the number is 0, then the result will be the remaining digits multiplied by 2. For example, the number 40 ends in a zero (0), so take the remaining digits (4) and multiply that by two (4 × 2 = 8). The result is the same as the result of 40 divided by 5(40/5 = 8).</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
If the last digit in the number is 5, then the result will be the remaining digits multiplied by two (2), plus one (1). For example, the number 125 ends in a 5, so take the remaining digits (12), multiply them by two (12 × 2 = 24), then add one (24 + 1 = 25). The result is the same as the result of 125 divided by 5 (125/5=25).</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<b>Ex.</b><br /><b>If the last digit is 0</b></div>
<ol style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; list-style-image: none; margin: 0.3em 0px 0px 3.2em; padding: 0px;">
<li style="margin-bottom: 0.1em;">110 (The original number)</li>
<li style="margin-bottom: 0.1em;"><s>11</s> <u>0</u> (Take the last digit of the number, and check if it is 0 or 5)</li>
<li style="margin-bottom: 0.1em;"><u>11</u> <s>0</s> (If it is 0, take the remaining digits, discarding the last)</li>
<li style="margin-bottom: 0.1em;">11 × 2 = 22 (Multiply the result by 2)</li>
<li style="margin-bottom: 0.1em;">110 ÷ 5 = 22 (The result is the same as the original number divided by 5)</li>
</ol>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<b>If the last digit is 5</b></div>
<ol style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; list-style-image: none; margin: 0.3em 0px 0px 3.2em; padding: 0px;">
<li style="margin-bottom: 0.1em;">85 (The original number)</li>
<li style="margin-bottom: 0.1em;"><s>8</s> <u>5</u> (Take the last digit of the number, and check if it is 0 or 5)</li>
<li style="margin-bottom: 0.1em;"><u>8</u> <s>5</s> (If it is 5, take the remaining digits, discarding the last)</li>
<li style="margin-bottom: 0.1em;">8 × 2 = 16 (Multiply the result by 2)</li>
<li style="margin-bottom: 0.1em;">16 + 1 = 17 (Add 1 to the result)</li>
<li style="margin-bottom: 0.1em;">85 ÷ 5 = 17 (The result is the same as the original number divided by 5)</li>
</ol>
<div>
<span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 19.200000762939453px;"><br /></span></span></div>
</div>
<div>
<h3 style="background-color: white; background-image: none; border-bottom-style: none; font-family: sans-serif; font-size: 17px; line-height: 19.200000762939453px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Divisibility_by_6">Divisibility by 6</span></h3>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
Divisibility by 6 is determined by checking the original number to see if it is both an even number (<a href="http://en.wikipedia.org/wiki/Divisibility_rule#Divisibility_by_2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;">divisible by 2</a>) and <a href="http://en.wikipedia.org/wiki/Divisibility_rule#Divisibility_by_3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;">divisible by 3</a>.<sup class="reference" id="cite_ref-product-of-coprimes_5-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Divisibility_rule#cite_note-product-of-coprimes-5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial; white-space: nowrap;">[5]</a></sup> This is the best test to use.</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
Alternatively, one can check for divisibility by six by taking the number (246), dropping the last digit in the number (<u>24</u> <s>6</s>, adding together the remaining number (24 becomes 2 + 4 = 6), multiplying that by four (6 × 4 = 24), and adding the last digit of the original number to that (24 + 6 = 30). If this number is divisible by six, the original number is divisible by 6.</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
If the number is divisible by six, take the original number (246) and divide it by two (246 ÷ 2 = 123). Then, take that result and divide it by three (123 ÷ 3 = 41). This result is the same as the original number divided by six (246 ÷ 6 = 41).</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<b>Ex.</b><br /><b>General rule</b></div>
<ol style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; list-style-image: none; margin: 0.3em 0px 0px 3.2em; padding: 0px;">
<li style="margin-bottom: 0.1em;">324 (The original number)</li>
<li style="margin-bottom: 0.1em;">324 ÷ 3 = 108 (Check to see if the original number is divisible by 3)</li>
<li style="margin-bottom: 0.1em;">324 ÷ 2 = 162 <b>OR</b> 108 ÷ 2 = 54 (Check to see if either the original number or the result of the previous equation is divisible by 2)</li>
<li style="margin-bottom: 0.1em;">324 ÷ 6 = 54 (If either of the tests in the last step are true, then the original number is divisible by 6. Also, the result of the second test returns the same result as the original number divided by 6)</li>
</ol>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
<br /><b>Finding a remainder of a number when divided by 6</b><br />6 − (1, −2, −2, −2, −2, and −2 goes on for the rest) No period.<br />Minimum magnitude sequence<br />(1, 4, 4, 4, 4, and 4 goes on for the rest)<br />Positive sequence<br />Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on. Next, compute the sum of all the values and take the remainder on division by 6.<br />Example: What is the remainder when 1036125837 is divided by 6?<br />Multiplication of the rightmost digit = 1 × 7 = 7<br />Multiplication of the second rightmost digit = 3 × −2 = −6<br />Third rightmost digit = −16<br />Fourth rightmost digit = −10<br />Fifth rightmost digit = −4<br />Sixth rightmost digit = −2<br />Seventh rightmost digit = −12<br />Eighth rightmost digit = −6<br />Ninth rightmost digit = 0<br />Tenth rightmost digit = −2<br />Sum = −51<br />−51 modulo 6 = 3<br />Remainder = 3</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
</div>
<h3 style="background-image: none; border-bottom-style: none; font-size: 17px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="mw-headline" id="Divisibility_by_7">Divisibility by 7</span></h3>
<div>
<span class="mw-headline"><br /></span></div>
<br />
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.200000762939453px; margin-bottom: 0.5em; margin-top: 0.4em;">
</div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
Divisibility by 7 can be tested by a recursive method. A number of the form 10<i>x</i> + <i>y</i> is divisible by 7 if and only if <i>x</i> − 2<i>y</i> is divisible by 7. In other words, subtract twice the last digit from the number formed by the remaining digits. Continue to do this until a small number (below 20 in <a href="http://en.wikipedia.org/wiki/Absolute_value" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;" title="Absolute value">absolute value</a>) is obtained. The original number is divisible by 7 if and only if the number obtained using this procedure is divisible by 7. For example, the number 371: 37 − (2×1) = 37 − 2 = 35; 3 − (2 × 5) = 3 − 10 = −7; thus, since −7 is divisible by 7, 371 is divisible by 7.</div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
Another method is multiplication by 3. A number of the form 10<i>x</i> + <i>y</i> has the same remainder when divided by 7 as 3<i>x</i> + <i>y</i>. One must multiply the leftmost digit of the original number by 3, add the next digit, take the remainder when divided by 7, and continue from the beginning: multiply by 3, add the next digit, etc. For example, the number 371: 3×3 + 7 = 16 remainder 2, and 2×3 + 1 = 7. This method can be used to find the remainder of division by 7.</div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
A more complicated algorithm for testing divisibility by 7 uses the fact that 10<sup style="line-height: 1em;">0</sup> ≡ 1, 10<sup style="line-height: 1em;">1</sup> ≡ 3, 10<sup style="line-height: 1em;">2</sup> ≡ 2, 10<sup style="line-height: 1em;">3</sup> ≡ 6, 10<sup style="line-height: 1em;">4</sup> ≡ 4, 10<sup style="line-height: 1em;">5</sup> ≡ 5, 10<sup style="line-height: 1em;">6</sup> ≡ 1, ... (mod 7). Take each digit of the number (371) in reverse order (173), multiplying them successively by the digits <b>1</b>, <b>3</b>, <b>2</b>, <b>6</b>, <b>4</b>, <b>5</b>, repeating with this sequence of multipliers as long as necessary (1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, ...), and adding the products (1×<b>1</b> + 7×<b>3</b> + 3×<b>2</b> = 1 + 21 + 6 = 28). The original number is divisible by 7 if and only if the number obtained using this procedure is divisible by 7 (hence 371 is divisible by 7 since 28 is).<sup class="reference" id="cite_ref-7Divis1_8-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Divisibility_rule#cite_note-7Divis1-8" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial; white-space: nowrap;">[8]</a></sup></div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
This method can be simplified by removing the need to multiply. All it would take with this simplification is to memorize the sequence above (132645...), and to add and subtract, but always working with one-digit numbers.</div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
The simplification goes as follows:</div>
<ul style="list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;">Take for instance the number <b>371</b></li>
<li style="margin-bottom: 0.1em;">Change all occurrences of <b>7</b>, <b>8</b> or <b>9</b> into <b>0</b>, <b>1</b> and <b>2</b>, respectively. In this example, we get: <b>301</b>. This second step may be skipped, except for the left most digit, but following it may facilitate calculations later on.</li>
<li style="margin-bottom: 0.1em;">Now convert the first digit (3) into the following digit in the sequence <b>13264513...</b> In our example, 3 becomes <b>2</b>.</li>
<li style="margin-bottom: 0.1em;">Add the result in the previous step (2) to the second digit of the number, and substitute the result for both digits, leaving all remaining digits unmodified: 2 + 0 = 2. So <i>30</i>1 becomes <b><i>2</i>1</b>.</li>
<li style="margin-bottom: 0.1em;">Repeat the procedure until you have a recognizable multiple of 7, or to make sure, a number between 0 and 6. So, starting from 21 (which is a recognizable multiple of 7), take the first digit (2) and convert it into the following in the sequence above: 2 becomes 6. Then add this to the second digit: 6 + 1 = <b>7</b>.</li>
<li style="margin-bottom: 0.1em;">If at any point the first digit is 8 or 9, these become 1 or 2, respectively. But if it is a 7 it should become 0, only if no other digits follow. Otherwise, it should simply be dropped. This is because that 7 would have become 0, and numbers with at least two digits before the decimal dot do not begin with 0, which is useless. According to this, our 7 becomes <b>0</b>.</li>
</ul>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
If through this procedure you obtain a <b>0</b> or any recognizable multiple of 7, then the original number is a multiple of 7. If you obtain any number from <b>1</b> to <b>6</b>, that will indicate how much you should subtract from the original number to get a multiple of 7. In other words, you will find the <a href="http://en.wikipedia.org/wiki/Remainder" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: initial;" title="Remainder">remainder</a> of dividing the number by 7. For example take the number <b>186</b>:</div>
<ul style="list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;">First, change the 8 into a 1: <b>116</b>.</li>
<li style="margin-bottom: 0.1em;">Now, change 1 into the following digit in the sequence (3), add it to the second digit, and write the result instead of both: 3 + 1 = <i>4</i>. So <i>11</i>6 becomes now <b><i>4</i>6</b>.</li>
<li style="margin-bottom: 0.1em;">Repeat the procedure, since the number is greater than 7. Now, 4 becomes 5, which must be added to 6. That is <b>11</b>.</li>
<li style="margin-bottom: 0.1em;">Repeat the procedure one more time: 1 becomes 3, which is added to the second digit (1): 3 + 1 = <b>4</b>.</li>
</ul>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
Now we have a number lower than 7, and this number (4) is the remainder of dividing 186/7. So 186 minus 4, which is 182, must be a multiple of 7.</div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
Note: The reason why this works is that if we have: <b>a+b=c</b> and <b>b</b> is a multiple of any given number <b>n</b>, then <b>a</b> and <b>c</b> will necessarily produce the same remainder when divided by <b>n</b>. In other words, in 2 + 7 = 9, 7 is divisible by 7. So 2 and 9 must have the same reminder when divided by 7. The remainder is 2.</div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
Therefore, if a number <i>n</i> is a multiple of 7 (i.e.: the remainder of <i>n</i>/7 is 0), then adding (or subtracting) multiples of 7 cannot possibly change that property.</div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
What this procedure does, as explained above for most divisibility rules, is simply subtract little by little multiples of 7 from the original number until reaching a number that is small enough for us to remember whether it is a multiple of 7. If 1 becomes a 3 in the following decimal position, that is just the same as converting 10×10<sup style="line-height: 1em;"><i>n</i></sup> into a 3×10<sup style="line-height: 1em;"><i>n</i></sup>. And that is actually the same as subtracting 7×10<sup style="line-height: 1em;"><i>n</i></sup> (clearly a multiple of 7) from 10×10<sup style="line-height: 1em;"><i>n</i></sup>.</div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
Similarly, when you turn a 3 into a 2 in the following decimal position, you are turning 30×10<sup style="line-height: 1em;"><i>n</i></sup> into 2×10<sup style="line-height: 1em;"><i>n</i></sup>, which is the same as subtracting 30×10<sup style="line-height: 1em;"><i>n</i></sup>−28×10<sup style="line-height: 1em;">n</sup>, and this is again subtracting a multiple of 7. The same reason applies for all the remaining conversions:</div>
<ul style="list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;">20×10<sup style="line-height: 1em;"><i>n</i></sup> − 6×10<sup style="line-height: 1em;"><i>n</i></sup>=<b>14</b>×10<sup style="line-height: 1em;"><i>n</i></sup></li>
<li style="margin-bottom: 0.1em;">60×10<sup style="line-height: 1em;"><i>n</i></sup> − 4×10<sup style="line-height: 1em;"><i>n</i></sup>=<b>56</b>×10<sup style="line-height: 1em;"><i>n</i></sup></li>
<li style="margin-bottom: 0.1em;">40×10<sup style="line-height: 1em;"><i>n</i></sup> − 5×10<sup style="line-height: 1em;"><i>n</i></sup>=<b>35</b>×10<sup style="line-height: 1em;"><i>n</i></sup></li>
<li style="margin-bottom: 0.1em;">50×10<sup style="line-height: 1em;"><i>n</i></sup> − 1×10<sup style="line-height: 1em;"><i>n</i></sup>=<b>49</b>×10<sup style="line-height: 1em;"><i>n</i></sup></li>
</ul>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
<b>First method example</b><br />1050 → 105 − 0=105 → 10 − 10 = 0. ANSWER: 1050 is divisible by 7.</div>
<div style="margin-bottom: 0.5em; margin-top: 0.4em;">
<b>Second method example</b><br />1050 → 0501 (reverse) → 0×<b>1</b> + 5×<b>3</b> + 0×<b>2</b> + 1×<b>6</b> = 0 + 15 + 0 + 6 = 21 (multiply and add). ANSWER: 1050 is divisible by 7.</div>
</div>
</div>
Anonymoushttp://www.blogger.com/profile/12500986417656529636noreply@blogger.com0Naraina, New Delhi, Delhi, India28.6302834 77.139887828.6163464 77.1201468 28.6442204 77.1596288tag:blogger.com,1999:blog-7072204672677043641.post-31629333609098239582012-11-30T22:12:00.000-08:002017-08-05T05:08:43.969-07:00BASIC MENSURATION FORMULA <div dir="ltr" style="text-align: left;" trbidi="on">
<br />
<br />
<div>
<u><b>UNIT</b>-</u></div>
<div>
<b> </b>You should be familiar with the following units:</div>
<ol style="text-align: left;">
<li> Length: mm, cm, m, km</li>
<li> Area: mm2, cm2, m2, ha, km2</li>
<li> Volume: mm3, cm3, m3</li>
<li> Capacity: ml, cl, l</li>
<li> Mass: g, kg</li>
</ol>
<ul style="text-align: left;">
<li> To convert from smaller to larger units we divide by the conversion factor.</li>
<li> To convert from larger to smaller units we multiply by the conversion factor</li>
</ul>
<div>
<b><u>LENGTH-</u></b></div>
<div>
<b><u><br /></u></b></div>
<div>
<div>
The perimeter of a figure is the measurement of the distance</div>
<div>
around its boundary.</div>
<div>
For a polygon the perimeter is the sum of the lengths of all</div>
<div>
sides</div>
</div>
<div>
<br /></div>
<div>
UNIT-METER</div>
<div>
<b><u>AREA-</u></b></div>
<div>
The area of a figure is the amount of surface within its</div>
<div>
boundaries.</div>
<div>
You should be able to use these formulae for area</div>
<div>
<br /></div>
<div>
Rectangles- Area = (length * width)</div>
<div>
<br /></div>
<div>
Triangles-Area = 1/2 (base * height)</div>
<div>
Parallelograms-Area = base * height</div>
<div>
Trapezia-Area = 1/2 (a + b) * h</div>
<div>
<br /></div>
<div>
<b><u>VOLUME-</u></b></div>
<div>
<b> </b>The volume of a solid is the amount of space it occupies.You should be able to use these formulae for volume:Solids of uniform cross-section--</div>
<div>
<div>
<br /></div>
<div>
Volume of uniform solid = area of end * height</div>
<div>
<br /></div>
</div>
<div>
Pyramids and cones= 1/3 (area of base * height)</div>
<div>
<div>
Volume of a sphere = 4/3(pi)r3</div>
</div>
<div>
<b><u><br /></u></b></div>
<div>
<b><u>SURFACE AREA</u></b></div>
<div>
<b><u><br /></u></b></div>
<div>
<div>
<b>Solids with plane faces</b></div>
<div>
The surface area of a three dimensional figure with plane facesis the sum of the areas of the faces.</div>
<div>
To assist in your calculations, you can draw a net of the solid,</div>
<div>
correctly labelling the dimensions.</div>
<div>
<b>Solids with curved surfaces</b></div>
<div>
You should be able to use these formulae for surface area:</div>
</div>
<br />
<br />
<br /></div>
Anonymoushttp://www.blogger.com/profile/12500986417656529636noreply@blogger.comNaraina, New Delhi, Delhi, India28.6302834 77.139887828.6163464 77.1201468 28.6442204 77.1596288tag:blogger.com,1999:blog-7072204672677043641.post-91707220337540407482012-11-30T21:18:00.001-08:002012-11-30T22:58:38.843-08:00BASIC MATHS FORMULA..<div dir="ltr" style="text-align: left;" trbidi="on">
<b> ALGEBRA FORMULA</b><br />
<b><br /></b>
<b><br /></b>
<b></b><br />
<b>1. (a + b)2 = a2 + 2ab + b2; a2 + b2 = (a+b)2 −2ab</b><br />
<b>2. (a − b)2 = a2 − 2ab + b2; a2 + b2 = (a−b)2 + 2ab</b><br />
<b>3. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)</b><br />
<b>4. (a + b)3 = a3 + b3 + 3ab(a + b); a3 + b3 = (a+b)3 −3ab(a + b)</b><br />
<b>5. (a − b)3 = a3 − b3 − 3ab(a − b); a3 − b3 = (a−b)3 + 3ab(a − b)</b><br />
<b>6. a2 − b2 = (a+b)(a − b)</b><br />
<b>7. a3 − b3 = (a−b)(a2 + ab + b2)</b><br />
<b>8. a3 + b3 = (a+b)(a2 − ab + b2)</b><br />
<b>9. an − bn = (a−b)(an−1 + an−2b + an−3b2 + +bn−1)</b><br />
<b>10. an = a:a:a : : : n times</b><br />
<br /></div>
Anonymoushttp://www.blogger.com/profile/12500986417656529636noreply@blogger.comNaraina, New Delhi, Delhi, India28.6302834 77.139887828.6163464 77.1201468 28.6442204 77.1596288