List of trigonometric identities

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.
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.

Useful trig formulas  for learning trigonometric concepts.



Triangle ABC is any triangle with side lengths a,b,c
Law of Cosines

Law of Sines

 


Pythagorean identity

The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity:

    \cos^2\theta + \sin^2\theta = 1\!
where cos2 θ means (cos(θ))2 and sin2 θ means (sin(θ))2.

 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:

 \sin\theta = \pm \sqrt{1-\cos^2\theta} \quad \text{and} \quad \cos\theta = \pm \sqrt{1 - \sin^2\theta}. \,


Related identities

Dividing the Pythagorean identity through by either cos2 θ or sin2 θ yields two other identities:

1 + \tan^2\theta = \sec^2\theta\quad\text{and}\quad 1 + \cot^2\theta = \csc^2\theta.\!


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):

in terms of \sin \theta\! \cos \theta\! \tan \theta\! \csc \theta\! \sec \theta\! \cot \theta\!
\sin \theta =\! \sin \theta\ \pm\sqrt{1 - \cos^2 \theta}\! \pm\frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}}\! \frac{1}{\csc \theta}\! \pm\frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta}\! \pm\frac{1}{\sqrt{1 + \cot^2 \theta}}\!
\cos \theta =\!\pm\sqrt{1 - \sin^2\theta}\! \cos \theta\! \pm\frac{1}{\sqrt{1 + \tan^2 \theta}}\! \pm\frac{\sqrt{\csc^2 \theta - 1}}{\csc \theta}\! \frac{1}{\sec \theta}\! \pm\frac{\cot \theta}{\sqrt{1 + \cot^2 \theta}}\!
\tan \theta =\!\pm\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}\! \pm\frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta}\! \tan \theta\! \pm\frac{1}{\sqrt{\csc^2 \theta - 1}}\! \pm\sqrt{\sec^2 \theta - 1}\! \frac{1}{\cot \theta}\!
\csc \theta =\! \frac{1}{\sin \theta}\! \pm\frac{1}{\sqrt{1 - \cos^2 \theta}}\! \pm\frac{\sqrt{1 + \tan^2 \theta}}{\tan \theta}\! \csc \theta\! \pm\frac{\sec \theta}{\sqrt{\sec^2 \theta - 1}}\! \pm\sqrt{1 + \cot^2 \theta}\!
\sec \theta =\!\pm\frac{1}{\sqrt{1 - \sin^2 \theta}}\! \frac{1}{\cos \theta}\! \pm\sqrt{1 + \tan^2 \theta}\! \pm\frac{\csc \theta}{\sqrt{\csc^2 \theta - 1}}\! \sec \theta\! \pm\frac{\sqrt{1 + \cot^2 \theta}}{\cot \theta}\!
\cot \theta =\!\pm\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}\! \pm\frac{\cos \theta}{\sqrt{1 - \cos^2 \theta}}\! \frac{1}{\tan \theta}\! \pm\sqrt{\csc^2 \theta - 1}\! \pm\frac{1}{\sqrt{\sec^2 \theta - 1}}\! \cot \theta\!

Angel

1 full circle  = 360 degrees = 2\pi radians  =  400 grads.
The following table shows the conversions for some common angles:

Degrees30°60°120°150°210°240°300°330°
Radians\frac\pi6\!\frac\pi3\!\frac{2\pi}3\!\frac{5\pi}6\!\frac{7\pi}6\!\frac{4\pi}3\!\frac{5\pi}3\!\frac{11\pi}6\!
Grads33⅓ grad66⅔ grad133⅓ grad166⅔ grad233⅓ grad266⅔ grad333⅓ grad366⅔ grad
Degrees45°90°135°180°225°270°315°360°
Radians\frac\pi4\!\frac\pi2\!\frac{3\pi}4\!\pi\!\frac{5\pi}4\!\frac{3\pi}2\!\frac{7\pi}4\!2\pi\!
Grads50 grad100 grad150 grad200 grad250 grad300 grad350 grad400 grad

Trigonometric functions


The primary trigonometric functions are the sine and cosine of an angle. These are sometimes abbreviated sin(θ) and cos(θ), respectively, where θ is the angle, but the parentheses around the angle are often omitted, e.g., sin θ and cos θ.
The tangent (tan) of an angle is the ratio of the sine to the cosine:
\tan\theta = \frac{\sin\theta}{\cos\theta}.
Finally, the reciprocal functions secant (sec), cosecant (csc), and cotangent (cot) are the reciprocals of the cosine, sine, and tangent:
\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}.

Inverse functions


Main article: Inverse trigonometric functions
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

\sin(\arcsin x) = x\quad\text{for} \quad |x| \leq 1

and

\arcsin(\sin x) = x\quad\text{for} \quad |x| \leq \pi/2.

This article uses the notation below for inverse trigonometric functions:

Functionsincostanseccsccot
Inversearcsinarccosarctanarcsecarccscarccot

Symmetry, shifts, and periodicity


Symmetry-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:

Reflected in \theta=0 [Reflected in \theta= \pi/2
(co-function identities)[		Reflected in \theta= \pi
\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} \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} \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}

Shifts and periodicity--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.

Shift by π/2Shift by π
Period for tan and cot		Shift by 2π
Period for sin, cos, csc and sec[
\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} \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} \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}

Angle sum and difference identities-These are also known as the addition and subtraction theorems or formulæ. They were originally established by the 10th century Persian mathematician Abū al-Wafā' Būzjānī. One method of proving these identities is to apply Euler's formula. The use of the symbols \pm and \mp is described in the article plus-minus sign.


Sine\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \!
Cosine\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta\,
Tangent\tan(\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}
Arcsine\arcsin\alpha \pm \arcsin\beta = \arcsin\left(\alpha\sqrt{1-\beta^2} \pm \beta\sqrt{1-\alpha^2}\right)
Arccosine\arccos\alpha \pm \arccos\beta = \arccos\left(\alpha\beta \mp \sqrt{(1-\alpha^2)(1-\beta^2)}\right)
Arctangent\arctan\alpha \pm \arctan\beta = \arctan\left(\frac{\alpha \pm \beta}{1 \mp \alpha\beta}\right)

Matrix form-The sum and difference formulae for sine and cosine can be written in matrix form as:


\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}
This shows that these matrices form a representation of the rotation group in the plane (technically, the special orthogonal group SO(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.

Reciprocal identities
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Pythagorean Identities
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Quotient Identities




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Co-Function Identities




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Even-Odd Identities




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Sum-Difference Formulas


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Double Angle Formulas

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Power-Reducing/Half Angle Formulas
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Sum-to-Product Formulas
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Product-to-Sum Formulas



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Addition Formulas


cos(X + Y) = cosX cosY - sinX sinY

cos(X - Y) = cosX cosY + sinX sinY

sin(X + Y) = sinX cosY + cosX sinY

sin(X - Y) = sinX cosY - cosX sinY

tan(X + Y) = [ tanX + tanY ] / [ 1 - tanX tanY]

tan(X - Y) = [ tanX - tanY ] / [ 1 + tanX tanY] 

cot(X + Y) = [ cotX cotY - 1 ] / [ cotX + cotY]

cot(X - Y) = [ cotX cotY + 1 ] / [ cotX - cotY]


Difference of Squares Formulas

sin 2X - sin 2Y = sin(X + Y)sin(X - Y) 
cos 2X - cos 2Y = - sin(X + Y)sin(X - Y) 
cos 2X - sin 2Y = cos(X + Y)cos(X - Y)


Location: Naraina, New Delhi, Delhi, India

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