# Trigonometric Formulas and Identities



## Trigonometric Functions of Sums/Difference of Angles

• $\cos(A \pm B) = \cos A \cos B \mp \sin A \cos B$
• $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
• $\tan(A \pm B) = \dfrac{\tan A \pm tan B}{1 \mp \tan A \tan B}$

## Sums of Trigonometric Functions to Product of Trigonometric Functions

• $\sin A + \sin B = 2 \sin \left( \dfrac {A+B}{2} \right) \cos \left( \dfrac {A-B}{2} \right)$
• $\sin A - \sin B = 2 \sin \left( \dfrac {A - B}{2} \right) \cos \left( \dfrac {A + B}{2} \right)$
• $\cos A + \cos B = 2 \cos \left( \dfrac {A+B}{2} \right) \cos \left( \dfrac {A-B}{2} \right)$
• $\cos A - \cos B = - 2 \sin \left( \dfrac {A+B}{2} \right) \sin \left( \dfrac {A-B}{2} \right)$

## Product of Trigonometric Functions to Sum of Trigonometric Functions

• $\sin A \cos B = \dfrac{1}{2} \; [ \sin(A+B) + \sin(A-B) ]$
• $\cos A \cos B = \dfrac{1}{2} \; [ \cos(A+B) + \cos(A-B) ]$
• $\sin A \sin B = - \dfrac{1}{2} \; [ \cos(A+B) - \cos(A-B) ]$

## Trigonometric Functions of Multiple Angles

• $\sin (2 A) = 2 \sin A \cos A$
• $\cos (2 A) = 1 - 2 \sin^2 A = 2 \cos^2 A -1$
• $\sin (3 A) = 3 \sin A - 4 \sin^3 A$
• $\cos (3 A) = 4\cos^3 A - 3 \cos A$

## Power Reducing Identities

• $\sin^2 A = \dfrac{1}{2} [1 - \cos (2A)]$
• $\cos^2 A = \dfrac{1}{2} [1 + \cos (2A)]$

## Half Angle Formula

• $\sin (A/2) = \pm \sqrt {\dfrac{1- \cos A}{2}}$
• $\cos (A/2) = \pm \sqrt {\dfrac{1 + \cos A}{2}}$
• $\tan (A/2) = \dfrac{1 - \cos A}{\sin A} = \dfrac{\sin A}{1+\cos A}$

## Derivatives of Trigonometric Functions and Their Inverses

### $\dfrac{d f(x)}{dx}$

$\sin x$ $\cos x$
$\cos x$ $- \sin x$
$\tan x$ $\sec^2 x$
$\cot x$ $- \csc^2 x$
$\sec x$ $\sec x \tan x$
$\csc x$ $- \csc x \cot x$
$\sin^{-1} x$ $\dfrac{1}{\sqrt{1-x^2}}$
$\cos^{-1} x$ $- \dfrac{1}{\sqrt{1-x^2}}$
$\tan^{-1} x$ $\dfrac{1}{1+x^2}$

## Indefinite Integrals of Trigonometric Functions

Note that in all cases of indefinite integrals, the constant of integration is omitted here but should be added whenever necessary.

### $\displaystyle \int f(x) dx$

$\sin x$ $-\cos x$
$\cos x$ $\sin x$
$\tan x$ $-\ln |\cos x|$
$\cot x$ $\ln |\sin x|$
$\sec x$ $\ln( \sec x + \tan x )$
$\csc x$ $\ln(\csc x - \cot x)$

## More References and Links

Handbook of Mathematical Functions Engineering Mathematics with Examples and Solutions