Table of Contents

Trigonometric Formulas and Identities

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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

\( f(x) \)	\( \dfrac{d f(x)}{dx} \)
\( \sin x \)	\( \cos x \)
\( \cos x \)	\( - \sin x \)
\( \tan x \)	\( \sec^2 x \)
\( \cot x \)	\( - \csc^2 x \)
\( \sec x \)	\( \sec x \tan x \)
\( \csc x \)	\( - \csc x \cot x\)
\( \sin^{-1} x\)	\( \dfrac{1}{\sqrt{1-x^2}} \)
\( \cos^{-1} x\)	\( - \dfrac{1}{\sqrt{1-x^2}} \)
\( \tan^{-1} x\)	\( \dfrac{1}{1+x^2} \)

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.

\( f(x) \)	\( \displaystyle \int f(x) dx \)
\( \sin x \)	\( -\cos x \)
\( \cos x \)	\( \sin x \)
\( \tan x \)	\( -\ln \|\cos x\| \)
\( \cot x \)	\( \ln \|\sin x\| \)
\( \sec x \)	\( \ln( \sec x + \tan x ) \)
\( \csc x \)	\( \ln(\csc x - \cot x) \)

More References and Links

Handbook of Mathematical Functions Engineering Mathematics with Examples and Solutions