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.

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

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