Trigonometric Functions


Collection of functions and identities to help my memory in a few courses I’m taking.

Derivatives of Trigonometric Functions

\[ \frac{d}{dx}\sin\left(x\right) = \cos\left(x\right) \]

\[ \frac{d}{dx}\cos\left(x\right) = -\sin\left(x\right) \]

\[ \frac{d}{dx}\tan\left(x\right) = \sec^2\left(x\right) \]

\[ \frac{d}{dx}\cot\left(x\right) = -\csc^2\left(x\right) \]

\[ \frac{d}{dx}\sec\left(x\right) = \sec\left(x\right)\tan\left(x\right) \]

\[ \frac{d}{dx}\csc\left(x\right) = -\csc\left(x\right)\cot\left(x\right) \]

Derivatives of Inverse Trigonometric Functions

\[ \frac{d}{dy}\arcsin\left(y\right) = \frac{1}{\sqrt{1-y^2}} \]

\[ \frac{d}{dy}\arccos\left(y\right) = \frac{-1}{\sqrt{1-y^2}} \]

\[ \frac{d}{dy}\arctan\left(y\right) = \frac{1}{1+y^2} \]

Pythagorean Identity

\[ \sin^2\left(x\right) + \cos^2\left(x\right) = 1 \]

Angle Sum and Difference Identities

\[ \sin\left(A+B\right) = \sin\left(A\right)\cos\left(B\right) + \cos\left(A\right)\sin\left(B\right) \]

\[ \sin\left(A-B\right) = \sin\left(A\right)\cos\left(B\right) - \cos\left(A\right)\sin\left(B\right) \]

\[ \cos\left(A+B\right) = \cos\left(A\right)\cos\left(B\right) - \sin\left(A\right)\sin\left(B\right) \]

\[ \cos\left(A-B\right) = \cos\left(A\right)\cos\left(B\right) + \sin\left(A\right)\sin\left(B\right) \]

\[ \tan\left(A+B\right) = \frac{\tan\left(A\right) + \tan\left(B\right)}{1 - \tan\left(A\right)\tan\left(B\right)} \]

\[ \tan\left(A-B\right) = \frac{\tan\left(A\right) - \tan\left(B\right)}{1 + \tan\left(A\right)\tan\left(B\right)} \]

Double Angle Formulas

\[ \sin\left(2x\right) = 2 \sin\left(x\right)\cos\left(x\right) \]

\[ \tag{2} \tan\left(2x\right) = \frac{2 \tan\left(x\right)}{1 - \tan^2\left(x\right)} \]