Trigonometric

← Functions

Definitions

• $$\begin{array}{rl}\sin z& =\frac{\exp (iz)-\exp (-iz)}{2i}\\ \cos z& =\frac{\exp (iz)+\exp (-iz)}{2i}\\ \tan z& =\frac{\sin z}{\cos z}\\ \cot z& =\frac{\cos z}{\sin z}\\ \sec z& =\frac{1}{\cos z}\\ \csc z& =\frac{1}{\sin z}\end{array}$$

Fundamental Properties

$$\begin{array}{rl}\sin (z+2\pi )& =\sin z\\ \cos (z+2\pi )& =\cos z\\ \tan (z+\pi )& =\tan z\end{array}$$

Identities

For complex numbers $a,b,z$ the following holds: $$\begin{array}{rl}{(\sin z)}^2+{(\cos z)}^2& =1\end{array}$$ $$\begin{array}{rl}\sin (a+b)& =\sin a\cos b+\cos a\sin b\\ \cos (a+b)& =\cos a\cos b-\cos a\cos b\\ \sin a+\sin b& =2\sin \left(\frac{a+b}{2}\right)\cos \left(\frac{a-b}{2}\right)\\ \cos a+\cos b& =2\cos \left(\frac{a+b}{2}\right)\cos \left(\frac{a-b}{2}\right)\end{array}$$