Complex Trig Formulae

Level 1 - Math I (Physics) topic page in Complex Form.

Complex Trigonometric Formulae

Connections between complex exponentials and trig functions:

Euler's

\[e^{i\theta} = \cos\theta + i\sin\theta\]

Cosine from Exponential

\[\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}\]

Sine from Exponential

\[\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}\]

Hyperbolic Cosine

\[\cosh x = \frac{e^x + e^{-x}}{2}\]

Hyperbolic Sine

\[\sinh x = \frac{e^x - e^{-x}}{2}\]

Identity 1

\[\cos^2\theta + \sin^2\theta = 1\]

Identity 2

\[\cosh^2 x - \sinh^2 x = 1\]