< Up
\( \newcommand{\dif}{\,\textrm{d}} \newcommand{\abs}[1]{\left\lvert #1 \right\rvert} \newcommand{\norm}[1]{\left\lVert #1 \right\rVert} \)

Trigonometry

A list of common formulas, not in any way complete.

Most, if not all, trigonometric problems become trivial using following substitutions. \[ \begin{align} e^{i \theta} &= \cos(\theta) + i \sin(\theta) \tag{Euler's formula} \\ \sin(\theta) &= \frac{e^{i \theta} - e^{-i \theta}}{2i} \\ \cos(\theta) &= \frac{e^{i \theta} + e^{-i \theta}}{2} \\ \tan(\theta) &= \frac{i(e^{-i \theta} - e^{i \theta})}{e^{i \theta} + e^{-i \theta}} \\ \end{align} \]