This is a memo. To prevent me from forgetting some trigonometric formulas.

Basic Formula#

Unit circle

Pythagorean Theotheorm

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

Rotate matrix (counterclockwise rotation)

\left [ \begin{matrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{matrix} \right ] \times \left [ \begin{matrix} x \\ y \end{matrix} \right ]

Sum Formula#

\begin{aligned} & \sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta\\ & \cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta\\ & \tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1-\tan\alpha\tan\beta} \end{aligned}

Double-angle

For sin.

\sin 2\theta = 2\sin\theta\cos\theta

For cos.

\begin{aligned} \cos 2\theta &= \cos^2\theta - \sin^2\theta \\ &= 1 - 2\sin^2\theta\\ &= 2\cos^2\theta -1 \end{aligned}

For tan.

\tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta}

Half-angle

For sin & cos.

\begin{aligned} \sin \frac{\theta}{2} &= \pm \sqrt{\frac{1-\cos\theta}{2}} \\ \cos \frac{\theta}{2} &= \pm \sqrt{\frac{1+\cos\theta}{2}} \end{aligned}

For tan.

\begin{aligned} \tan \frac{\theta}{2} &= \frac{\sin\theta}{1 + \cos\theta}\\ &= \frac{1 - \cos\theta}{\sin\theta}\\ &= \pm \sqrt{\frac{1-\cos\theta}{1+\cos\theta}} \end{aligned}

The sign is depend on which quardrant $\theta/2$ is in.

Also, there are some other trigonometric identities but they all can be prove by above formula so it’s all for this memo.