Identities

$\sin \theta =\cos \left(\frac{\pi }{2}-\theta \right)$	$\cos \theta =\sin \left(\frac{\pi }{2}-\theta \right)$	$\tan \theta =\frac{\sin \theta }{\cos \theta }$
$\csc \theta =\frac{1}{\sin \theta }$	$\sec \theta =\frac{1}{\cos \theta }$	$\cot \theta =\frac{\cos \theta }{\sin \theta }$

Pythagorean identities

$$\begin{array}{r}{\sin }^2\alpha +{\cos }^2\alpha =1\\ 1+{\tan }^2\alpha ={\sec }^2\alpha \\ 1+{\cot }^2\alpha ={\csc }^2\alpha \end{array}$$

Angle sum and difference identities

\sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ \cos(\alpha \pm \beta) &= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \\ \tan(\alpha\pm \beta) &=\frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta} \end{align}$$ ### Law of sines $$\frac{a}{\sin{\alpha}} \,=\, \frac{b}{\sin{\beta}} \,=\, \frac{c}{\sin{\gamma}} \,=\, 2R$$ ### Law of cosines $$a^2=b^2+c^2-2bc\cos\alpha$$ ### Even/Odd Identities

\begin{align} \sin(-\alpha)&=-\sin(\alpha) \ \cos(-\alpha)&=\cos(\alpha) \ \tan(-\alpha)&=-\tan(\alpha) \ \end{align}

### Power-Reducing Formulas

\begin{align} \sin^2\alpha=\frac{1-\cos{2}\alpha}{2} \ \cos^2\alpha=\frac{1+\cos{2}\alpha}{2} \ \tan^2\alpha=\frac{1-\cos{2}\alpha}{1+\cos{2}\alpha} \ \end{align}

- Multiple-angle formula - Double-angle formula - $\sin(2\alpha)=2\sin\alpha\cos\alpha=(\sin \alpha+\cos \alpha)^2-1$ - $\cos(2\alpha)=\cos^2\alpha-\sin^2\alpha-1=1-2\sin^2\alpha$ - Triple-angle formula #todo - Half-angle formula #todo - Product-to-sum and sum-to-product identities #todo ## Common Trigonometric Values  ___  ___ | | $\sin$ | $\cos$ | | ---- | --------------------- | ---------------------------------------------- | | $1$ | $2k\pi+\frac{\pi}{2}$ | $2k\pi$ | | $0$ | $k\pi$ | $(2k\pm 1)\frac{\pi}{2}=\pi k\pm\frac{\pi}{2}$ | | $-1$ | $2k\pi-\frac{\pi}{2}$ | $(2k + 1)\pi$ | ### Inverse trigonometric functions - $\arcsin(x),~ \arccos(x), ~\arctan(x)$