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# System prepended metadata

title: trig

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## Double Angle Forumlas
$\sin 2x=2\sin x \cos x$
$\cos 2x=\cos^2x-sin^2x=1-2\sin^2x=2\cos^2x-1=\frac{1-\tan^2x}{1+\tan^2x}$
$\tan 2x=\frac{2\tan x}{1-\tan^2x}$
## Addition and Subtraction Formulas
$\sin(\alpha + \beta)=\sin \alpha \cos \beta + \cos \alpha \sin \beta$
$\sin(\alpha - \beta)=\sin \alpha \cos \beta - \cos \alpha \sin \beta$
$\cos(\alpha + \beta)=\cos \alpha \cos \beta - \sin \alpha \sin \beta$
$\cos(\alpha - \beta)=\cos \alpha \cos \beta + \sin \alpha \sin \beta$
$\tan(\alpha + \beta)=\frac{\tan \alpha + \tan \beta}{1-\tan \alpha \tan \beta}$
$\tan(\alpha - \beta)=\frac{\tan \alpha - \tan \beta}{1+\tan \alpha \tan \beta}$
## Triple Angle Formulas
$\sin 3x=3\sin x-4\sin^3 x$
$\cos 3x=\cos^3x-3\cos x$
## Forumlas for Lowering Powers
$\sin^2x = \frac{1-\cos 2x}{2}$
$\cos^2x = \frac{1+\cos 2x}{2}$
$\tan^2x = \frac{1-\cos 2x}{1+\cos 2x}$
## Half-Angle Forumlas
$\sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}}$
$\cos \frac{x}{2}=\pm \sqrt{\frac{1+\cos x}{2}}$
$\tan \frac{x}{2}=\frac{1-\cos x}{\sin x}=\frac{\sin x}{1+\cos x}$
## Product-to-Sum Formulas
$\sin(x+y)+\sin(x-y)=2\sin x\cos y$
$\sin(x+y)-\sin(x-y)=2\cos x\sin y$
$\cos(x+y)+\cos(x-y)=2\cos x\cos y$
$\cos(x+y)-\cos(x-y)=-2\sin x\sin y$
## Sum-to-Product Forumlas
$\sin x + \sin y = 2 \sin \frac{x+y}{2} \cos \frac{x-y}{2}$
$\sin x - \sin y = 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2}$
$\cos x + \cos y = 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2}$
$\cos x - \cos y = -2 \sin \frac{x+y}{2} \sin \frac{x-y}{2}$