Trig Functions Trig Functions
Basics
\begin{align*}
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r} \\
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r} \\
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x} \\
\end{align*}
Basics (Inverse)
\begin{align*}
\sin^{-1}\left(\frac{y}{r}\right) = \theta \\
\cos^{-1}\left(\frac{x}{r}\right) = \theta \\
\tan^{-1}\left(\frac{y}{x}\right) = \theta \\
\end{align*}
Reciprocals
\begin{align*}
\csc(x) = \frac{1}{\sin(x)} \\
\sec(x) = \frac{1}{\cos(x)} \\
\cot(x) = \frac{1}{\tan(x)} \\
\end{align*}
Pythagorean Identities
\begin{align*}
\sin^2(x) + \cos^2(x) &= 1 \\
1 + \tan^2(x) &= \sec^2(x) \\
1 + \cot^2(x) &= \csc^2(x) \\
\end{align*}
Angle Addition & Subtraction
\begin{align*}
\sin(a + b) &= \sin(a)\cos(b) + \cos(a)\sin(b) \\
\cos(a + b) &= \cos(a)\cos(b) - \sin(a)\sin(b) \\
\tan(a + b) &= \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \\
\end{align*}
Double & Half Angle Formulas
\begin{align*}
\sin(2x) &= 2\sin(x)\cos(x) \\
\cos(2x) &= \cos^2(x) - \sin^2(x) \\
&= 2\cos^2(x) - 1 \\
&= 1 - 2\sin^2(x) \\
\tan(2x) &= \frac{2\tan(x)}{1 - \tan^2(x)} \\
\\
\sin\left(\frac{x}{2}\right) &= \pm\sqrt{\frac{1 - \cos(x)}{2}} \\
\cos\left(\frac{x}{2}\right) &= \pm\sqrt{\frac{1 + \cos(x)}{2}} \\
\tan\left(\frac{x}{2}\right) &= \frac{\sin(x)}{1 + \cos(x)} = \frac{1 - \cos(x)}{\sin(x)}
\end{align*}
Power-Reduction Formulas
\begin{align*}
\sin^2(x) &= \frac{1 - \cos(2x)}{2} \\
\cos^2(x) &= \frac{1 + \cos(2x)}{2} \\
\tan^2(x) &= \frac{1 - \cos(2x)}{1 + \cos(2x)} \\
\end{align*}
Sum-to-Product & Product-to-Sum
\begin{align*}
\sin(a) + \sin(b) &= 2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right) \\
\sin(a) - \sin(b) &= 2\cos\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right) \\
\cos(a) + \cos(b) &= 2\cos\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right) \\
\cos(a) - \cos(b) &= -2\sin\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right) \\
\\
\sin(a)\cos(b) &= \frac{1}{2}[\sin(a+b) + \sin(a-b)] \\
\cos(a)\cos(b) &= \frac{1}{2}[\cos(a+b) + \cos(a-b)] \\
\sin(a)\sin(b) &= \frac{1}{2}[\cos(a-b) - \cos(a+b)] \\
\end{align*}
Co-function Identities
\begin{align*}
\sin\left(\frac{\pi}{2} - x\right) &= \cos(x) \\
\cos\left(\frac{\pi}{2} - x\right) &= \sin(x) \\
\tan\left(\frac{\pi}{2} - x\right) &= \cot(x) \\
\end{align*}
Even-Odd Properties
\begin{align*}
\sin(-x) &= -\sin(x) \\
\cos(-x) &= \cos(x) \\
\tan(-x) &= -\tan(x) \\
\end{align*}
Periodicity
\begin{align*}
\sin(x + 2\pi) &= \sin(x) \\
\cos(x + 2\pi) &= \cos(x) \\
\tan(x + \pi) &= \tan(x) \\
\end{align*}