Trigonometric Cheat Sheet: Essential Formulas & Identities

Sin$(\theta) = \dfrac{y}{1} = y \space \space \space \space$ Csc$(\theta) = \dfrac{1}{y}$

Cos$(\theta) = \dfrac{x}{1} = x \space \space \space$ Sec$(\theta) = \dfrac{1}{x}$

tan$(\theta) = \dfrac{y}{x} \space \space \space \space \space \space \space \space \space \space \space \space \space$Cot$(\theta) = \dfrac{x}{y}$

2. Facts and Properties:

Domain

The domain is all the values of $\theta$ that can be plugged into the function.

$\sin(\theta)$, $\theta$ can be any angle

$\cos(\theta)$, $\theta$ can be any angle

$\tan(\theta)$, $\theta \neq \left(n + \dfrac{1}{2}\right)\pi$, $n = 0, \pm 1, \pm 2, \dots$

$\csc(\theta)$, $\theta \neq n\pi$, $n = 0, \pm 1, \pm 2, \dots$

$\sec(\theta)$, $\theta \neq \left(n + \dfrac{1}{2}\right)\pi$, $n = 0, \pm 1, \pm 2, \dots$

$\cot(\theta)$, $\theta \neq n\pi$, $n = 0, \pm 1, \pm 2, \dots$

Period

The period of a function is the number, $T$, such that $f(\theta + T) = f(\theta)$.
So, if $\omega$ is a fixed number and $\theta$ is any angle we have the following periods.

$sin(\omega \theta) \rightarrow T = \dfrac{2\pi}{\omega}$

$cos(\omega \theta) \rightarrow T = \dfrac{2\pi}{\omega}$

$tan(\omega \theta) \rightarrow T = \dfrac{\pi}{\omega}$

$csc(\omega \theta) \rightarrow T = \dfrac{2\pi}{\omega}$

$sec(\omega \theta) \rightarrow T = \dfrac{2\pi}{\omega}$

$cot(\omega \theta) \rightarrow T = \dfrac{\pi}{\omega}$

Range

The range is all possible values to get out of the function.

$\space \space -1 \leq \sin(\theta) \leq 1 \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space -1 \leq \cos(\theta) \leq 1$

$\space \space -\infty < \tan(\theta) < \infty \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space -\infty < \cot(\theta) < \infty$

$\space \space \space \sec(\theta) \geq 1$ & $\sec(\theta) \leq -1 \space \space \space \space \space \space \csc(\theta) \geq 1$ & $\csc(\theta) \leq -1$

3. Formulas and Identities:

Basic Identities

$\tan(\theta) = \dfrac{\sin(\theta)}{\cos(\theta)} \space \space \space \space \space \space \space \cot(\theta) = \dfrac{\cos(\theta)}{\sin(\theta)}$

$\csc(\theta) = \dfrac{1}{\sin(\theta)} \space \ \ \ \space \space \space \space \sin(\theta) = \dfrac{1}{\csc(\theta)}$

$\sec(\theta) = \dfrac{1}{\cos(\theta)} \space \space \space \space \ \ \ \cos(\theta) = \dfrac{1}{\sec(\theta)}$

Pythagorean Identities

$\sin^2(\theta) + \cos^2(\theta) = 1$

$\tan^2(\theta) + 1 = \sec^2(\theta)$

$1 + \cot^2(\theta) = \csc^2(\theta)$

Even/Odd Formulas

$\sin(-\theta) = - \sin(\theta) \space \space \ \ \ \csc(-\theta) = - \csc(\theta)$
$\cos(-\theta) = \cos(\theta) \space \space \space \space \space \ \ \ \sec(-\theta) = \sec(\theta)$
$\tan(-\theta) = -\tan(\theta) \space \ \ \ \cot(-\theta) = -\cot(\theta)$

Periodic Formulas

If $n$ is an integer, then,

$\sin(\theta + 2\pi n) = \sin(\theta) \space \space \space \space \space \space \space \csc(\theta + 2\pi n) = \csc(\theta)$
$\cos(\theta + 2\pi n) = \cos(\theta) \space \space \space \ \ \ \sec(\theta + 2\pi n) = \sec(\theta)$
$\tan(\theta + \pi n) = \tan(\theta) \space \space \space \space \space \space \space \cot(\theta + \pi n) = \cot(\theta)$

Degrees to Radians Formulas

Degree to radian Calculator
If $x$ is an angle in degrees and $t$ is an angle in radians, then

$\dfrac{\pi}{180} = \dfrac{t}{x} \Rightarrow t = \dfrac{\pi x}{180}$ and $x = \dfrac{180t}{\pi}$

Double Angle Formulas

Cosine Calculator
$\sin(2\theta) = 2 \sin(\theta) \cos(\theta) = \dfrac{2tan(\theta)}{1+\tan^2(\theta)}$

$\cos(2\theta)$ $= \begin{cases} \cos^2(\theta) - \sin^2(\theta) \\ 2 \cos^2(\theta) - 1 \\ 1 - 2 \sin^2(\theta) \\ \dfrac{1-\tan^2(\theta)}{1+\tan^2(\theta)} \end{cases}$

$\tan(2\theta) = \dfrac{2 \tan(\theta)}{1 - \tan^2(\theta)}$

Triple Angle Formulas

$\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)$

$\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$

$\tan(3\theta) = \dfrac{3\tan(\theta) - \tan^3(\theta)}{1-3\tan^2(\theta)}$

Half Angle Formulas

$\sin \left(\dfrac{\theta}{2}\right) = \pm \sqrt{\dfrac{1 - \cos(\theta)}{2}}$

$\cos \left(\dfrac{\theta}{2}\right) = \pm \sqrt{\dfrac{1 + \cos(\theta)}{2}}$

$\tan \left(\dfrac{\theta}{2}\right) = \pm \sqrt{\dfrac{1 - \cos(\theta)}{1 + \cos(\theta)}}$

Half Angle Formulas (alternate form)

$\sin^2(\theta) = \dfrac{1 - \cos(2\theta)}{2}$

$\cos^2(\theta) = \dfrac{1 + \cos(2\theta)}{2}$

$\tan^2(\theta) = \dfrac{1 - \cos(2\theta)}{1 + \cos(2\theta)}$

Sum and Difference Formulas

Sine Calculator
$\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)$

$\sin(a - b) = \sin(a) \cos(b) - \cos(a) \sin(b)$

$\cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b)$

$\cos(a - b) = \cos(a) \cos(b) + \sin(a) \sin(b)$

$\tan(a + b) = \dfrac{\tan(a) + \tan(b)}{1 - \tan(a) \tan(b)}$

$\tan(a - b) = \dfrac{\tan(a) - \tan(b)}{1 + \tan(a) \tan(b)}$

$\cot(a + b) = \dfrac{\cot(a).\cot(b) -1}{\cot(b) + \cot(a)}$

$\cot(a - b) = \dfrac{\cot(a).\cot(b) +1}{\cot(b) - \cot(a)}$

Product to Sum Formulas

$\sin(a) \sin(b) = \dfrac{1}{2}\bigg(\cos(a - b) - \cos(a + b)\bigg)$

$\cos(a) \cos(b) = \dfrac{1}{2}\bigg(\cos(a - b) + \cos(a + b)\bigg)$

$\sin(a) \cos(b) = \dfrac{1}{2}\bigg(\sin(a + b) + \sin(a - b)\bigg)$

$\cos(a) \sin(b) = \dfrac{1}{2}\bigg(\sin(a + b) - \sin(a - b)\bigg)$

Sum to Product Formulas

$\sin(a) + \sin(b) = 2 \sin \left(\dfrac{a + b}{2}\right) \cos \left(\dfrac{a - b}{2}\right)$

$\sin(a) - \sin(b) = 2 \cos \left(\dfrac{a + b}{2}\right) \sin \left(\dfrac{a - b}{2}\right)$

$\cos(a) + \cos(b) = 2 \cos\left(\dfrac{a + b}{2}\right) \cos \left(\dfrac{a - b}{2}\right)$

$\cos(a) - \cos(b) = -2 \sin \left(\dfrac{a + b}{2}\right) \sin \left(\dfrac{a - b}{2}\right)$

Cofunction Formulas

$\sin \left(\dfrac{\pi}{2} - \theta\right) = \cos(\theta) \space \space \space \space \ \ \ \cos \left(\dfrac{\pi}{2} - \theta\right) = \sin(\theta)$

$\csc \left(\dfrac{\pi}{2} - \theta\right) = \sec(\theta) \space \space \space \space \ \ \ \sec \left(\dfrac{\pi}{2} - \theta\right) = \csc(\theta)$

$\tan \left(\dfrac{\pi}{2} - \theta\right) = \cot(\theta) \space \space \space \ \ \ \cot \left(\frac{\pi}{2} - \theta\right) = \tan(\theta)$

Unit Circle Chart

For any ordered pair on the unit circle $(x, y)$: $\space \cos(\theta) = x$ and $\sin(\theta) = y$