Trigonometry is the branch of mathematics studying relationships between angles and side lengths in triangles, particularly through functions like sine, cosine, and tangent. Born from practical needs in astronomy and surveying, it extends far beyond triangles—powering wave analysis, signal processing, physics, and engineering applications. The key mental model: angles as circular motion—every trigonometric concept traces back to points traveling around the unit circle, where position, rotation, and periodicity intertwine. In calculus and complex analysis, trigonometric functions appear in derivatives, integrals, substitutions, and polar representations of complex numbers, making them one of the most pervasive tools in all of mathematics.
What This Cheat Sheet Covers
This topic spans 33 focused tables and 205 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
A jump-to index of every table row in this cheat sheet.
An interactive map of every table and concept in this topic.
Table 1: Right Triangle Definitions
Everything in trigonometry starts here: the six functions defined as ratios of a right triangle's sides. Sine, cosine, and tangent — locked in by the mnemonic SOH-CAH-TOA — are the three you reach for constantly, and the remaining three (cosecant, secant, cotangent) are simply their reciprocals.
| Function | Example | Description |
|---|---|---|
$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ | • Ratio of opposite side to hypotenuse • mnemonic SOH (sine = opposite/hypotenuse). | |
$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ | • Ratio of adjacent side to hypotenuse • mnemonic CAH (cosine = adjacent/hypotenuse). | |
$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin \theta}{\cos \theta}$ | • Ratio of opposite to adjacent • mnemonic TOA • undefined when $\cos \theta = 0$. |