---
tag: H313BRXEJS
kind: concept
name: Trigonometric functions
msc: [33B10, 26A09]
sources:
  - url: https://en.wikipedia.org/wiki/Trigonometric_functions
same_as:
  wikidata: Q93344
  wikipedia: Trigonometric functions
---

## Summary

The trigonometric functions are the six functions sine, cosine, tangent, cotangent, secant and cosecant of a real angle.
[[Sine and cosine|08WFZP6P89]] give the coordinates of a point on the unit circle.
The other four are quotients built from them, such as the [[tangent|05CG0S20S3]] $\tan = \sin / \cos$.
They are periodic, they describe rotations and oscillations, and they are the building blocks of Fourier series.
In analysis they are defined by power series, which makes the geometric facts into theorems.

## Explanations

### L1 human

Walk an angle $x$ (in radians) around the circle of radius $1$ starting from the point $(1, 0)$.
The point you reach is $(\cos x, \sin x)$.
The tangent $\tan x = \sin x / \cos x$ is the slope of the line from the centre to that point.
The secant, cosecant and cotangent are the reciprocals $1/\cos x$, $1/\sin x$ and $\cos x / \sin x$.
For example, at $x = \pi/4$ the point is $(\frac{\sqrt2}{2}, \frac{\sqrt2}{2})$, so $\tan(\pi/4) = 1$.