---
tag: C73TGW752D
kind: concept
name: Sine and cosine addition equations
msc: [39B22, 39B72]
specializes: [SRAN4EKWS0]
sources:
  - url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
---

## Summary

The sine and cosine addition equations are the system of [[functional equations|SRAN4EKWS0]] $S(x + y) = S(x) C(y) + C(x) S(y)$ and $C(x + y) = C(x) C(y) - S(x) S(y)$ for two unknown functions $S, C \colon \R \to \R$.
They turn the angle addition formulas of [[sine and cosine|08WFZP6P89]] into equations.
They are equivalent to Cauchy's exponential equation for the complex-valued function $C + iS$.
So they characterize sine and cosine, up to a scaling of the variable and an exponential factor, among continuous functions.

## Explanations

### L1 human

Rotating by an angle $x$ and then by an angle $y$ is the same as rotating by $x + y$.
Writing this in coordinates gives the addition formulas for sine and cosine.
The equations ask which pairs of functions obey these formulas.
Besides $\sin$ and $\cos$, the pair $S(x) = \sin(2x)$, $C(x) = \cos(2x)$ also works, which corresponds to rotating twice as fast.