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  "body": "\n## Summary\n\nThe 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 $S, C \\colon \\R \\to \\R$.\nThey turn the angle addition formulas of [[sine and cosine|08WFZP6P89]] into equations.\nThey are equivalent to Cauchy's exponential equation for the complex-valued function $C + iS$.\nSo they characterize sine and cosine, up to a scaling of the variable and an exponential factor, among continuous functions.\n\n## Explanations\n\n### L1 human\n\nRotating by an angle $x$ and then by an angle $y$ is the same as rotating by $x + y$.\nWriting this in coordinates gives the addition formulas for sine and cosine.\nThe equations ask which pairs of functions obey these formulas.\nBesides $\\sin$ and $\\cos$, the pair $S(x) = \\sin(2x)$, $C(x) = \\cos(2x)$ also works, which corresponds to rotating twice as fast.\n",
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