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  "body": "\n## Definition\n\nThe sine and cosine functions $\\sin, \\cos \\colon \\mathbb{R} \\to \\mathbb{R}$ are defined by the power series\n$$\\sin x = \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n+1}}{(2n+1)!}, \\qquad \\cos x = \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n}}{(2n)!}.$$\n\n## Remarks\n\nBoth [[series|02KTRHQMHN]] [[converge|08W5RN375X]] absolutely for every real $x$.\nSine is odd and cosine is even. Also $\\sin 0 = 0$ and $\\cos 0 = 1$.\nGeometrically, $(\\cos x, \\sin x)$ is the point of the unit circle at arc length $x$ from $(1, 0)$, counterclockwise.\nMathlib defines `Real.sin` and `Real.cos` from the complex exponential, with $\\cos x + i \\sin x = e^{i x}$.\n\n## Explanations\n\n### L1 human\n\nSine and cosine describe motion around the unit circle.\nIn analysis, we define them by power series, and we then prove their geometric properties.\n",
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