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  "body": "\n## Statement\n\n$$\\cos \\pi = -1, \\qquad \\sin \\pi = 0.$$\n\n## Remarks\n\nBy the [[definition of $\\pi$|0C1VYVNV2N]], $\\cos(\\pi/2) = 0$. The [[Pythagorean identity|03H7WZ4T3K]] gives $\\sin(\\pi/2) = 1$, since $\\sin \u003e 0$ on $(0, 2]$.\nThe [[double angle formulas|028XEQCE88]] then give $\\cos \\pi = 2 \\cos^2(\\pi/2) - 1 = -1$ and $\\sin \\pi = 2 \\sin(\\pi/2) \\cos(\\pi/2) = 0$.\nTogether with $\\sin \\pi = 0$, it gives Euler's identity $e^{i \\pi} = -1$ in complex form. Mathlib states the first claim as `Real.cos_pi`.\n\n## Explanations\n\n### L1 human\n\nHalf a turn around the unit circle takes the point $(1, 0)$ to the point $(-1, 0)$.\nSo $\\cos \\pi = -1$ and $\\sin \\pi = 0$.\n",
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