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  "body": "\n## Statement\n\nFor every real $x$,\n$$\\sin(x + 2\\pi) = \\sin x, \\qquad \\cos(x + 2\\pi) = \\cos x.$$\nThe number $2\\pi$ is the smallest positive period of $\\sin$ and of $\\cos$.\n\n## Remarks\n\nFrom [[$\\cos \\pi = -1$ and $\\sin \\pi = 0$|01A8JGR4JM]], the [[addition formulas|028XEQCE88]] give $\\cos 2\\pi = 1$ and $\\sin 2\\pi = 0$. Apply the addition formulas again with $y = 2\\pi$.\nSo $\\sin$ and $\\cos$ are [[periodic|089KWVYD69]] functions.\nAlso $\\sin(x + \\pi) = -\\sin x$ and $\\sin(x + \\pi/2) = \\cos x$. Mathlib states the periodicity as `Real.sin_periodic`.\n\n## Explanations\n\n### L1 human\n\nA full turn around the unit circle has length $2\\pi$, and it brings the point back to where it started.\nSo the graphs of sine and cosine repeat after each interval of length $2\\pi$.\n",
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