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  "body": "\n## Statement\n\nFor every complex number $z$,\n$$e^{i z} = \\cos z + i \\sin z.$$\nHere $e^{w} = \\sum_{n=0}^{\\infty} w^n / n!$ is the complex exponential function, and $\\cos$ and $\\sin$ are given by their power series.\n\n## Remarks\n\nThe main case is a real number $z = x$. Then $\\cos x$ and $\\sin x$ are the real and imaginary parts of $e^{i x}$, and $|e^{i x}| = 1$.\nFor $x = \\pi$, the formula gives [[Euler's identity|09285XX7AE]].\nA proof compares the power series: the even terms of $e^{i z}$ give $\\cos z$ and the odd terms give $i \\sin z$.\n\n## Explanations\n\n### L1 human\n\nAs the real number $x$ grows, the point $e^{i x}$ moves around the unit circle at constant speed.\nIts coordinates are $\\cos x$ and $\\sin x$, as in trigonometry.\nThe formula links the exponential function with the trigonometric functions.\n",
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