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  "body": "\n## Definition\n\nFor $z \\in \\mathbb{C}$, the complex sine and the complex cosine of $z$ are\n$$\\sin z = \\frac{e^{iz} - e^{-iz}}{2i}, \\qquad \\cos z = \\frac{e^{iz} + e^{-iz}}{2},$$\nwhere $e^w$ is the [[complex exponential|054VXHMCA3]] of $w$.\n\n## Remarks\n\nFor real $z$, these are the usual sine and cosine, by [[Euler's formula|04R3CY41VJ]].\nThe functions $\\sin$ and $\\cos$ are [[entire|0419794R0T]], with $\\sin' = \\cos$ and $\\cos' = -\\sin$. The identity $\\sin^2 z + \\cos^2 z = 1$ holds for each $z \\in \\mathbb{C}$.\nThey are not bounded on $\\mathbb{C}$. For real $y$, $\\cos(iy) = \\cosh y = \\frac{e^y + e^{-y}}{2}$.\nThe [[power series of the cosine|0Z1G2F8DGY]] is the same as in the real case.\n\n## Explanations\n\n### L1 human\n\nEuler's formula $e^{i\\theta} = \\cos \\theta + i \\sin \\theta$ lets us solve for $\\cos \\theta$ and $\\sin \\theta$ in terms of exponentials.\nThese formulas make sense for every complex number, so we use them to define $\\sin z$ and $\\cos z$.\nOn the imaginary axis they grow very fast: $\\cos(10 i)$ is more than $10000$.\n",
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