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  "body": "\n## Definition\n\nThe hyperbolic functions are the functions $\\sinh$, $\\cosh$ and $\\tanh$ on $\\mathbb{R}$, together with their reciprocals:\n$$\\coth x = \\frac{\\cosh x}{\\sinh x} \\ (x \\ne 0), \\qquad \\operatorname{sech} x = \\frac{1}{\\cosh x}, \\qquad \\operatorname{csch} x = \\frac{1}{\\sinh x} \\ (x \\ne 0).$$\nHere [[$\\sinh$|0GWD21EWRW]], [[$\\cosh$|0EQFWKJ37J]] and [[$\\tanh$|0MWYQTZ07Z]] are the hyperbolic sine, cosine and tangent.\n\n## Remarks\n\nThey satisfy identities similar to the trigonometric identities, often with a change of sign.\nThe point $(\\cosh t, \\sinh t)$ lies on the hyperbola $x^2 - y^2 = 1$, as $(\\cos t, \\sin t)$ lies on the circle.\nVincenzo Riccati introduced the hyperbolic functions in 1757.\n\n## Explanations\n\n### L1 human\n\nThe hyperbolic functions are built from $e^x$ and $e^{-x}$.\nThey play the role for the hyperbola that sine and cosine play for the circle.\n",
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