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Hyperbolic functions
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, hyperbolic functions are analogues of the ordinary <a href="/facts/Trigonometric_function/czej7ur0">trigonometric functions</a>, but defined using the <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a> rather than the <a href="/facts/Circle/9fy5ggKn">circle</a>. Just as the points (cos <i>t</i>, sin <i>t</i>) form a <a href="/facts/Unit_circle/VsTffjPK">circle with a unit radius</a>, the points (cosh <i>t</i>, sinh <i>t</i>) form the right half of the <a href="/facts/Unit_hyperbola/AvwG6gJ9">unit hyperbola</a>. Also, similarly to how the derivatives of sin(<i>t</i>) and cos(<i>t</i>) are cos(<i>t</i>) and –sin(<i>t</i>) respectively, the derivatives of sinh(<i>t</i>) and cosh(<i>t</i>) are cosh(<i>t</i>) and sinh(<i>t</i>) respectively. </p><p>Hyperbolic functions are used to express the <a href="/facts/Angle_of_parallelism/gsCvkgeh">angle of parallelism</a> in <a href="/facts/Hyperbolic_geometry/DjTMKdgy">hyperbolic geometry</a>. They are used to express <a href="/facts/Lorentz_boost/jotJL06G">Lorentz boosts</a> as <a href="/facts/Hyperbolic_rotation/CYxzaaKR">hyperbolic rotations</a> in <a href="/facts/Special_relativity/vsFFc63E">special relativity</a>. They also occur in the solutions of many linear <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a> (such as the equation defining a <a href="/facts/Catenary/fWVU770E">catenary</a>), <a href="/facts/Cubic_equation/Eqm3HCMx">cubic equations</a>, and <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equation</a> in <a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a>. <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equations</a> are important in many areas of <a href="/facts/Physics/a7aH2y08">physics</a>, including <a href="/facts/Electromagnetic_theory/oAkmALHT">electromagnetic theory</a>, <a href="/facts/Heat_transfer/1IV3WbFL">heat transfer</a>, and <a href="/facts/Fluid_dynamics/y7uzSCfg">fluid dynamics</a>. </p><p>The basic hyperbolic functions are: </p> <ul><li>hyperbolic sine "sinh" (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/),</li> <li>hyperbolic cosine "cosh" (/ˈkɒʃ, ˈkoʊʃ/),</li></ul> <p>from which are derived: </p> <ul><li>hyperbolic tangent "tanh" (/ˈtæŋ, ˈtæntʃ, ˈθæn/),</li> <li>hyperbolic cotangent "coth" (/ˈkɒθ, ˈkoʊθ/),</li> <li>hyperbolic secant "sech" (/ˈsɛtʃ, ˈʃɛk/),</li> <li>hyperbolic cosecant "csch" or "cosech" (/ˈkoʊsɛtʃ, ˈkoʊʃɛk/)</li></ul> <p>corresponding to the derived trigonometric functions. </p><p>The <a href="/facts/Inverse_hyperbolic_functions/myk0wJty">inverse hyperbolic functions</a> are: </p> <ul><li>inverse hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh")</li> <li>inverse hyperbolic cosine "arcosh" (also denoted "cosh−1", "acosh" or sometimes "arccosh")</li> <li>inverse hyperbolic tangent "artanh" (also denoted "tanh−1", "atanh" or sometimes "arctanh")</li> <li>inverse hyperbolic cotangent "arcoth" (also denoted "coth−1", "acoth" or sometimes "arccoth")</li> <li>inverse hyperbolic secant "arsech" (also denoted "sech−1", "asech" or sometimes "arcsech")</li> <li>inverse hyperbolic cosecant "arcsch" (also denoted "arcosech", "csch−1", "cosech−1","acsch", "acosech", or sometimes "arccsch" or "arccosech")</li></ul> <p>The hyperbolic functions take a <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Argument_of_a_function/QkuPc28I">argument</a> called a <a href="/facts/Hyperbolic_angle/m4IaIh0g">hyperbolic angle</a>. The magnitude of a hyperbolic angle is the <a href="/facts/Area/KsNG1G9t">area</a> of its <a href="/facts/Hyperbolic_sector/PCEFy8fd">hyperbolic sector</a> to <i>xy</i> = 1. The hyperbolic functions may be defined in terms of the <a href="/facts/Hyperbolic_sector/PCEFy8fd">legs of a right triangle</a> covering this sector. </p><p>In <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are <a href="/facts/Entire_function/XNluKzu7">entire functions</a>. As a result, the other hyperbolic functions are <a href="/facts/Meromorphic_function/LeNBd6Sh">meromorphic</a> in the whole complex plane. </p><p>By <a href="/facts/Lindemann%25E2%2580%2593Weierstrass_theorem/BafvlpWm">Lindemann–Weierstrass theorem</a>, the hyperbolic functions have a <a href="/facts/Transcendental_number/IrB7JppM">transcendental value</a> for every non-zero <a href="/facts/Algebraic_number/MkH1PzTK">algebraic value</a> of the argument. </p>