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Trigonometry

Hyperbolic

Definitions of hyperbolic functions, fundamental identities, inverses, and relations to trigonometric functions.

Hyperbolic functions are exponential analogues of the trigonometric functions, parameterising the unit hyperbola the way sine and cosine parameterise the unit circle. Their identities mirror trigonometric ones with sign changes, notably cosh2 - sinh2 = 1 in place of cos2 + sin2 = 1. Inverse hyperbolic functions have closed-form logarithmic expressions, useful in calculus, special-relativity rapidity, and the catenary. Imaginary arguments link the two families: cosh(ix) = cos(x) and sinh(ix) = i sin(x). They are a single function viewed from two axes of the complex plane.

(cos t, sin t)x² + y² = 1(cosh t, sinh t)x² - y² = 1
Trig functions parameterize the unit circle; hyperbolic functions parameterize the unit hyperbola.

Hyperbolic Definitions

The hyperbolic functions are built from the exponential, each one corresponding to a trigonometric counterpart.

Hyperbolic sine, the odd part of ex.

sinh⁡(x)=(ex−e−x)2\sinh\left(x\right)=\frac{\left(e^{x}-e^{-x}\right)}{2}

Hyperbolic cosecant, reciprocal of sinh.

csch(x)=1sinh⁡(x)=2(ex−e−x)csch\left(x\right)=\frac{1}{\sinh\left(x\right)}=\frac{2}{\left(e^{x}-e^{-x}\right)}

Hyperbolic cosine, the even part of ex.

cosh⁡(x)=(ex−e−x)2\cosh\left(x\right)=\frac{\left(e^{x}-e^{-x}\right)}{2}

Hyperbolic secant, reciprocal of cosh.

sech(x)=1cosh⁡(x)=2(ex−e−x)sech\left(x\right)=\frac{1}{\cosh\left(x\right)}=\frac{2}{\left(e^{x}-e^{-x}\right)}

Hyperbolic tangent, bounded in (-1, 1).

tanh⁡(x)=sinh⁡(x)cosh⁡(x)=(ex−e−x)(ex+e−x)\tanh\left(x\right)=\frac{\sinh\left(x\right)}{\cosh\left(x\right)}=\frac{\left(e^{x}-e^{-x}\right)}{\left(e^{x}+e^{-x}\right)}

Hyperbolic cotangent, reciprocal of tanh.

coth⁡(x)=1tanh⁡(x)=(ex+e−x)(ex−e−x)\coth\left(x\right)=\frac{1}{\tanh\left(x\right)}=\frac{\left(e^{x}+e^{-x}\right)}{\left(e^{x}-e^{-x}\right)}

Sum and Difference

The hyperbolic Pythagorean identity cosh2 - sinh2 = 1 corresponds to the unit hyperbola. The other identities follow by dividing through.

Hyperbolic identity for the unit hyperbola x2 - y2 = 1.

cosh⁡2(x)−sinh⁡2(x)=1\cosh^{2}\left(x\right)-\sinh^{2}\left(x\right)=1

Divide by cosh2.

tanh⁡2(x)−sech2(x)=1\tanh^{2}\left(x\right)-sech^{2}\left(x\right)=1

Divide by sinh2.

coth⁡2(x)−csch2(x)=1\coth^{2}\left(x\right)-csch^{2}\left(x\right)=1

Inverse Hyperbolic Definitions

Each inverse hyperbolic function has a closed form in natural logarithms, a property the trigonometric inverses lack on the reals.

Inverse hyperbolic sine.

arcsinh(z)=ln⁡(z+z2+1)arcsinh\left(z\right)=\ln\left(z+\sqrt{z^{2}+1}\right)

Inverse hyperbolic cosine (± gives two branches).

arccosh(z)=ln⁡(z±z2−1)arccosh\left(z\right)=\ln\left(z\pm\sqrt{z^{2}-1}\right)

Inverse hyperbolic tangent, for |z| < 1.

arctanh(z)=12ln⁡(1+z1−z)arctanh\left(z\right)=\frac{1}{2\ln\left(\frac{1+z}{1-z}\right)}

Inverse hyperbolic cosecant.

arccsch(z)=ln⁡((1+(1+z2))z)arccsch\left(z\right)=\ln\left(\frac{\left(1+\left(\sqrt{1+z^{2}}\right)\right)}{z}\right)

Inverse hyperbolic secant.

arcsech(z)=ln⁡((1±(1−z2))z)arcsech\left(z\right)=\ln\left(\frac{\left(1\pm\left(\sqrt{1-z^{2}}\right)\right)}{z}\right)

Inverse hyperbolic cotangent, for |z| > 1.

arccoth(z)=12ln⁡(z+1z−1)arccoth\left(z\right)=\frac{1}{2\ln\left(\frac{z+1}{z-1}\right)}

Relations to Trig Functions

Imaginary arguments connect the hyperbolic and circular families. They are the same function in different orientations of the complex plane.

Sinh as sine of an imaginary argument.

sinh⁡(z)=−i sin⁡(iz)\sinh\left(z\right)=-i\text{ }\sin\left(iz\right)

Cosecant counterpart.

csch(z)=i csc⁡(iz)csch\left(z\right)=i\text{ }\csc\left(iz\right)

Cosh in terms of cosine (no factor of i, since cosine is even).

cosh⁡(z)=cos⁡(iz)\cosh\left(z\right)=\cos\left(iz\right)

Sech in terms of sec.

sech(z)=sec⁡(iz)sech\left(z\right)=\sec\left(iz\right)

Tanh in terms of tangent.

tanh⁡(z)=−i tan⁡(iz)\tanh\left(z\right)=-i\text{ }\tan\left(iz\right)