Hyperbolic functions — chapter review — 9231 FP2

Notes and key methods

Definitions are \(\sinh x=(e^x-e^{-x})/2\), \(\cosh x=(e^x+e^{-x})/2\), \(\tanh x=\sinh x/\cosh x\), and \(\operatorname{sech}x=1/\cosh x\). They imply cosh²x−sinh²x=1.

Unlike cos x, cosh x is at least one for real x. Sinh is odd and strictly increasing; cosh is even. Work from exponentials when proving a hyperbolic identity rather than borrowing an unchecked circular-trigonometric sign.

For real x, \(\operatorname{arsinh}x=\ln(x+\sqrt{x^2+1})\). On x≥1, \(\operatorname{arcosh}x=\ln(x+\sqrt{x^2-1})\) is nonnegative. For |x|<1, \(\operatorname{artanh}x=\frac12\ln[(1+x)/(1-x)]\).

These are inverse functions, not reciprocals. Solving an equation involving cosh requires choosing a branch unless its argument is explicitly restricted to be nonnegative.

Check the conditions

• The argument is real unless stated otherwise.
• Cosh is never zero on the real axis.
• Arcosh has real domain [1,∞), with range [0,∞).
• Artanh has real domain (−1,1).