Hyperbolic Functions

Hyperbolic functions (sinh, cosh, tanh and their relatives) are analogs of trigonometric functions defined using exponentials rather than circles. This topic covers evaluating hyperbolic functions, verifying their identities, computing their derivatives, and working with inverse hyperbolic functions.

Hyperbolic functions describe the shape of hanging cables (catenaries), appear in special relativity for rapidity, and arise naturally when solving certain differential equations in engineering.

Practice Tips

• 1Learn the exponential definitions: sinh(x) = (e^x - e^(-x))/2 and cosh(x) = (e^x + e^(-x))/2, since most identities follow directly from these.
• 2Hyperbolic identities mirror trig identities but with sign changes; for example, cosh^2(x) - sinh^2(x) = 1 (note the minus instead of plus).
• 3The derivatives of sinh and cosh are cosh and sinh respectively, which is simpler than their trigonometric counterparts since there are no sign flips.