Question

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

- the definitions of the hyperbolic functions, $\sin hx,\cos hx$and$\tan hx$in terms of exponential functions

Short answer

$\sin hx=\frac{e^x-e^{-x}}{2}$

$\cos hx=\frac{e^x+e^{-x}}{2}$

role="math" localid="1650917163313" $\tan hx=\frac{e^x-e^{-x}}{e^x+e^{-x}}$

Step-by-step solution

Given Information

The given expressions are$\sin hx.\cos hx,\tan hx$

Simplification

Hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle.

$\sin hx=\frac{e^x-e^{-x}}{2}$

$\cos hx=\frac{e^x+e^{-x}}{2}$

$\tan hx=\frac{e^x-e^{-x}}{e^x+e^{-x}}$