Hyperbolic function

Question

Find the numerical value of the expression: cosh^-1 (3)=

please help

Answer 1

If you do not use a calculator, then use the definition of cosh(x) = (e^x+e^-x)/2.

To find the number that has cosh = 3, i.e. cosh^-1(3) , solve this equation:

(e^x+e^-x)/2 = 3

e^x+e^-x = 6,

e^x + 1/e^x = 6

e^2x + 1 = 6e^x

e^2x-6e^x+1 = 0

This is quadratic in e^x. Try using quadratic formula:

e^x = (6 ±√(36-4(1)(1)))/2 = (6±√32)/2 = (6±4√2)/2 = 3±√2.

Finally take ln of both sides to give x = ln(3±√2).

Here is some validation:

desmos.com/calculator/pabwwdyhf7