Question

If cosh x = 13 5 and x > 0, find the values of the other hyperbolic functions at x.

Answer 1

Recall that


`\cosh^2x-\sinh^2x=1\implies \cosh x=\pm√(1+\sinh^2x)`

You're given that
`\cosh x=\frac{13}5>0`, so omit the negative root. Then


`\frac{13}5=√(1+\sinh^2x)\implies\sinh^2x=(144)/(25)\implies\sinh x=\pm\frac{12}5`

Because
`x>0`, you have
`\sinh x>0` too, so omit the negative root again. Then
`\sinh x=\frac{12}5`.

Now,


`\mbox{sech }x=\frac1{\cosh x}=\frac5{13}`

`\mbox{csch }x=\frac1{\sinh x}=\frac5{12}`

`\tanh x=(\sinh x)/(\cosh x)=\frac{\frac{12}5}{\frac{13}5}=(12)/(13)`

`\coth x=\frac1{\tanh x}=(13)/(12)`