Question

Let tanh(x) = -12/13 Find the values of the other
hyperbolic functions. My lecturer said I should use Pythagoras? Since tanh(x) = opposite/adjacent. But I am still stuck, any help please?

Answer 1

Definition of hyperbolic tangent:
\[\tanh x=\frac{\sinh x}{\cosh x}\]
Inverse identities:
\[\coth x=\frac{1}{\tanh x}\\\text{csch}x=\frac{1}{\sinh x}\\\text{sech}x=\frac{1}{\cosh x}\]
And the Pythagorean identities:
\[\cosh^2x=1+\sinh^2x\\
\tanh^2x+\text{sech}^2x=1\\
\coth^2x=1+\text{csch}^2x\]
Given that \(\tanh x=-\dfrac{12}{13}\), you know that \(\sinh x=-12\) and \(\cosh x=13\) because \(\cosh x>0\). Knowing this, you can solve for all the other ratios.