Calculate cosh(x) given sinh(x)

Use the identity $\cosh^2x-\sinh^2x \equiv 1$. If $\sinh x = \frac{3}{2}$ then $$\cosh^2x - \left(\frac{3}{2}\right)^{\! 2} = 1$$ $$\cosh^2x - \frac{9}{4} = 1$$ $$\cosh^2x = \frac{13}{4}$$ It follows that $\cosh x = \pm\frac{1}{2}\sqrt{13}$. Since $\cosh x \ge 1$ for all $x \in \mathbb{R}$ we have $\cosh x = \frac{1}{2}\sqrt{13}$.

The trick is....

$$\cosh^2x-\sinh^2x=1$$