Solution
= Solution
For a <real number> $x$, the <hyperbolic functions> satisfy $\cosh x>0$ and $1-\tanh^2x=\operatorname{sech}^2x$. Hence $\cosh x=1/\sqrt{1-u^2}$, and multiplying it by $\tanh x=u$ gives
$$
\boxed{\sinh x=\frac{u}{\sqrt{1-u^2}},\qquad-1<u<1}.
$$
The sign of the numerator preserves the sign of $\sinh x$.