Half Angle Formulas/Hyperbolic Sine

Theorem

Let $x \in \R$.

Then:

$\ds \sinh \frac x 2$

$=$

$\ds +\sqrt {\frac {\cosh x - 1} 2}$

for $x \ge 0$

$\ds \sinh \frac x 2$

$=$

$\ds -\sqrt {\dfrac {\cosh x - 1} 2}$

for $x \le 0$

where $\sinh$ denotes hyperbolic sine and $\cosh$ denotes hyperbolic cosine.

$\ds \cosh x$

$=$

$\ds 1 + 2 \ \sinh^2 \frac x 2$

$\ds \leadsto \ \ $

$\ds 2 \ \sinh^2 \frac x 2$

$=$

$\ds \cosh x - 1$

$\ds \leadsto \ \ $

$\ds \sinh \frac x2$

$=$

$\ds \pm \sqrt {\frac {\cosh x - 1} 2}$

We also have that:

when $x \ge 0$, $\sinh x \ge 0$

when $x \le 0$, $\sinh x \le 0$.

$\blacksquare$