Half Angle Formulas/Hyperbolic Sine
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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.
Proof
\(\ds \cosh x\) | \(=\) | \(\ds 1 + 2 \ \sinh^2 \frac x 2\) | Double Angle Formula for Hyperbolic Cosine: Corollary $2$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \ \sinh^2 \frac x 2\) | \(=\) | \(\ds \cosh x - 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sinh \frac x2\) | \(=\) | \(\ds \pm \sqrt {\frac {\cosh x - 1} 2}\) |
We also have that:
- when $x \ge 0$, $\sinh x \ge 0$
- when $x \le 0$, $\sinh x \le 0$.
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.27$: Double Angle Formulas