Half Angle Formulas/Hyperbolic Tangent
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Theorem
Let $x \in \R$.
Then:
\(\ds \tanh \frac x 2\) | \(=\) | \(\ds +\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }\) | for $x \ge 1$ | |||||||||||
\(\ds \tanh \frac x 2\) | \(=\) | \(\ds -\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }\) | for $x \le 1$ |
where $\tanh$ denotes hyperbolic tangent and $\cosh$ denotes hyperbolic cosine.
Corollary 1
- $\tanh \dfrac x 2 = \dfrac {\sinh x} {\cosh x + 1}$
Corollary 2
For $x \ne 0$:
- $\tanh \dfrac x 2 = \dfrac {\cosh x - 1} {\sinh x}$
Proof
\(\ds \tanh \frac x 2\) | \(=\) | \(\ds \frac {\sinh \frac x 2} {\cosh \frac x 2}\) | Definition 2 of Hyperbolic Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\pm \sqrt {\frac {\cosh x - 1} 2} } {\pm \sqrt {\frac {\cosh x + 1} 2} }\) | Half Angle Formula for Hyperbolic Sine and Half Angle Formula for Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \pm \sqrt {\frac {\cosh x - 1} {\cosh x + 1} }\) |
We also have that:
- when $x \ge 0$, $\tanh \dfrac x 2 \ge 0$
- when $x \le 0$, $\tanh \dfrac x 2 \le 0$.
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.29$: Double Angle Formulas