Power Reduction Formulas/Hyperbolic Sine Cubed
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Theorem
- $\sinh^3 x = \dfrac {\sinh 3 x - 3 \sinh x} 4$
where $\sin$ denotes hyperbolic sine.
Proof
\(\ds \sinh 3 x\) | \(=\) | \(\ds 3 \sinh x + 4 \sinh^3 x\) | Triple Angle Formula for Hyperbolic Sine | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 4 \sinh^3 x\) | \(=\) | \(\ds \sinh 3 x - 3 \sinh x\) | rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sinh^3 x\) | \(=\) | \(\ds \frac {\sinh 3 x - 3 \sinh x} 4\) | dividing both sides by $4$ |
$\blacksquare$
Also see
- Square of Hyperbolic Sine
- Square of Hyperbolic Cosine
- Cube of Hyperbolic Cosine
- Fourth Power of Hyperbolic Sine
- Fourth Power of Hyperbolic Cosine
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.38$: Powers of Hyperbolic Functions