Power Reduction Formulas/Hyperbolic Sine Cubed

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