Triple Angle Formulas/Hyperbolic Sine
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Theorem
- $\sinh 3 x = 3 \sinh x + 4 \sinh^3 x$
where $\sinh$ denotes hyperbolic sine.
Proof
\(\ds \sinh {3 x}\) | \(=\) | \(\ds \map \sinh {2 x + x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sinh 2 x \cosh x + \cosh 2 x \sinh x\) | Hyperbolic Sine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 \sinh x \cosh x} \cosh x + \cosh 2 x \sinh x\) | Double Angle Formula for Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 \sinh x \cosh x} \cosh x + \paren {\cosh^2 x + \sinh^2 x} \sinh x\) | Double Angle Formula for Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sinh x \cosh^2 x + \cosh^2 x \sinh x + \sinh^3 x\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sinh x \paren {1 + \sinh^2 x} + \paren {1 + \sinh^2 x} \sinh x + \sinh^3 x\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sinh x + 2 \sinh^3 x + \sinh x + \sinh^3 x + \sinh^3 x\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 \sinh x + 4 \sinh^3 x\) | gathering terms |
$\blacksquare$
Also see
- Quadruple Angle Formula for Hyperbolic Sine
- Quadruple Angle Formula for Hyperbolic Cosine
- Quadruple Angle Formula for Hyperbolic Tangent
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.30$: Multiple Angle Formulas