Quadruple Angle Formulas/Hyperbolic Sine
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Theorem
- $\sinh 4 x = 8 \sinh^3 x \cosh x + 4 \sinh x \cosh x$
where $\sinh$ and $\cosh$ denote hyperbolic sine and hyperbolic cosine respectively.
Proof
| \(\ds \sinh 4 x\) | \(=\) | \(\ds \map \sinh {3 x + x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sinh 3 x \cosh x + \cosh 3 x \sinh x\) | Hyperbolic Sine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {3 \sinh x + 4 \sinh^3 x} \cosh x + \cosh 3 x \sinh x\) | Triple Angle Formula for Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {3 \sinh x + 4 \sinh^3 x} \cosh x + \paren {4 \cosh^3 x - 3 \cosh x} \sinh x\) | Triple Angle Formula for Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3 \sinh x \cosh x + 4 \sinh^3 x \cosh x + 4 \cosh^3 x \sinh x - 3 \cosh x \sinh x\) | multiplying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3 \sinh x \cosh x + 4 \sinh^3 x \cosh x\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 4 \cosh x \paren {1 + \sinh^2 x} \sinh x - 3 \cosh x \sinh x\) | Difference of Squares of Hyperbolic Cosine and Sine | ||||||||||
| \(\ds \) | \(=\) | \(\ds 8 \sinh^3 x \cosh x + 4 \sinh x \cosh x\) | multiplying out and gathering terms |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.33$: Multiple Angle Formulas