Triple Angle Formulas/Hyperbolic Cosine
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Theorem
- $\cosh 3 x = 4 \cosh^3 x - 3 \cosh x$
where $\cosh$ denotes hyperbolic cosine.
Proof
| \(\ds \cosh 3 x\) | \(=\) | \(\ds \cosh \paren {2 x + x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cosh 2 x \cosh x + \sinh 2 x \sinh x\) | Hyperbolic Cosine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\cosh^2 x + \sinh^2 x} \cosh x + \sinh 2 x \sinh x\) | Double Angle Formula for Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\cosh^2 x + \sinh^2 x} \cosh x + \paren {2 \sinh x \cosh x} \sinh x\) | Double Angle Formula for Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cosh^3 x + \sinh^2 x \cosh x + 2 \sinh^2 x \cosh x\) | multiplying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cosh^3 x + \paren {\cosh^2 x - 1} \cosh x + 2 \paren {\cosh^2 x - 1} \cosh x\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cosh^3 x + \cosh^3 x - \cosh x + 2 \cosh^3 x - 2 \cosh x\) | multiplying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds 4 \cosh^3 x - 3 \cosh x\) | 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: $8.31$: Multiple Angle Formulas