Power Reduction Formulas/Hyperbolic Cosine Cubed
< Power Reduction Formulas(Redirected from Cube of Hyperbolic Cosine)
Jump to navigation
Jump to search
Theorem
- $\cosh^3 x = \dfrac {\cosh 3 x + 3 \cosh x} 4$
where $\cosh$ denotes hyperbolic cosine.
Proof 1
| \(\ds \cosh 3 x\) | \(=\) | \(\ds 4 \cosh^3 x - 3 \cosh x\) | Triple Angle Formula for Hyperbolic Cosine | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 4 \cosh^3 x\) | \(=\) | \(\ds \cosh 3 x + 3 \cosh x\) | rearranging | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cosh^3 x\) | \(=\) | \(\ds \dfrac {\cosh 3 x + 3 \cosh x} 4\) | dividing both sides by $4$ |
$\blacksquare$
Proof 2
| \(\ds \cosh^3 x\) | \(=\) | \(\ds \frac 1 {2^3} \paren {e^x + e^{-x} }^3\) | Definition of Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 8 \paren {e^{3x} + e^{-3x} + 3e^{x} + 3e^{-x} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 4 \paren {\frac{ e^{3x} + e^{-3x} } 2} + \frac 3 4 \paren {\frac{e^{x} + e^{-x} } 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\cosh 3x} 4 + \frac {3 \cosh x} 4\) | Definition of Hyperbolic Cosine |
$\blacksquare$
Proof 3
| \(\ds \cosh^3 x\) | \(=\) | \(\ds \cos^3 i x\) | Hyperbolic Cosine in terms of Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\map \cos {3 i x} + 3 \cos i x} 4\) | Cube of Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\cosh 3 x + 3 \cosh x} 4\) | Hyperbolic Cosine in terms of Cosine |
$\blacksquare$
Also see
- Square of Hyperbolic Sine
- Square of Hyperbolic Cosine
- Cube of Hyperbolic Sine
- 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: $8.39$: Powers of Hyperbolic Functions