Double Angle Formulas/Hyperbolic Sine
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Theorem
- $\sinh 2 x = 2 \sinh x \cosh x$
where $\sinh$ and $\cosh$ denote hyperbolic sine and hyperbolic cosine respectively.
Corollary
- $\map \sinh {2 \theta} = \dfrac {2 \tanh \theta} {1 - \tanh^2 \theta}$
Proof 1
| \(\ds \sinh 2 x\) | \(=\) | \(\ds \map \sinh {x + x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sinh x \cosh x + \cosh x \sinh x\) | Hyperbolic Sine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \sinh x \cosh x\) |
$\blacksquare$
Proof 2
| \(\ds \sinh 2 x\) | \(=\) | \(\ds \frac 1 2 \paren {e^{2 x} - e^{-2 x} }\) | Definition of Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {e^x + e^{-x} } \paren {e^x - e^{-x} }\) | Difference of Two Squares | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \paren {\frac{e^x + e^{-x} } 2 \cdot \frac {e^x - e^{-x} } 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \sinh x \cosh x\) | Definition of Hyperbolic Sine, Definition of Hyperbolic Cosine |
$\blacksquare$
Proof 3
| \(\ds \sinh 2 x\) | \(=\) | \(\ds -i \sin 2 i x\) | Hyperbolic Sine in terms of Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds -2 i \sin i x \cos i x\) | Double Angle Formula for Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \sinh x \cosh x\) | Hyperbolic Sine in terms of Sine, Hyperbolic Cosine in terms of Cosine |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.24$: Double Angle Formulas
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): double-angle formula
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): hyperbolic function