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


Sources