Multiple angles and Osborn’s rule

This post was motivated by an exercise in [1] that says

Prove that for the hyperbolic functions … formulas hold similar to those in Section 2.3 with all the minuses replaced by pluses.

My first thought was that this sounds like Osborn’s rule, a heuristic for translating between (circular) trig identities and hyperbolic trig identities. As explained in that post, Osborn’s rule is an easy consequence of Euler’s identity.

Now what are the formulas the exercise refers to?

Sine to hyperbolic sine

Here’s the identity for sine.

$\sin n\theta = \binom{n}{1} \cos^{n-1}\theta \sin\theta - \binom{n}{3} \cos^{n-3}\theta \sin^3\theta + \binom{n}{5} \cos^{n-5}\theta \sin^5\theta -\cdots$

Osborn’s rule says to change sin to sinh and cos to cosh, and flip signs whenever two sinh terms are multiplied together. The term with sin³ θ loses its minus sign because two sines are multiplied together. The term with sin⁵ θ changes sign twice, and so the net result is that it doesn’t change sign. So we have the following.

$\sinh n\theta = \binom{n}{1} \cosh^{n-1}\theta \sinh\theta + \binom{n}{3} \cosh^{n-3}\theta \sinh^3\theta + \binom{n}{5} \cosh^{n-5}\theta \sinh^5\theta + \cdots$

Cosine to hyperbolic cosine

The cosine identity

$\cos n\theta = \cos^n \theta - \binom{n}{2} \cos^{n-2}\theta \sin^2 \theta + \binom{n}{4} \cos^{n-4} \theta \sin^4\theta - \cdots$

becomes

$\cosh n\theta = \cosh^n \theta - \binom{n}{2} \cosh^{n-2}\theta \sinh^2 \theta + \binom{n}{4} \cosh^{n-4} \theta \sinh^4\theta - \cdots$

by similar reasoning.

Tangent to hyperbolic tangent

Osborn’s rule applies to tan and tanh as well, if you imagine each tangent as sin/cos.

Thus

$\tan n\theta = \frac{\binom{n}{1} \tan \theta - \binom{n}{3} \tan^3 \theta + \binom{n}{5} \tan^5 \theta - \cdots}{1 - \binom{n}{2} \tan^2 \theta + \binom{n}{4} \tan^4\theta - \cdots}$

becomes

$\tanh n\theta = \frac{\binom{n}{1} \tanh \theta + \binom{n}{3} \tanh^3 \theta + \binom{n}{5} \tanh^5 \theta + \cdots}{1 - \binom{n}{2} \tanh^2 \theta + \binom{n}{4} \tanh^4\theta + \cdots}$

[1] Dmitry Fuchs and Serge Tabachnikov. Mathematical Omnibus: Thirty Lectures on Classical Mathematics.

Multiple angles and Osborn’s rule

Sine to hyperbolic sine

Cosine to hyperbolic cosine

Tangent to hyperbolic tangent

Related posts

Leave a Reply