8. Multiple-angle formulae

Multiple-angle formulae

Double-angle, triple-angle, and half-angle formulae

Double-angle formulae

Triple-angle formulae

Half-angle formulae

Also

* Table

These can be shown by using either the sum and difference identities or the multiple-angle formulae.

The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools, by field theory.

A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x³ − 3x + d = 0, where x is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle. However, the discriminant of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle). None of these solutions is reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.

Sine, cosine, and tangent of multiple angles

For specific multiples, these follow from the angle addition formulas, while the general formula was given by 16th-century French mathematician François Viète.

${\displaystyle {\begin{aligned}\sin(n\theta )&=\sum _{k{\text{ odd}}}(-1)^{\frac {k-1}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta ,\\\cos(n\theta )&=\sum _{k{\text{ even}}}(-1)^{\frac {k}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta \end{aligned}}}$

In each of these two equations, the first parenthesized term is a binomial coefficient, and the final trigonometric function equals one or minus one or zero so that half the entries in each of the sums are removed. tan nθ can be written in terms of tan θ using the recurrence relation:

${\displaystyle \tan \,{\big (}(n{+}1)\theta {\big )}={\frac {\tan(n\theta )+\tan \theta }{1-\tan(n\theta )\,\tan \theta }}.}$

cot nθ can be written in terms of cot θ using the recurrence relation:

${\displaystyle \cot \,{\big (}(n{+}1)\theta {\big )}={\frac {\cot(n\theta )\,\cot \theta -1}{\cot(n\theta )+\cot \theta }}.}$

Chebyshev method

The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula knowing the (n − 1)th and (n − 2)th formulae.

cos(nx) can be computed from the cosine of (n − 1)x and (n − 2)x as follows:

${\displaystyle \cos(nx)=2\cdot \cos x\cdot \cos {\big (}(n-1)x{\big )}-\cos {\big (}(n-2)x{\big )}\,}$

Similarly sin(nx) can be computed from the sines of (n − 1)x and (n − 2)x

${\displaystyle \sin(nx)=2\cdot \cos x\cdot \sin {\big (}(n-1)x{\big )}-\sin {\big (}(n-2)x{\big )}\,}$

For the tangent, we have:

${\displaystyle \tan(nx)={\frac {H+K\tan x}{K-H\tan x}}\,}$

where H/K = tan(n − 1)x.

Tangent of an average

${\displaystyle \tan \left({\frac {\alpha +\beta }{2}}\right)={\frac {\sin \alpha +\sin \beta }{\cos \alpha +\cos \beta }}=-\,{\frac {\cos \alpha -\cos \beta }{\sin \alpha -\sin \beta }}}$

Setting either α or β to 0 gives the usual tangent half-angle formulae.

Viète's infinite product

${\displaystyle \cos {\theta \over 2}\cdot \cos {\theta \over 4}\cdot \cos {\theta \over 8}\cdots =\prod _{n=1}^{\infty }\cos {\theta \over 2^{n}}={\sin \theta \over \theta }=\operatorname {sinc} \,\theta .}$

(Refer to sinc function.)