Multiple angle formulae

Expressing \(\cos(n\theta)\) and \(\sin(n\theta)\) as polynomials of \(\sin \theta\) and \(\cos \theta\).

Let \(n\) be a positive integer. On one hand, by the De Moivre's Theorem,

\[(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta).\]

On the other hand,

\[(\cos \theta + i \sin \theta)^n = \sum_{k=0}^{n} \binom{n}{k} \cos^{n-k} \theta (i \sin \theta)^k.\]

Separating the real and imaginary parts and comparing with the real and imaginary parts of \((\cos \theta + i \sin \theta)^n\), we have that

\[\cos(n\theta)=\sum_{j=0}^{\lfloor n/2\rfloor}(-1)^j\binom n{2j}\cos^{n-2j}\theta\sin^{2j}\theta.\]

and

\[\sin(n\theta)=\sum_{j=0}^{\lfloor(n-1)/2\rfloor}(-1)^j\binom n{2j+1}\cos^{n-2j-1}\theta\sin^{2j+1}\theta.\]

Every power of \(\sin\theta\) in the cosine expression is even; the powers of \(\cos\theta\) have the same parity as \(n\). Replacing \(\sin^2\theta\) by \(1-\cos^2\theta\) expresses \(\cos(n\theta)\) as a polynomial \(T_n\) in \(\cos\theta\):

\[T_n(\cos \theta) = \cos(n\theta).\]

The name Chebyshev polynomial of the first kind is optional terminology; the binomial derivation of multiple-angle identities is the mathematical focus.

For the expression of \(\sin(n\theta)\), we may factor out \(\sin \theta\) and see that

\[\sin(n\theta)=\left[\sum_{j=0}^{\lfloor(n-1)/2\rfloor}(-1)^j\binom n{2j+1}\cos^{n-2j-1}\theta\sin^{2j}\theta\right]\sin\theta.\]

Thus, by convert all the occurrences of \(\sin^2 \theta\) into \(1 - \cos^2 \theta\) we again can express

\[\sin(n\theta)=U_{n-1}(\cos\theta)\sin\theta.\]

The polynomial \(U_{n-1}\) has degree \(n-1\); it is called a Chebyshev polynomial of the second kind. These names and the general family are enrichment, not required AA HL terminology.