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Multiple angles

The double angle formulas can be extended to larger multiples. For example, to find \(\cos 3\theta\), we write \(3\theta\) as \(2\theta + \theta\) and expand:

\[ \cos 3\theta = \cos(2\theta + \theta) = \cos2\theta\,\cos\theta - \sin2\theta\,\sin\theta. \]

We can now apply the double angle formulas to obtain

\[ \cos 3\theta = \bigl(\cos^2\theta - \sin^2\theta\bigr)\,\cos\theta - 2\sin\theta\,\cos\theta\,\sin\theta = \cos^3\theta - 3\sin^2\theta\,\cos\theta. \]

Replacing \(\sin^2\theta\) with \(1-\cos^2\theta\), we have

\[ \cos 3\theta = 4\cos^3\theta - 3\cos\theta. \]

Exercise 18

Use the method above to find a formula for \(\sin 3\theta\).

A more general approach is obtained using complex numbers. The complex number \(i\) is defined by \(i^2=-1\).

De Moivre's theorem says that, if \(n\) is a positive integer, then

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

Hence, for any given \(n\), we can expand \((\cos \theta + i\sin \theta)^n\) using the binomial theorem and equate the real and imaginary parts to find formulas for \(\cos n\theta\) and \(\sin n\theta\).

For example, in the case of \(n=3\),

\[ \bigl(\cos \theta + i\sin \theta\bigr)^3 = \cos 3\theta +i\sin 3\theta \] and \[ \bigl(\cos \theta + i\sin \theta\bigr)^3 = \cos^3\theta + 3i\cos^2\theta \sin\theta - 3\cos\theta \sin^2\theta - i\sin^3\theta. \]

Equating real and imaginary parts, plus some algebraic manipulation, will produce the triple angle formulas.

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