Exercise 828Algebra › Complex numbersHigh schoolAdded 3 Nov 2026

Linearizing powers of cosine

#De Moivre#linearization#cos

Statement

Express $\cos^3\theta$ as a linear combination of $\cos\theta$ and $\cos 3\theta$ .

Hint

Use $\cos\theta=(e^{i\theta}+e^{-i\theta})/2$ and expand.

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Solution

Expansion.
$\cos^3\theta=\left(\frac{e^{i\theta}+e^{-i\theta}}{2}\right)^3=\frac18(e^{3i\theta}+3e^{i\theta}+3e^{-i\theta}+e^{-3i\theta}).$

Regroup.
$=\frac18[(e^{3i\theta}+e^{-3i\theta})+3(e^{i\theta}+e^{-i\theta})]=\frac18[2\cos3\theta+6\cos\theta]=\frac{\cos3\theta+3\cos\theta}{4}.$

Conclusion. $\cos^3\theta=\frac{3\cos\theta+\cos3\theta}{4}$ .

Linearizing powers of cosine

Statement

Solution

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