For ω=cosα+i sinα, prove the cosine/sine power formulae and deduce the displayed product identity.

01 / Standard#Your turn

\cos n\alpha=\frac{\omega^n+\omega^{-n}}2,\quad\sin n\alpha=\frac{\omega^n-\omega^{-n}}{2i};\quad\sin^2\alpha\cos^3\alpha=\frac{2\cos\alpha-\cos3\alpha-\cos5\alpha}{16}

Official paper · jm02-2024 · 4(b) · PDF 6

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use De Moivre for positive and negative powers.

Hint 2

Expand (ω−ω⁻¹)²(ω+ω⁻¹)³ and group conjugate powers.

Worked solution

The two powers have equal real parts and opposite imaginary parts.
$\omega^{\pm n}=\cos n\alpha\pm i\sin n\alpha$
Adding and subtracting proves both requested formulae.
$\omega^n+\omega^{-n}=2\cos n\alpha,\quad\omega^n-\omega^{-n}=2i\sin n\alpha$
Expand the product using those formulae.
$\sin^2\alpha\cos^3\alpha=-\frac1{32}(\omega^2-2+\omega^{-2})(\omega^3+3\omega+3\omega^{-1}+\omega^{-3})$
Collect symmetric powers and convert back to cosine.
$=-\frac1{32}\big[(\omega^5+\omega^{-5})+(\omega^3+\omega^{-3})-2(\omega+\omega^{-1})\big]=\frac{2\cos\alpha-\cos3\alpha-\cos5\alpha}{16}$

Both power formulae and the product identity follow.

Checks and common pitfalls: The symmetric first-power pair is ω+ω⁻¹, which converts to 2cosα.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The two powers have equal real parts and opposite imaginary parts.
$\omega^{\pm n}=\cos n\alpha\pm i\sin n\alpha$
Adding and subtracting proves both requested formulae.
$\omega^n+\omega^{-n}=2\cos n\alpha,\quad\omega^n-\omega^{-n}=2i\sin n\alpha$
Expand the product using those formulae.
$\sin^2\alpha\cos^3\alpha=-\frac1{32}(\omega^2-2+\omega^{-2})(\omega^3+3\omega+3\omega^{-1}+\omega^{-3})$
Collect symmetric powers and convert back to cosine.
$=-\frac1{32}\big[(\omega^5+\omega^{-5})+(\omega^3+\omega^{-3})-2(\omega+\omega^{-1})\big]=\frac{2\cos\alpha-\cos3\alpha-\cos5\alpha}{16}$

Think first. Reveal a hint when the class is ready.

2024 JM02