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01 / Standard#Your turn

2\cos\alpha-\cos3\alpha-\cos5\alpha=0

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the product identity from part (b).

Hint 2

A product is zero when at least one factor is zero.

Worked solution

Reduce to sine and cosine zeros.
$16\sin^2\alpha\cos^3\alpha=0\iff\sin\alpha=0\ \text{or}\ \cos\alpha=0$
Merge the two families.
$\alpha=k\pi\ \text{or}\ \alpha=\pi/2+k\pi\iff\alpha=n\pi/2,\ n\in\mathbb Z$

α=nπ/2, n∈ℤ.

Checks and common pitfalls: Dividing by sine or cosine would discard a whole solution family.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Reduce to sine and cosine zeros.
$16\sin^2\alpha\cos^3\alpha=0\iff\sin\alpha=0\ \text{or}\ \cos\alpha=0$
Merge the two families.
$\alpha=k\pi\ \text{or}\ \alpha=\pi/2+k\pi\iff\alpha=n\pi/2,\ n\in\mathbb Z$

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

2024 JM02