Find cos(α+2β).

01 / Standard#Your turn

0<\alpha,\beta<\frac\pi2,\quad\tan\alpha=\frac15,\quad\cos\beta=\frac{3\sqrt{13}}{13}

Official paper · jm01-2024 · II.2(b) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use part (a) and 0<α+β<π to identify the sum.

Hint 2

Write α+2β=(α+β)+β.

Worked solution

Tangent equals one only at π/4 in this range.
$\alpha+\beta=\pi/4$
Apply cosine addition.
$\cos(\alpha+2\beta)=\frac{\sqrt2}2\left(\frac{3\sqrt{13}}{13}-\frac{2\sqrt{13}}{13}\right)=\frac{\sqrt{26}}{26}$

√26/26.

Checks and common pitfalls: Do not infer an angle from tangent without its range.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Tangent equals one only at π/4 in this range.
$\alpha+\beta=\pi/4$
Apply cosine addition.
$\cos(\alpha+2\beta)=\frac{\sqrt2}2\left(\frac{3\sqrt{13}}{13}-\frac{2\sqrt{13}}{13}\right)=\frac{\sqrt{26}}{26}$

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

2024 JM01