If 0≤θ≤π and √3cosθ−sinθ=−1/2, find sin(θ/2+π/12).

01 / Standard#Your turn

Official paper · jm01-2022 · II.4(c) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Set u=θ/2+π/12.

Hint 2

Use cos2u=1−2sin²u, then choose the sign from u’s range.

Worked solution

The auxiliary-angle expression gives cos2u=−1/4.
$2\cos(\theta+\pi/6)=-1/2\implies\sin^2u=\frac{1+1/4}2=5/8$
u lies in [π/12,7π/12], where sine is positive.
$\sin u=\sqrt{5/8}=\frac{\sqrt{10}}4$

√10/4.

Checks and common pitfalls: This subpart has right side −1/2, different from part (b).

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The auxiliary-angle expression gives cos2u=−1/4.
$2\cos(\theta+\pi/6)=-1/2\implies\sin^2u=\frac{1+1/4}2=5/8$
u lies in [π/12,7π/12], where sine is positive.
$\sin u=\sqrt{5/8}=\frac{\sqrt{10}}4$

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

2022 JM01