Write √3cosθ−sinθ as r cos(θ+β), and give its range.

01 / Standard#Your turn

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

The amplitude is √(3+1).

Hint 2

Match cosine and sine coefficients in the addition formula.

Worked solution

2cos(θ+π/6), range [−2,2].

Checks and common pitfalls: The minus sine coefficient corresponds to a positive phase inside cosine.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Choose r=2 and β=π/6.
$2\cos(\theta+\pi/6)=\sqrt3\cos\theta-\sin\theta$
Use the full cosine range.
$-2\le\sqrt3\cos\theta-\sin\theta\le2$

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

2022 JM01