Determine f from its least positive period 3π. The printed question does not state the sign of w.

01 / Standard#Your turn

f(x)=\sqrt3\sin(2wx)-2\cos^2(wx)

Official paper · jm01-2023 · II.4(a) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use 2cos²t=1+cos2t.

Hint 2

A positive period uses the absolute frequency |2w|.

Worked solution

Combine the sine and cosine.
$f(x)=\sqrt3\sin(2wx)-\cos(2wx)-1=2\sin(2wx-\pi/6)-1$
The period determines two possible signs.
$\frac{2\pi}{|2w|}=3\pi\implies w=\pm1/3,\quad f(x)=2\sin(\pm2x/3-\pi/6)-1$

w=±1/3; the official answer selects the positive branch.

Checks and common pitfalls: A period alone determines |w|, not its sign.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Combine the sine and cosine.
$f(x)=\sqrt3\sin(2wx)-\cos(2wx)-1=2\sin(2wx-\pi/6)-1$
The period determines two possible signs.
$\frac{2\pi}{|2w|}=3\pi\implies w=\pm1/3,\quad f(x)=2\sin(\pm2x/3-\pi/6)-1$

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

2023 JM01