a. [4 marks]

Use \( \cos 3\theta = \cos(2\theta + \theta) = \cos 2\theta \cos \theta – \sin 2\theta \sin \theta \) (M1).
Apply double angle identities: \( \cos 2\theta = 2 \cos^2 \theta – 1 \), \( \sin 2\theta = 2 \sin \theta \cos \theta \) (A1).
Substitute: \( \cos 3\theta = (2 \cos^2 \theta – 1) \cos \theta – (2 \sin \theta \cos \theta) \sin \theta \).
Simplify: \( = 2 \cos^3 \theta – \cos \theta – 2 \sin^2 \theta \cos \theta \).
Use \( \sin^2 \theta = 1 – \cos^2 \theta \): \( = 2 \cos^3 \theta – \cos \theta – 2 (1 – \cos^2 \theta) \cos \theta \) (A1).
Combine: \( = 2 \cos^3 \theta – \cos \theta – 2 \cos \theta + 2 \cos^3 \theta = 4 \cos^3 \theta – 3 \cos \theta \) (A1).
Answer: \( \cos 3\theta = 4 \cos^3 \theta – 3 \cos \theta \).

b. [4 marks]

Use \( \sin 3\theta = \sin(2\theta + \theta) = \sin 2\theta \cos \theta + \cos 2\theta \sin \theta \) (M1).
Apply double angle identities: \( \sin 2\theta = 2 \sin \theta \cos \theta \), \( \cos 2\theta = 1 – 2 \sin^2 \theta \) (A1).
Substitute: \( \sin 3\theta = (2 \sin \theta \cos \theta) \cos \theta + (1 – 2 \sin^2 \theta) \sin \theta \).
Simplify: \( = 2 \sin \theta \cos^2 \theta + \sin \theta – 2 \sin^3 \theta \).
Use \( \cos^2 \theta = 1 – \sin^2 \theta \): \( = 2 \sin \theta (1 – \sin^2 \theta) + \sin \theta – 2 \sin^3 \theta \) (A1).
Combine: \( = 2 \sin \theta – 2 \sin^3 \theta + \sin \theta – 2 \sin^3 \theta = 3 \sin \theta – 4 \sin^3 \theta \) (A1).
Answer: \( \sin 3\theta = 3 \sin \theta – 4 \sin^3 \theta \).

c. [3 marks]

Given \( \cos 3\theta = 11 \cos^2 \theta \), substitute \( \cos 3\theta = 4 \cos^3 \theta – 3 \cos \theta \) (M1).
Equation becomes: \( 4 \cos^3 \theta – 3 \cos \theta = 11 \cos^2 \theta \).
Rearrange: \( 4 \cos^3 \theta – 11 \cos^2 \theta – 3 \cos \theta = 0 \).
Factor: \( \cos \theta (4 \cos^2 \theta – 11 \cos \theta – 3) = 0 \) (A1).
Solutions: \( \cos \theta = 0 \) or solve \( 4 \cos^2 \theta – 11 \cos \theta – 3 = 0 \).
Quadratic formula for \( u = \cos \theta \): \( u = \frac{11 \pm \sqrt{121 + 48}}{8} = \frac{11 \pm 13}{8} \), so \( u = 3 \) (invalid, as \( | \cos \theta | \leq 1 \)) or \( u = -\frac{1}{4} \) (A1).
Answer: \( \cos \theta = 0, -\frac{1}{4} \).

d. [3 marks]

From part (c), use \( \cos \theta = 0, -\frac{1}{4} \) for \( 0 < \theta < \pi \).
If \( \cos \theta = 0 \), then \( \theta = \frac{\pi}{2} \) (A1).
If \( \cos \theta = -\frac{1}{4} \), then \( \theta = \cos^{-1} \left(-\frac{1}{4}\right) \approx 1.823 \) radians (A1).
Verify: For \( \theta = \frac{\pi}{2} \), \( \cos 3\theta = \cos \frac{3\pi}{2} = 0 \), \( 11 \cos^2 \theta = 11 \cdot 0 = 0 \). For \( \theta = \cos^{-1} \left(-\frac{1}{4}\right) \), \( \cos 3\theta = 4 \left(-\frac{1}{4}\right)^3 – 3 \left(-\frac{1}{4}\right) = -\frac{1}{16} + \frac{3}{4} = \frac{11}{16} \), and \( 11 \cos^2 \theta = 11 \cdot \frac{1}{16} = \frac{11}{16} \) (M1).
Answer: \( \theta = \frac{\pi}{2}, \cos^{-1} \left(-\frac{1}{4}\right) \).