MRSM Trial 2025 Add Maths Paper 2, Question 4Trigonometric Functions

Solved on video in English · Form 5, Trigonometric Functions · 7:50 video

The question

(a)(i)

(a)(i) Prove that (2 sin θ cos³θ)/(1 − sin²θ) = sin 2θ.

(a)(ii)

Given that (2 sin θ cos³θ)/(1 − sin²θ) = sin 2θ:

(a)(ii) Hence, solve the equation 8 sin θ cos³θ = 3 − 3 sin²θ for 0° ≤ θ ≤ 270°.

(b)(i)

Diagram 3 shows a trigonometric function graph over 0 ≤ x ≤ 2π, with a maximum of 6, a minimum of −4 and a y-intercept of 1.

(b)(i) State (a) the range for the domain 0 ≤ x ≤ 2π, (b) the function represented by the graph.

(b)(ii)

(b)(ii) If a straight line y = k is drawn on the same axes, where k is a constant, state the possible number of solutions.

MRSM Trial 2025 Add Maths Paper 2, Question 4: the question as drawn in Eduly's video7:50

Part (a)(i)

Step 1

Use the identity 1 − sin²θ = cos²θ on the denominator: LHS = (2 sin θ cos³θ)/cos²θ.

Why this step

The denominator is an identity in disguise. Swapping it for cos²θ lets it cancel with the cos³θ on top.

2 more steps, each with the reason for it, in the full solution

Final answer

Shown

Where students lose marks

Watch out: recognise 1 − sin²θ = cos²θ, then cancel fully to 2 sin θ cos θ before applying the double-angle identity.

Part (a)(ii)

5 more steps, each with the reason for it, in the full solution

Final answer

θ = 24.30°, 65.71°, 204.30°, 245.71°

Where students lose marks

Watch out: 2θ goes up to 540°, not 270°. Extend the range first or you miss θ = 204.30° and 245.71°.

Part (b)(i)

3 more steps, each with the reason for it, in the full solution

Final answer

Range: −4 ≤ y ≤ 6; y = 5 sin(3/2 x) + 1

Where students lose marks

Watch out: 1.5 waves in 2π means the coefficient of x is 3/2, so sin(3/2 x), not sin x or sin 2x.

Part (b)(ii)

2 more steps, each with the reason for it, in the full solution

Final answer

0, 1, 2, 4

Where students lose marks

Watch out: 3 is impossible. Include the single case k = −4 (1 solution) and k = 6 (2 solutions) in your list.

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