SPM 2019 Add Maths Paper 2, Question 10Trigonometric Functions

Solved on video in English and Bahasa Melayu · 10 marks · Form 5, Trigonometric Functions · 5:28 video

The question

(a)(i) Prove that tan(A/2) = (1 - cos A)/sin A. (a)(ii) Hence, without using a calculator, find the value of tan 15 degrees. State your answer in the form p - sqrt(q), where p and q are constants. (b)(i) Sketch the graph of y = -(3/2)sin A for 0 <= A <= 2pi. (b)(ii) Hence, using the same axes, sketch a suitable straight line to find the number of solutions for the equation (cot(A/2))(1-cos A) = -A/(2pi) for 0 <= A <= 2*pi. State the number of solutions.

[10 marks]

SPM 2019 Add Maths Paper 2, Question 10: the question as drawn in Eduly's video5:28

Step 1

RHS = (1-cos A)/sin A. Using cos A = 1 - 2sin²(A/2) and sin A = 2 sin(A/2)cos(A/2):

Why this step

Pick the sin² form of cos A: the 1s cancel and 2sin²(A/2) appears at once. Mixing up 1 − 2sin²(A/2) with 2cos²(A/2) − 1 is a common slip.

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

Final answer

(a)(i) Proved; (a)(ii) tan 15° = 2 − √3 (p = 2, q = 3); (b)(i) Sketch (see video); (b)(ii) 3 solutions

Where students lose marks

Watch out: use the right double-angle form of cos A, and multiply the equation by −3/2 so it matches the sketch before counting crossings.

Watch every step of this question solved on video (5:28), with the reason behind each step and a box that checks your own answer.

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