SPM 2022 Add Maths Paper 1, Question 13Trigonometric Functions

Solved on video in English and Bahasa Melayu · Form 5, Trigonometric Functions · 6:44 video

The question

(a)(i)

13 (a) Diagram 9 shows angle theta on a Cartesian plane (the line from the origin O to the point (p, q) in the first quadrant makes an angle theta with the positive x-axis). (i) On Diagram 9, label the position of −theta.

(a)(ii)

13 (a)(ii) State the value of tan(−theta) in terms of p and q (where the line from O makes angle theta with the positive x-axis through the point (p, q)).

(b)

13 (b) Solve the equation 2 cos x = sqrt(3) cot x for 0° ≤ x ≤ 360°.

(c)

13 (c) It is given that tan m = p, such that m is a reflex angle. Express cos(pi/3 − m) in terms of p.

SPM 2022 Add Maths Paper 1, Question 13: the question as drawn in Eduly's video6:44

Part (a)(i)

Step 1

The angle −theta is measured clockwise from the positive x-axis (below the x-axis), the reflection of theta in the x-axis.

Why this step

The minus sign means turn the opposite way. Same size as θ, but clockwise, so it mirrors θ across the x-axis.

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

Final answer

−θ below the x-axis, point (p, −q)

Where students lose marks

Watch out: label the point (p, −q) as well as the angle −θ, not just the angle.

Part (a)(ii)

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

Final answer

tan(−θ) = −q/p

Where students lose marks

Watch out: −θ only flips the sign, so tan(−θ) = −q/p, not the reciprocal p/q.

Part (b)

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

Final answer

x = 60°, 90°, 120°, 270°

Where students lose marks

Watch out: never divide both sides by cos x. You lose x = 90° and 270°. Factorise instead.

Part (c)

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

Final answer

cos(π/3 − m) = (−1 − √3·p) / (2√(p² + 1))

Where students lose marks

Watch out: in the third quadrant sin m and cos m are both negative, so check every sign in your triangle.

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