ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Trigonometry (Set 4): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30 ECAT…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. Evaluate sin 22.5° cos 22.5°.
Answer: B. √2/4
Explanation: sin 22.5° cos 22.5° = ½ sin 45° = ½ × √2/2 = √2/4.
Q2. The exact value of tan 22.5° is:
Answer: C. √2 − 1
Explanation: tan(θ/2) = (1 − cos θ)/sin θ with θ = 45° gives (1 − 1/√2)/(1/√2) = √2 − 1.
Q3. The sum of all solutions of 2sin²θ − 3sin θ + 1 = 0 in the interval [0, 2π) is:
Answer: C. 3π/2
Explanation: Factorising (2sin θ − 1)(sin θ − 1) = 0 gives sin θ = 1/2 or 1, so θ = π/6, 5π/6, π/2. The sum is π/6 + 5π/6 + π/2 = 3π/2.
Q4. The number of solutions of cos θ + sin θ = 1 in [0, 2π) is:
Answer: A. 2
Explanation: Dividing by √2: sin(θ + π/4) = 1/√2, so θ + π/4 = π/4 or 3π/4, giving θ = 0 or π/2.
Q5. The value of tan(cos⁻¹(4/5) + tan⁻¹(3/4)) is:
Answer: D. 24/7
Explanation: Both angles equal θ with tan θ = 3/4 (cos θ = 4/5). So the sum is 2θ and tan 2θ = 2(3/4)/(1 − 9/16) = (3/2)/(7/16) = 24/7.
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Q6. The value of tan⁻¹ 2 + tan⁻¹ 3 is:
Q7. In triangle ABC, a = 10, A = 30° and B = 45°. The length of side b is:
Q8. A triangle has sides 3, 5 and 7. Find the angle opposite the longest side.
Q9. A triangle has sides 13, 14 and 15. Its inradius is:
Q10. A triangle has sides 13, 14 and 15. Its circumradius is:
Q11. A sector of a circle has radius 6 cm and central angle 150°. Its area is:
Q12. The minute hand of a clock is 14 cm long. Taking π = 22/7, the distance covered by its tip in 15 minutes is:
Q13. From two points on level ground in a line with the foot of a tower and 40 m apart, the angles of elevation of its top are 30° and 60°. The height of the tower is:
Q14. If sin θ = 1/3, then sin 3θ equals:
Q15. Evaluate sin 75° − sin 15°.
Q16. The value of 2 sin 105° cos 15° is:
Q17. Evaluate cos 36° − cos 72°.
Q18. The value of tan 20° tan 40° tan 80° is:
Q19. The period of f(x) = sin 4x + cos 6x is:
Q20. The period of f(x) = cos(πx/3) is:
Q21. The range of f(x) = 3 sin x − 4 cos x + 2 is:
Q22. The maximum value of 5 cos x + 12 sin x + 3 is:
Q23. The expression sin(π + θ)cos(π/2 + θ) ÷ [cos(π − θ)sin(π/2 − θ)] simplifies to:
Q24. If tan θ = −5/12 and θ lies in the second quadrant, the value of sin θ − cos θ is:
Q25. If sec θ + tan θ = 3, then sec θ equals:
Q26. The sum of all solutions of cos 2θ + 3 cos θ + 2 = 0 in [0, 2π) is:
Q27. In triangle ABC, sin A : sin B : sin C = 3 : 4 : 5. The triangle is:
Q28. A triangle has sides 8 and 5 including an angle of 150°. Its area is:
Q29. In triangle ABC, b = 3, c = 5 and A = 60°. The length of side a is:
Q30. The principal value of sin⁻¹(sin 7π/6) is: