ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Trigonometry (Set 3): 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. The exact value of tan 75° is:
Answer: B. 2 + √3
Explanation: tan(45° + 30°) = (1 + 1/√3)/(1 − 1/√3) = (√3 + 1)/(√3 − 1) = (√3 + 1)²/2 = 2 + √3.
Q2. If sin A = 3/5 and cos B = 5/13 with A and B both acute, then sin(A + B) equals:
Answer: C. 63/65
Explanation: cos A = 4/5, sin B = 12/13. sin(A + B) = (3/5)(5/13) + (4/5)(12/13) = 15/65 + 48/65 = 63/65.
Q3. If tan θ = 1/3, then cos 2θ equals:
Answer: D. 4/5
Explanation: cos 2θ = (1 − tan²θ)/(1 + tan²θ) = (8/9)/(10/9) = 4/5.
Q4. The value of sin⁴(π/8) + cos⁴(π/8) is:
Answer: D. 3/4
Explanation: sin⁴x + cos⁴x = 1 − ½sin²2x. With 2x = π/4, sin²2x = ½, so the value is 1 − ¼ = 3/4.
Q5. The value of sin 20° · sin 40° · sin 80° is:
Answer: C. √3/8
Explanation: Using sin x·sin(60° − x)·sin(60° + x) = ¼ sin 3x with x = 20°: ¼ sin 60° = √3/8.
Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.
Q6. The value of cos 20° · cos 40° · cos 80° is:
Q7. The period of f(x) = sin 3x + cos 2x is:
Q8. The maximum value of 7 sin x − 24 cos x is:
Q9. The range of the function f(x) = 5 + 3 cos 2x is:
Q10. An arc subtends an angle of 150° at the centre of a circle of radius 12 cm. The arc length is:
Q11. A sector of a circle of radius 4 cm has an arc length of 10 cm. Its area is:
Q12. The value of cos⁻¹(cos 7π/6) is:
Q13. The value of tan⁻¹(1/2) + tan⁻¹(1/5) + tan⁻¹(1/8) is:
Q14. The value of tan(tan⁻¹(1/2) + tan⁻¹(1/3)) is:
Q15. The value of cos(2 sin⁻¹(3/5)) is:
Q16. The sum of all solutions of sin 2x = ½ in the interval [0, π] is:
Q17. The number of solutions of 2cos²θ + 3 sin θ = 3 in the interval [0, 2π) is:
Q18. In triangle ABC, a = 10, A = 30° and B = 45°. Then side b equals:
Q19. Two sides of a triangle measure 6 cm and 10 cm and the angle between them is 120°. The third side is:
Q20. A triangle has sides 9, 10 and 17. Its in-radius is:
Q21. The circum-radius of the triangle with sides 9, 10 and 17 is:
Q22. In a triangle with sides 7, 8 and 13, the angle opposite the longest side is:
Q23. From a point on level ground the angle of elevation of the top of a tower is 60°. From a point 40 m farther away in the same line the angle of elevation is 30°. The height of the tower is:
Q24. If sin x = 1/3, then sin 3x equals:
Q25. If sin x + cos x = ½, then sin 2x equals:
Q26. In a triangle ABC, tan A = 1 and tan B = 2. Then tan C equals:
Q27. If cos θ = −3/5 and θ lies in the third quadrant, then sin 2θ equals:
Q28. If cos θ = 3/5 and θ is acute, then tan(θ/2) equals:
Q29. The value of sin 75° + sin 15° is:
Q30. The minimum value of 2sin²x + 3cos²x is: