1. The value of sin 30° is:
A. 1/2
B. √3/2
C. 1/√2
D. 1
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Answer: A. 1/2
Explanation: Standard trigonometric angle value: sin 30° = 1/2.
2. If sin θ = 3/5, then cos θ is equal to:
A. 4/5
B. 5/3
C. 3/4
D. 4/3
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Answer: A. 4/5
Explanation: cos θ = √(1 - sin² θ) = √(1 - (3/5)²) = √(1 - 9/25) = √(16/25) = 4/5.
3. The value of sin² 60° + cos² 60° is:
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Answer: B. 1
Explanation: By fundamental identity sin² θ + cos² θ = 1 for any angle θ.
4. The value of tan 45° is:
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Answer: C. 1
Explanation: Standard value: tan 45° = sin 45° / cos 45° = (1/√2) / (1/√2) = 1.
5. If tan A = 4/3, then sec A is equal to:
A. 5/3
B. 3/5
C. 5/4
D. 4/5
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Answer: A. 5/3
Explanation: sec² A = 1 + tan² A = 1 + (4/3)² = 1 + 16/9 = 25/9 ⇒ sec A = 5/3.
6. The maximum value of sin θ for 0° ≤ θ ≤ 90° is:
A. 0
B. 1/2
C. 1
D. Infinite
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Answer: C. 1
Explanation: In the interval 0° ≤ θ ≤ 90°, the maximum value reached by sin θ is 1 (at θ = 90°).
7. The value of (1 + tan² θ)(1 - sin θ)(1 + sin θ) is:
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Answer: B. 1
Explanation: (1 + tan² θ)(1 - sin² θ) = sec² θ × cos² θ = (1/cos² θ) × cos² θ = 1.
8. If 15 cot A = 8, then sin A is equal to:
A. 15/17
B. 8/17
C. 17/15
D. 17/8
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Answer: A. 15/17
Explanation: cot A = 8/15 (Adjacent/Opposite) ⇒ Hypotenuse = √(8² + 15²) = √(64 + 225) = √289 = 17. sin A = 15/17.
9. The value of 9 sec² A - 9 tan² A is:
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Answer: B. 9
Explanation: 9(sec² A - tan² A) = 9(1) = 9 using identity sec² A - tan² A = 1.
10. If cos θ = 1/2, then the value of 2 sec θ is:
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Answer: C. 4
Explanation: sec θ = 1/cos θ = 2. So, 2 sec θ = 2 × 2 = 4.
11. The value of (sin 30° + cos 60°) is:
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Answer: B. 1
Explanation: sin 30° + cos 60° = 1/2 + 1/2 = 1.
12. Which of the following is true for 0° ≤ θ ≤ 90°?
A. cos θ increases as θ increases
B. sin θ increases as θ increases
C. tan θ decreases as θ increases
D. cot θ increases as θ increases
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Answer: B. sin θ increases as θ increases
Explanation: sin θ goes from 0 to 1 as θ increases from 0° to 90°.
13. If sin A = cos A, then the angle A is equal to:
A. 0°
B. 30°
C. 45°
D. 60°
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Answer: C. 45°
Explanation: sin A / cos A = 1 ⇒ tan A = 1 ⇒ A = 45°.
14. The value of (1 - tan² 45°) / (1 + tan² 45°) is:
A. tan 90°
B. 1
C. sin 45°
D. 0
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Answer: D. 0
Explanation: Since tan 45° = 1, (1 - 1²) / (1 + 1²) = 0 / 2 = 0.
15. If sin 2A = 2 sin A is true, then A must be equal to:
A. 0°
B. 30°
C. 45°
D. 60°
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Answer: A. 0°
Explanation: At A = 0°, sin(0°) = 0 and 2 sin(0°) = 0, so RHS = LHS.
16. The value of 2 tan 30° / (1 + tan² 30°) is equal to:
A. sin 60°
B. cos 60°
C. tan 60°
D. sin 30°
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Answer: A. sin 60°
Explanation: [2(1/√3)] / [1 + (1/3)] = (2/√3) / (4/3) = (2/√3) × (3/4) = √3/2 = sin 60°.
17. Value of 2 tan 30° / (1 - tan² 30°) is equal to:
A. cos 60°
B. sin 60°
C. tan 60°
D. sin 30°
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Answer: C. tan 60°
Explanation: [2(1/√3)] / [1 - (1/3)] = (2/√3) / (2/3) = √3 = tan 60°.
18. If tan (A + B) = √3 and tan (A - B) = 1/√3 (0° < A + B ≤ 90°, A > B), then values of A and B are:
A. 45°, 15°
B. 60°, 30°
C. 30°, 15°
D. 45°, 30°
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Answer: A. 45°, 15°
Explanation: A + B = 60° and A - B = 30°. Adding gives 2A = 90° ⇒ A = 45° and B = 15°.
19. Trigonometric ratio cosec θ is defined as the reciprocal of:
A. cos θ
B. tan θ
C. sin θ
D. sec θ
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Answer: C. sin θ
Explanation: cosec θ = 1 / sin θ.
20. The identity 1 + cot² A = cosec² A holds true for all angles A such that:
A. 0° < A ≤ 90°
B. 0° ≤ A < 90°
C. 0° ≤ A ≤ 90°
D. 0° < A < 90°
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Answer: A. 0° < A ≤ 90°
Explanation: cot A and cosec A are undefined at A = 0°, so it holds for 0° < A ≤ 90°.
21. If cos A = 4/5, then tan A is:
A. 3/5
B. 3/4
C. 4/3
D. 5/3
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Answer: B. 3/4
Explanation: sin A = √(1 - 16/25) = 3/5. tan A = sin A / cos A = (3/5) / (4/5) = 3/4.
22. The value of (sin 0° × cos 30° × tan 45° × cosec 60°) is equal to:
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Answer: B. 0
Explanation: Since sin 0° = 0, product of any finite values with sin 0° equals 0.
23. In ΔABC right angled at B, if tan A = 1, then 2 sin A cos A is equal to:
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Answer: A. 1
Explanation: tan A = 1 ⇒ A = 45°. 2 sin 45° cos 45° = 2 × (1/√2) × (1/√2) = 2 × 1/2 = 1.
24. (sec A + tan A)(1 - sin A) is equal to:
A. sec A
B. sin A
C. cosec A
D. cos A
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Answer: D. cos A
Explanation: [(1 + sin A)/cos A] (1 - sin A) = (1 - sin² A)/cos A = cos² A / cos A = cos A.
25. (1 + tan² A) / (1 + cot² A) is equal to:
A. sec² A
B. -1
C. cot² A
D. tan² A
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Answer: D. tan² A
Explanation: (1 + tan² A) / (1 + 1/tan² A) = (1 + tan² A) / [(tan² A + 1)/tan² A] = tan² A.
26. If sin θ + cos θ = √2 cos θ, then tan θ is equal to:
A. √2 - 1
B. √2 + 1
C. 1/√2
D. √2
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Answer: A. √2 - 1
Explanation: Divide by cos θ: tan θ + 1 = √2 ⇒ tan θ = √2 - 1.
27. If x = a cos θ and y = b sin θ, then (x²/a²) + (y²/b²) is equal to:
A. 1
B. 0
C. a² b²
D. a² + b²
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Answer: A. 1
Explanation: (a² cos² θ / a²) + (b² sin² θ / b²) = cos² θ + sin² θ = 1.
28. Value of cos 0° · cos 1° · cos 2° · ... · cos 90° is:
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Answer: C. 0
Explanation: Since cos 90° = 0, the product of all terms becomes 0.
29. In a right-angled triangle ABC, right-angled at C, the value of cos(A + B) is:
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Answer: A. 0
Explanation: Angle C = 90° ⇒ A + B = 180° - 90° = 90°. cos(A + B) = cos 90° = 0.
30. If sin α = 1/2 and cos β = 1/2, then value of (α + β) is:
A. 0°
B. 30°
C. 60°
D. 90°
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Answer: D. 90°
Explanation: sin α = 1/2 ⇒ α = 30°; cos β = 1/2 ⇒ β = 60°. Therefore, α + β = 30° + 60° = 90°.
