Basic Trigonometric Problems

  1. In a right triangle ABC right-angled at B, if AB = 6 cm and BC = 8 cm, find sin A, cos A, and tan A.
  2. If sin θ = 12/13, find the value of cos θ and tan θ (θ is acute).
  3. Evaluate: sin 60° × cos 30° + cos 60° × sin 30°.
  4. If cos A = 4/5, find sin A and sec A.
  5. Prove that: (sin θ + cos θ)² + (sin θ – cos θ)² = 2.
  6. If tan θ = √3, find the value of θ.
  7. Evaluate: tan 45° + cot 45°.
  8. If sin A = 3/5, find cosec A.
  9. Prove that: 1 + tan² θ = sec² θ.
  10. If cot θ = 1, find sin θ + cos θ.
  11. Simplify: sin⁴ θ – cos⁴ θ.
  12. If sec θ = 13/5, find tan θ.
  13. Evaluate: cos 30° / sin 60°.
  14. Prove that: sin(90° – θ) = cos θ.
  15. If sin θ + cos θ = √2, find tan θ.
  16. Find the value of sin 15° using angle formulas (or identity).
  17. If tan A + cot A = 2, find tan² A + cot² A.
  18. Prove that: cosec² θ – cot² θ = 1.
  19. If 3 cot θ = 4, find (5 sin θ – 3 cos θ)/(5 sin θ + 3 cos θ).
  20. Evaluate: sin² 30° + cos² 60° – tan² 45°.
  21. If sin θ = cos θ, find the value of θ (acute).
  22. Simplify: (1 – sin² θ)/(cos² θ).
  23. Prove that: tan θ / (1 – cot θ) + cot θ / (1 – tan θ) = 1 + sec θ cosec θ.
  24. If cosec θ – cot θ = 7/√53, find cos θ.
  25. Evaluate: (sin 67° + cos 23°).
  26. In △ABC, ∠B = 90°, AB = 7 cm, AC = 25 cm. Find tan C.
  27. If sin(A + B) = 1 and cos(A – B) = 1/√2, find A and B.
  28. Prove that: (cosec θ – cot θ)² = (1 – cos θ)/(1 + cos θ).
  29. If tan θ = 1/√3, find sin 3θ.
  30. Simplify: √[(1 – cos θ)/(1 + cos θ)].
  31. If √3 tan θ = 1, find cos 2θ.
  32. Prove that: sin θ / (1 + cos θ) + (1 + cos θ) / sin θ = 2 cosec θ.
  33. Evaluate: 2 sin 30° cos 30°.
  34. If sin θ = p/q, prove that √(1 – sin² θ) = √(q² – p²)/q.
  35. Find the value of cos 0° × cos 30° × cos 45° × cos 60° × cos 90°.
  36. If cot A = 12/5, find sin A + cos A.
  37. Prove that: (1 + cot² θ) sin² θ = 1.
  38. If tan θ + sec θ = 3, find sin θ.
  39. Simplify: (tan θ + cot θ) sin θ cos θ.
  40. If sin(A + B) = sin(A – B), prove that tan A tan B = 1 or A = B.
  41. Evaluate: sin² 18° + cos² 72°.
  42. If cosec θ = √2, find cot θ.
  43. Prove that: sin 3θ = 3 sin θ – 4 sin³ θ (for θ = 30° verify).
  44. If 4 sin² θ – 3 = 0, find cos 2θ.
  45. Simplify: (sin θ + cos θ)² – 2 sin θ cos θ.
  46. If tan θ = m/n, prove that sin θ = m/√(m² + n²).
  47. Evaluate: cot 30° – tan 60° + sec 90° (careful!).
  48. Prove that: tan(45° + θ) – tan(45° – θ) = 2 tan 2θ / (1 – tan² 2θ) wait, actually standard: use identity.
  49. If sin θ / cos θ + cos θ / sin θ = 5, find sin² θ + cos² θ (trick).
  50. From a point on ground, angle of elevation of top of pole is 60°. After moving 40 m towards pole, it becomes 30°. Find height of pole.