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TRIGONOMETRY REVISION QUESTIONS WITH ANSWERS

Trigonometry Practice

Trigonometry Questions for High School Students

Attempt the following 30 questions covering various trigonometry topics including definitions, Pythagoras' Theorem, compound angles, elevation and depression, sine rule, and cosine rule.

1. Define \( \sin, \cos, \tan \) in terms of the sides of a right triangle.
2. If \( \sin \theta = \frac{3}{5} \) and \( \theta \) is acute, find \( \cos \theta \) and \( \tan \theta \).
3. In a right triangle, one leg is 6 cm and the hypotenuse is 10 cm. Find the other leg.
4. Use Pythagoras' Theorem to determine the length of the diagonal of a rectangle of sides 5 cm and 12 cm.
5. Prove the identity: \( \sin^2\theta + \cos^2\theta = 1 \).
6. Find \( \sin(45^\circ) \), \( \cos(45^\circ) \), and \( \tan(45^\circ) \).
7. Use compound angle formula to find \( \sin(75^\circ) \).
8. Prove that \( \cos(A - B) = \cos A \cos B + \sin A \sin B \).
9. If \( \tan A = 3 \), find \( \tan(2A) \).
10. A ladder leans against a wall making a \( 60^\circ \) angle with the ground. The ladder is 8 m long. How high up the wall does it reach?
11. An observer sees the top of a tower at an angle of elevation of \( 30^\circ \) from a distance of 50 m. Find the height of the tower.
12. A kite is flying at a height of 30 m. The string makes an angle of \( 40^\circ \) with the ground. Find the length of the string.
13. A person standing on top of a hill observes a boat in the sea at a depression angle of \( 25^\circ \). If the hill is 100 m high, find the distance from the person to the boat.
14. In triangle ABC, \( AB = 7 \), \( AC = 9 \), and angle \( A = 60^\circ \). Use the cosine rule to find \( BC \).
15. In triangle PQR, side \( PQ = 10 \), \( PR = 6 \), and angle \( QPR = 45^\circ \). Use the cosine rule to find side \( QR \).
16. In triangle XYZ, angle \( X = 40^\circ \), angle \( Y = 60^\circ \), and side \( x = 15 \). Use the sine rule to find side \( y \).
17. A triangle has angles \( A = 30^\circ \), \( B = 45^\circ \), and side \( a = 10 \). Find side \( b \) using the sine rule.
18. Solve the triangle with \( a = 8 \), \( b = 10 \), and angle \( C = 90^\circ \) using sine and cosine rules.
19. Find the height of a tree if it casts a 20 m long shadow when the angle of elevation of the sun is \( 30^\circ \).
20. Express \( \tan(75^\circ) \) using the formula \( \tan(A+B) \).
21. Find \( \cos(105^\circ) \) using \( \cos(60^\circ + 45^\circ) \).
22. Use the identity \( \tan^2\theta + 1 = \sec^2\theta \) to find \( \sec\theta \) if \( \tan\theta = 2 \).
23. Simplify \( \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \).
24. If \( \cos A = \frac{4}{5} \), find \( \sin A \) and \( \tan A \).
25. A plane is flying at an altitude of 1000 m and observes an airport at a depression angle of \( 20^\circ \). Find the ground distance to the airport.
26. Prove: \( 1 + \tan^2x = \sec^2x \).
27. Prove: \( 1 + \cot^2x = \csc^2x \).
28. Find \( \sin(2x) \) if \( \sin x = \frac{5}{13} \) and \( x \) is acute.
29. Given \( \tan x = \frac{3}{4} \), find \( \sin(2x) \) and \( \cos(2x) \).
30. In triangle ABC, angle A = 90°, AB = 9, and AC = 12. Find BC.

Answers

  1. \( \sin = \frac{\text{opposite}}{\text{hypotenuse}}, \cos = \frac{\text{adjacent}}{\text{hypotenuse}}, \tan = \frac{\text{opposite}}{\text{adjacent}} \)
  2. \( \cos \theta = \frac{4}{5}, \tan \theta = \frac{3}{4} \)
  3. 8 cm
  4. 13 cm
  5. Using identity: \( \sin^2\theta + \cos^2\theta = 1 \)
  6. \( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 1 \)
  7. \( \sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin45^\circ \cos30^\circ + \cos45^\circ \sin30^\circ \)
  8. Using compound angle identity.
  9. \( \tan(2A) = \frac{2 \cdot 3}{1 - 3^2} = \frac{6}{-8} = -\frac{3}{4} \)
  10. \( 8 \sin 60^\circ = 6.93 \) m approx.
  11. \( h = 50 \tan 30^\circ = 28.87 \) m
  12. \( \frac{30}{\sin 40^\circ} \approx 46.7 \) m
  13. \( 100 \csc 25^\circ \approx 236.6 \) m
  14. Use cosine rule: \( BC^2 = AB^2 + AC^2 - 2 \cdot AB \cdot AC \cdot \cos A \)
  15. Use cosine rule formula.
  16. Use sine rule.
  17. Use sine rule.
  18. Use sine and cosine rules to find third side and angles.
  19. \( h = 20 \tan 30^\circ = 11.55 \) m
  20. \( \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \cdot \tan 30^\circ} \)
  21. Use compound angle formula.
  22. \( \sec\theta = \sqrt{1 + 4} = \sqrt{5} \)
  23. \( \frac{\sin^2x + \cos^2x}{\sin x \cos x} = \frac{1}{\sin x \cos x} \)
  24. \( \sin A = \frac{3}{5}, \tan A = \frac{3}{4} \)
  25. \( 1000 \cot 20^\circ \approx 2746.5 \) m
  26. Proven using identity
  27. Proven using identity
  28. \( \sin(2x) = 2 \cdot \frac{5}{13} \cdot \frac{12}{13} = \frac{120}{169} \)
  29. Use identities for \( \sin 2x \) and \( \cos 2x \)
  30. \( BC = \sqrt{9^2 + 12^2} = 15 \)

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