TRIGONOMETRY REVISION QUESTIONS WITH ANSWERS

Trigonometry Questions

1. Define sine, cosine, and tangent in a right-angled triangle.

2. If \( \sin \theta = \frac{3}{5} \), find \( \cos \theta \) and \( \tan \theta \).

3. A ladder is 10 m long and makes a 60° angle with the ground. How high does it reach on the wall?

4. Given \( \cos \theta = \frac{4}{5} \), find \( \sin \theta \) and \( \tan \theta \).

5. A ramp rises 3 m vertically and is 5 m long. Find the angle of elevation.

6. Use trig ratios to find the missing side: \( \theta = 30^\circ \), hypotenuse = 12 cm, find opposite side.

7. Evaluate \( \sin 45^\circ \), \( \cos 60^\circ \), and \( \tan 30^\circ \).

8. In a right triangle, adjacent = 7 cm and angle = 40°. Find the opposite side.

9. A building casts a 15 m shadow when the angle of elevation of the sun is 45°. How tall is the building?

10. Solve \( \tan \theta = \frac{5}{12} \) for \( \theta \).

11. Find the angle \( \theta \) such that \( \sin \theta = 0.5 \).

12. Solve \( \cos \theta = 0.866 \) for \( \theta \).

13. Prove that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

14. A right triangle has sides 5 cm (adjacent), 12 cm (opposite). Find \( \tan \theta \).

15. A pole leans at 25° and touches the ground 8 m from its base. Find the length of the pole.

16. In a triangle, if \( \theta = 60^\circ \) and opposite side = 10 cm, find the hypotenuse.

17. What is the value of \( \cos 0^\circ \)?

18. What is the angle when \( \tan \theta = 1 \)?

19. Given \( \theta = 37^\circ \) and adjacent = 9 m, find the hypotenuse.

20. Solve for side \( x \) in \( \sin 53^\circ = \frac{x}{10} \).

21. In a triangle, angle = 45°, hypotenuse = 10 cm. Find adjacent side.

22. If \( \cos \theta = 0.5 \), what is \( \theta \)?

23. Prove \( \sin^2 \theta + \cos^2 \theta = 1 \).

24. A triangle has opposite = 5 cm and angle = 60°. Find hypotenuse.

25. A line of sight from a building makes an angle of 35° with the horizontal. If the observer is 20 m high, how far is the object?

26. Solve \( \cos \theta = \frac{3}{5} \), find \( \theta \).

27. What is the opposite side if \( \tan 45^\circ = \frac{x}{7} \)?

28. Calculate \( \sin 60^\circ \) using a calculator.

29. A person sees the top of a tower at 30° angle of elevation and is 50 m away. Find the height of the tower.

30. Find all trig ratios of a 45° angle.

Answers

1. Sine = Opp/Hyp, Cosine = Adj/Hyp, Tangent = Opp/Adj
2. \( \cos \theta = \frac{4}{5} \), \( \tan \theta = \frac{3}{4} \)
3. \( \text{Height} = 10 \cdot \sin 60^\circ \approx 8.66 \) m
4. \( \sin \theta = \frac{3}{5} \), \( \tan \theta = \frac{3}{4} \)
5. \( \theta = \sin^{-1}(0.6) \approx 36.87^\circ \)
6. \( \text{Opp} = 12 \cdot \sin 30^\circ = 6 \) cm
7. \( \sin 45^\circ = \frac{\sqrt{2}}{2} \), \( \cos 60^\circ = 0.5 \), \( \tan 30^\circ \approx 0.577 \)
8. \( \text{Opp} = 7 \cdot \tan 40^\circ \approx 5.88 \) cm
9. \( \text{Height} = 15 \) m
10. \( \theta = \tan^{-1}(\frac{5}{12}) \approx 22.62^\circ \)
11. \( \theta = 30^\circ \)
12. \( \theta \approx 30^\circ \)
13. \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) by definition
14. \( \tan \theta = \frac{12}{5} = 2.4 \)
15. \( \text{Length} = \frac{8}{\cos 25^\circ} \approx 8.82 \) m
16. \( \text{Hyp} = \frac{10}{\sin 60^\circ} \approx 11.55 \) cm
17. \( \cos 0^\circ = 1 \)
18. \( \theta = 45^\circ \)
19. \( \text{Hyp} = \frac{9}{\cos 37^\circ} \approx 11.24 \) m
20. \( x = 10 \cdot \sin 53^\circ \approx 7.99 \)
21. \( \text{Adj} = 10 \cdot \cos 45^\circ \approx 7.07 \) cm
22. \( \theta = 60^\circ \)
23. By identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
24. \( \text{Hyp} = \frac{5}{\sin 60^\circ} \approx 5.77 \) cm
25. \( \text{Distance} = \frac{20}{\tan 35^\circ} \approx 28.56 \) m
26. \( \theta \approx 53.13^\circ \)
27. \( x = 7 \) cm
28. \( \sin 60^\circ \approx 0.866 \)
29. \( h = 50 \cdot \tan 30^\circ \approx 28.87 \) m
30. \( \sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2} \), \( \tan 45^\circ = 1 \)