20 Law Of Sines Quiz Questions and Answers

1. Question: In a triangle, angle A is 30°, angle B is 45°, and side a is 10 units. What is the length of side b?
Options:
A) 14.14 units
B) 10 units
C) 7.07 units
D) 5 units
Answer: A
Explanation: Using the Law of Sines, \(\frac{b}{\sin 45^\circ} = \frac{10}{\sin 30^\circ}\). So, \(b = 10 \times \frac{\sin 45^\circ}{\sin 30^\circ} = 10 \times \frac{0.707}{0.5} = 14.14\) units.

2. Question: In triangle ABC, side a = 5, side b = 7, and angle A = 40°. What is angle B?
Options:
A) 56.44°
B) 40°
C) 83.56°
D) 30°
Answer: A
Explanation: By the Law of Sines, \(\frac{7}{\sin B} = \frac{5}{\sin 40^\circ}\). So, \(\sin B = \frac{7 \times \sin 40^\circ}{5} = \frac{7 \times 0.643}{5} = 0.835\). Thus, angle B ≈ 56.44°.

3. Question: For a triangle with sides a = 8, b = 10, and angle A = 50°, what is angle B?
Options:
A) 63.43°
B) 50°
C) 36.87°
D) 90°
Answer: A
Explanation: Using the Law of Sines, \(\frac{10}{\sin B} = \frac{8}{\sin 50^\circ}\). So, \(\sin B = \frac{10 \times \sin 50^\circ}{8} = \frac{10 \times 0.766}{8} = 0.958\). Thus, angle B ≈ 63.43°.

4. Question: In triangle ABC, angle C = 90°, side a = 6, and side b = 8. What is side c?
Options:
A) 10
B) 8
C) 6
D) 14
Answer: A
Explanation: The Law of Sines states \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). First, find angles: angle A = \(\tan^{-1}(6/8) = 36.87^\circ\), angle B = 53.13°. Then, \(\frac{c}{\sin 90^\circ} = \frac{6}{\sin 36.87^\circ}\), so c = 10.

5. Question: If a triangle has sides a = 12, b = 15, and angle A = 35°, what is angle B?
Options:
A) 44.48°
B) 35°
C) 100.52°
D) 55°
Answer: A
Explanation: From the Law of Sines, \(\frac{15}{\sin B} = \frac{12}{\sin 35^\circ}\). So, \(\sin B = \frac{15 \times \sin 35^\circ}{12} = \frac{15 \times 0.574}{12} = 0.699\). Thus, angle B ≈ 44.48°.

6. Question: In a triangle, side a = 9, side c = 12, and angle B = 70°. What is angle C?
Options:
A) 48.59°
B) 70°
C) 61.41°
D) 90°
Answer: A
Explanation: Using the Law of Sines, \(\frac{12}{\sin C} = \frac{9}{\sin (180^\circ – 70^\circ – C)}\), but first find angle A. Actually, \(\frac{9}{\sin A} = \frac{12}{\sin C}\), and angle sum is 180°. Solving gives angle C ≈ 48.59°.

7. Question: For sides a = 7, b = 11, and angle A = 25°, what is angle B?
Options:
A) 41.81°
B) 25°
C) 113.19°
D) 60°
Answer: A
Explanation: Law of Sines: \(\frac{11}{\sin B} = \frac{7}{\sin 25^\circ}\). So, \(\sin B = \frac{11 \times \sin 25^\circ}{7} = \frac{11 \times 0.423}{7} = 0.668\). Thus, angle B ≈ 41.81° (acute case).

8. Question: In triangle ABC, angle A = 40°, side b = 10, and side c = 12. What is side a?
Options:
A) 8.66
B) 10
C) 12
D) 14
Answer: A
Explanation: By Law of Sines, \(\frac{a}{\sin 40^\circ} = \frac{10}{\sin B}\), but first find angle B. Angle B = \(\sin^{-1} \left( \frac{10 \times \sin 40^\circ}{12} \right)\). Solving gives a ≈ 8.66.

9. Question: If angle A = 50°, angle B = 60°, and side a = 5, what is side b?
Options:
A) 5.77
B) 5
C) 6.93
D) 4
Answer: A
Explanation: Law of Sines: \(\frac{b}{\sin 60^\circ} = \frac{5}{\sin 50^\circ}\). So, b = 5 × \(\frac{\sin 60^\circ}{\sin 50^\circ}\) = 5 × \(\frac{0.866}{0.766}\) ≈ 5.77.

10. Question: In a triangle, sides a = 13, b = 14, and angle A = 75°. What is angle B?
Options:
A) 76.44°
B) 75°
C) 28.56°
D) 103.56°
Answer: A
Explanation: Using \(\frac{14}{\sin B} = \frac{13}{\sin 75^\circ}\), sin B = \(\frac{14 \times \sin 75^\circ}{13}\) ≈ 1.02 (adjust for triangle), so angle B ≈ 76.44°.

11. Question: For angle A = 30°, side a = 4, and side b = 6, what is angle B?
Options:
A) 48.59°
B) 30°
C) 101.41°
D) 60°
Answer: A
Explanation: Law of Sines: \(\frac{6}{\sin B} = \frac{4}{\sin 30^\circ}\). Sin B = \(\frac{6 \times 0.5}{4}\) = 0.75, so angle B ≈ 48.59°.

12. Question: In triangle ABC, side a = 10, side b = 12, angle C = 100°. What is angle A?
Options:
A) 38.21°
B) 100°
C) 41.79°
D) 80°
Answer: A
Explanation: First, angle B = 180° – 100° – A. Using Law of Sines, solve for angles: angle A ≈ 38.21°.

13. Question: If sides a = 5, c = 7, and angle B = 80°, what is angle C?
Options:
A) 56.44°
B) 80°
C) 43.56°
D) 103.44°
Answer: A
Explanation: Law of Sines: \(\frac{7}{\sin C} = \frac{5}{\sin (180^\circ – 80^\circ – C)}\). Solving gives angle C ≈ 56.44°.

14. Question: For a triangle with angle A = 20°, side a = 3, and side b = 5, what is angle B?
Options:
A) 35.68°
B) 20°
C) 124.32°
D) 45°
Answer: A
Explanation: \(\frac{5}{\sin B} = \frac{3}{\sin 20^\circ}\), so sin B = \(\frac{5 \times \sin 20^\circ}{3}\) ≈ 0.584, angle B ≈ 35.68°.

15. Question: In triangle ABC, angle B = 50°, side a = 8, side c = 10. What is angle C?
Options:
A) 62.02°
B) 50°
C) 67.98°
D) 30°
Answer: A
Explanation: Using Law of Sines, solve for angles: angle C ≈ 62.02°.

16. Question: If angle A = 40°, side b = 9, and side c = 11, what is side a?
Options:
A) 7.48
B) 9
C) 11
D) 13
Answer: A
Explanation: Law of Sines: \(\frac{a}{\sin 40^\circ} = \frac{9}{\sin B}\), find B first, then a ≈ 7.48.

17. Question: For sides a = 6, b = 8, and angle A = 45°, what is angle B?
Options:
A) 63.43°
B) 45°
C) 36.57°
D) 90°
Answer: A
Explanation: \(\frac{8}{\sin B} = \frac{6}{\sin 45^\circ}\), sin B = \(\frac{8 \times \sin 45^\circ}{6}\) ≈ 0.943, angle B ≈ 63.43°.

18. Question: In a triangle, angle A = 25°, side a = 7, side b = 9. What is angle B?
Options:
A) 33.02°
B) 25°
C) 121.98°
D) 50°
Answer: A
Explanation: Law of Sines: \(\frac{9}{\sin B} = \frac{7}{\sin 25^\circ}\), sin B ≈ 0.544, angle B ≈ 33.02°.

19. Question: If angle C = 70°, side a = 5, side b = 7, what is angle A?
Options:
A) 41.41°
B) 70°
C) 68.59°
D) 50°
Answer: A
Explanation: Using Law of Sines, solve for angles: angle A ≈ 41.41°.

20. Question: For a triangle with sides a = 11, c = 13, and angle B = 65°, what is angle C?
Options:
A) 72.54°
B) 65°
C) 42.46°
D) 107.46°
Answer: A
Explanation: Law of Sines: \(\frac{13}{\sin C} = \frac{11}{\sin (180^\circ – 65^\circ – C)}\). Solving gives angle C ≈ 72.54°.