Trigonometry Laws of Sines and Cosines 2 — Questions and Answers
Question 1: In triangle ABC, a = 8, b = 11, and angle A = 35°. Using the Law of Sines, which equation correctly finds angle B?
- sin B / 8 = sin 35° / 11
- sin B / 11 = sin 35° / 8
- sin B = 11 · sin 35° / 8 (Correct answer)
- sin B = 8 · sin 35° / 11
Correct answer: sin B = 11 · sin 35° / 8
By the Law of Sines, sin B / b = sin A / a, so sin B = b · sin A / a = 11 · sin 35° / 8.
Question 2: A triangle has sides a = 5, b = 7, c = 9. Which formula correctly begins the calculation to find angle C using the Law of Cosines?
- c² = a² + b² + 2ab·cos C
- cos C = (a² + b² − c²) / (2ab) (Correct answer)
- cos C = (a² + b² + c²) / (2ab)
- cos C = (c² − a² − b²) / (2ab)
Correct answer: cos C = (a² + b² − c²) / (2ab)
The Law of Cosines gives cos C = (a² + b² − c²) / (2ab).
Question 3: In an obtuse triangle where angle B > 90°, the Law of Sines may produce an ambiguous case. When is angle B guaranteed to be obtuse?
- When sin B < 0.5
- When b > a and the calculated sin B yields two possible angles
- When b is the longest side and b > a (Correct answer)
- The Law of Sines always produces a unique obtuse angle
Correct answer: When b is the longest side and b > a
If b is the longest side and b > a, then angle B must be the largest angle, guaranteeing it is obtuse.
Question 4: Using the Law of Cosines, what is the length of side c in a triangle where a = 6, b = 8, and C = 60°?
- √148
- √28
- √52 (Correct answer)
- √100
Correct answer: √52
c² = 36 + 64 − 2(6)(8)cos 60° = 100 − 96(0.5) = 100 − 48 = 52, so c = √52.
Question 5: Triangle PQR has angle P = 50°, angle Q = 80°, and side p = 12. What is the length of side q?
- q = 12·sin 80° / sin 50° (Correct answer)
- q = 12·sin 50° / sin 80°
- q = sin 80° / (12·sin 50°)
- q = 12·sin 50° / sin 130°
Correct answer: q = 12·sin 80° / sin 50°
By the Law of Sines, q / sin Q = p / sin P, so q = p · sin Q / sin P = 12 · sin 80° / sin 50°.
Question 6: Two ships leave a port; one travels 30 km on a bearing of 40° and the other travels 45 km on a bearing of 100°. What angle between their paths should you use in the Law of Cosines?
- 40°
- 100°
- 60° (Correct answer)
- 140°
Correct answer: 60°
The angle between the two paths is 100° − 40° = 60°.
Question 7: For which given information set is the Law of Cosines required (Law of Sines cannot be used directly)?
- AAS (two angles and a non-included side)
- ASA (two angles and the included side)
- SSS (three sides known) (Correct answer)
- AAA (all three angles known)
Correct answer: SSS (three sides known)
SSS gives three sides with no angles, so the Law of Cosines is needed to find an angle first.
Trigonometry Laws of Sines and Cosines 2
In triangle ABC, a = 8, b = 11, and angle A = 35°.
Using the Law of Sines, which equation correctly finds angle B?