Geometry Trigonometry & Applications 3 — Questions and Answers
Question 1: Using the Law of Sines, if a = 8, angle A = 30°, and angle B = 45°, which sets up side b?
- b = 8 sin 45° / sin 30° (Correct answer)
- b = 8 sin 30° / sin 45°
- b = 8 / (sin 45° sin 30°)
- b = 8 sin 30° sin 45°
Correct answer: b = 8 sin 45° / sin 30°
By the Law of Sines a/sin A = b/sin B, so b = 8 sin 45°/sin 30°.
Question 2: The Law of Cosines c² = a² + b² − 2ab cos C reduces to the Pythagorean theorem when C equals what?
- 90° (Correct answer)
- 45°
- 60°
- 0°
Correct answer: 90°
At C = 90°, cos C = 0, leaving c² = a² + b².
Question 3: In triangle with sides a = 5, b = 7 and included angle C = 60°, what is c²?
- 39 (Correct answer)
- 74
- 109
- 35
Correct answer: 39
c² = 25 + 49 − 2(5)(7)(0.5) = 74 − 35 = 39.
Question 4: What is the area of a triangle with sides 6 and 10 and an included angle of 30°?
- 15 (Correct answer)
- 30
- 60
- 7.5
Correct answer: 15
Area = ½·6·10·sin 30° = ½·60·0.5 = 15.
Question 5: When is the Law of Cosines preferred over the Law of Sines?
- Given SAS or SSS (Correct answer)
- Given two angles and a side
- Given a right angle only
- Given only one side
Correct answer: Given SAS or SSS
The Law of Cosines works directly with SAS or SSS configurations.
Question 6: In triangle ABC, angle A = 40°, angle B = 60°. What is angle C?
- 80° (Correct answer)
- 100°
- 70°
- 60°
Correct answer: 80°
Angles sum to 180°, so C = 180° − 40° − 60° = 80°.
Question 7: The ambiguous case (two possible triangles) can occur with which given information?
- SSA (Correct answer)
- SAS
- ASA
- SSS
Correct answer: SSA
SSA can produce zero, one, or two triangles, making it ambiguous.
Geometry Trigonometry & Applications 3
Using the Law of Sines, if a = 8, angle A = 30°, and angle B = 45°, which sets up side b?