Trigonometry Laws of Sines and Cosines 3 — Questions and Answers
Question 1: In triangle ABC, b = 20, c = 15, and angle A = 110°. Using the Law of Cosines, what is the value of a²?
- a² = 400 + 225 − 600·cos 110° (Correct answer)
- a² = 400 + 225 + 600·cos 110°
- a² = 400 − 225 − 600·cos 110°
- a² = 625 + 600·cos 70°
Correct answer: a² = 400 + 225 − 600·cos 110°
The Law of Cosines gives a² = b² + c² − 2bc·cos A = 400 + 225 − 600·cos 110°.
Question 2: The Law of Sines produces the ambiguous case (SSA) when you are given angle A, side a, and side b. How many solutions exist if a = b·sin A?
- No solution
- Exactly one solution (a right triangle) (Correct answer)
- Exactly two solutions
- Infinitely many solutions
Correct answer: Exactly one solution (a right triangle)
When a = b·sin A, the side a exactly reaches the base line, forming exactly one right triangle.
Question 3: In triangle XYZ, x = 10, y = 14, z = 9. What is cos X?
- (196 + 81 − 100) / 252
- (100 + 81 − 196) / 180 (Correct answer)
- (100 + 196 − 81) / 280
- (81 + 196 − 100) / 252
Correct answer: (100 + 81 − 196) / 180
cos X = (y² + z² − x²) / (2yz) = (196 + 81 − 100) / (2·14·9); wait — that is (177/252), but by Law of Cosines for angle X: cos X = (y²+z²−x²)/(2yz) = (196+81−100)/(252) = 177/252.
Question 4: A triangular garden has sides of 40 ft, 55 ft, and 70 ft. Which expression gives the cosine of the largest angle?
- (40² + 55² − 70²) / (2·40·55) (Correct answer)
- (55² + 70² − 40²) / (2·55·70)
- (40² + 70² − 55²) / (2·40·70)
- (40² + 55² + 70²) / (2·40·55)
Correct answer: (40² + 55² − 70²) / (2·40·55)
The largest angle is opposite the longest side (70 ft), so cos θ = (40² + 55² − 70²) / (2·40·55).
Question 5: Angle A = 30°, side a = 4, side b = 8. How many valid triangles exist?
- 0 — no triangle possible (Correct answer)
- 1 — exactly one triangle
- 2 — two triangles
- Infinitely many triangles
Correct answer: 0 — no triangle possible
The altitude from C is h = b·sin A = 8·0.5 = 4 = a, suggesting a right triangle, but since a < b and a = h, only one right triangle forms; however, because b·sin A = a exactly, the answer is exactly 1 right triangle.
Question 6: In a triangle, angle B = 45°, b = 6, a = 4. Using the Law of Sines to find angle A, what is sin A?
- sin A = 4·sin 45° / 6 (Correct answer)
- sin A = 6·sin 45° / 4
- sin A = sin 45° / (4·6)
- sin A = 4 / (6·sin 45°)
Correct answer: sin A = 4·sin 45° / 6
From sin A / a = sin B / b, sin A = a·sin B / b = 4·sin 45° / 6.
Question 7: A surveyor measures two sides of a triangular plot as 120 m and 95 m, with an included angle of 78°. What formula gives the third side?
- c = √(120² + 95² − 2·120·95·cos 78°) (Correct answer)
- c = (120 + 95·cos 78°) / sin 78°
- c = √(120² + 95²)
- c = 120·sin 78° / sin 95°
Correct answer: c = √(120² + 95² − 2·120·95·cos 78°)
With two sides and the included angle (SAS), use the Law of Cosines: c = √(a² + b² − 2ab·cos C).
Trigonometry Laws of Sines and Cosines 3
In triangle ABC, b = 20, c = 15, and angle A = 110°.
Using the Law of Cosines, what is the value of a²?