Law of Sines Test Answer Key with Scoring – Precalculus

Question 1 Area of a Triangle (2 points)

Area = ½(16)(11) sin 47° = 88 × 0.7314 ≈ 64.4 ft².

Scoring: 1 point for the correct formula and substitution; 1 point for the answer.

Question 2 Heron’s Formula (2 points)

s = 48/2 = 24. Area = √(24 · 14 · 7 · 3) = √7056 = 84 cm².

Scoring: 1 point for s and the substitution; 1 point for the area.

Question 3 Law of Sines in a Triangle (3 points)

1. C = 180° − 58° − 67° = 55°.
2. b = 14 sin 67° ÷ sin 58° ≈ 15.2.
3. c = 14 sin 55° ÷ sin 58° ≈ 13.5.

Scoring: 1 point per part.

Question 4 Law of Cosines, Then Law of Sines (3 points)

c² = 81 + 169 − 2(9)(13) cos 71° = 250 − 234(0.3256) ≈ 173.82, so c ≈ 13.2.

sin A = 9 sin 71° ÷ 13.18 ≈ 0.6455, and A is acute because a < b, so A ≈ 40.2°.

Scoring: 2 points for c (1 for the setup, 1 for the value); 1 point for A.

Question 5 Ambiguous Case (2 points)

The height is 10 sin 36° ≈ 5.88 and 5.88 < 7 < 10, so two triangles fit.

sin B = 10 sin 36° ÷ 7 ≈ 0.8397, so B ≈ 57.1° or 122.9°.

Scoring: 1 point for two triangles with a justification; 1 point for both values of B.

Question 6 Navigation with Bearings (3 points)

1. 180° − (135° − 70°) = 115°.
2. d² = 25² + 18² − 2(25)(18) cos 115° = 949 + 380.4 ≈ 1329.4.
3. d ≈ 36.5 miles.

Scoring: 1 point per part.

Question 7 Largest Angle (2 points)

The largest angle is opposite 14: cos C = (64 + 121 − 196)/(2 · 8 · 11) = −11/176 = −0.0625.

C ≈ 93.6°.

Scoring: 1 point for the Law of Cosines setup; 1 point for the angle.

Question 8 River Crossing Survey (3 points)

PQ² = 480² + 360² − 2(480)(360) cos 41° = 360,000 − 345,600(0.7547) ≈ 99,172, so PQ ≈ 315 ft.

Area = ½(480)(360) sin 41° ≈ 56,684 ft².

Scoring: 2 points for PQ (1 for the setup, 1 for the value); 1 point for the area.