Law of Sines and Law of Cosines Test Answer Key with Scoring – Geometry

Question 1 Triangle Area (2 points)

Area = ½ · 15 · 11 · sin 47° = 82.5 · 0.7314 = 60.34.

≈ 60.3 square units

Scoring: 1 point for the formula with the correct numbers, 1 point for the answer.

Question 2 Find a Side with Sines (2 points)

bsin 33° = 22sin 81°, so b = 22 sin 33°sin 81° ≈ 12.13.

b ≈ 12.1

Scoring: 1 point for the proportion, 1 point for the answer.

Question 3 Find a Side with Cosines (2 points)

c² = 19² + 14² − 2(19)(14) cos 48° = 557 − 532 · 0.6691 = 201.02.

c ≈ 14.2

Scoring: 1 point for the substitution, 1 point for the answer.

Question 4 Largest Angle (3 points)

The largest angle C is opposite 13. cos C = 49 + 100 − 1692 · 7 · 10 = −20140 ≈ −0.1429.

C = cos−1(−0.1429) ≈ 98.21°.

C ≈ 98.2°

Scoring: 1 point for choosing the opposite angle, 1 point for the cosine equation, 1 point for the answer.

Question 5 Which Law First? (2 points)

(a) Law of Sines: two angles and a side opposite one of them.

(b) Law of Cosines: all three sides are known and no angle is known.

(a) Sines (b) Cosines

Scoring: 1 point each.

Question 6 Across the Canyon (3 points)

BT² = 310² + 275² − 2(310)(275) cos 84° = 171,725 − 170,500 · 0.1045 = 153903.

BT ≈ 392.3.

≈ 392 ft

Scoring: 1 point for the Law of Cosines setup, 1 point for the arithmetic, 1 point for the answer.

Question 7 Two Lookouts (3 points)

Angle F = 180° − 58° − 49° = 73°.

AF is opposite the 49° angle at B: AFsin 49° = 5.4sin 73°.

AF = 5.4 sin 49°sin 73° ≈ 4.26.

≈ 4.3 mi

Scoring: 1 point for angle F, 1 point for the proportion, 1 point for the answer.

Question 8 Hikers at Different Bearings (3 points)

The angle between the bearings is 115° − 40° = 75°.

d² = 9² + 14² − 2(9)(14) cos 75° = 277 − 252 · 0.2588 = 211.78.

d ≈ 14.55.

≈ 14.6 mi

Scoring: 1 point for the angle, 1 point for the setup, 1 point for the answer.