Answer key: Law of Sines & Law of Cosines Worksheet
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Goal: Use the law of cosines and the law of sines to find missing sides and angles, and discover when two different triangles fit the same measurements.
Completed results table
| Setting | Side c | Angle A | Angle B | Area |
|---|---|---|---|---|
| SAS: a = 6, b = 8, C = 60° | 7.211 units | 46.102° | 73.898° | 20.785 units² |
| SAS: a = 6, b = 8, C = 90° | 10 units | 36.87° | 53.13° | 24 units² |
| SAS: a = 6, b = 8, C = 120° | 12.166 units | 25.285° | 34.715° | 20.785 units² |
| SSS: a = 7, b = 8, c = 9 | 9 units | 48.19° | 58.412° | 26.833 units² |
| SSA: a = 7, b = 10, A = 30° (triangle 1) | 13.559 units | 30° | 45.585° | 33.898 units² |
| SSA: a = 7, b = 10, A = 30° (triangle 2) | 3.761 units | 30° | 134.415° | 9.403 units² |
Answers
For a = 6, b = 8 and C = 60°, use the law of cosines c² = a² + b² − 2ab cos C to find c by hand. Does your answer match the solver?
c² = 36 + 64 − 2 × 6 × 8 × cos 60° = 100 − 96 × 0.5 = 52, so c = √52 ≈ 7.211. This matches the solver.
What happens to the law of cosines when C = 90°? Which famous theorem do you get?
cos 90° = 0, so the last term disappears and c² = a² + b² = 36 + 64 = 100: the Pythagorean theorem. Side c = 10.
As angle C grew from 60° to 90° to 120°, how did side c change? Use the sign of cos C to explain why.
c grew from 7.211 to 10 to 12.166. When C is acute, cos C is positive, so 2ab cos C is subtracted and c² < a² + b². When C is obtuse, cos C is negative, so the term is added: c² = 100 + 48 = 148 when C = 120°.
In the 7–8–9 triangle, which angle is the largest, and why? Find it with the law of cosines.
Angle C, because it is opposite the longest side (9). cos C = (7² + 8² − 9²) ÷ (2 × 7 × 8) = 32 ÷ 112 ≈ 0.2857, so C ≈ 73.398°.
The SSA measurements a = 7, b = 10, A = 30° gave two triangles. Use the law of sines to find both possible values of angle B, and explain why there are two.
sin B = b sin A ÷ a = 10 × 0.5 ÷ 7 ≈ 0.7143. Two angles have this sine: B ≈ 45.585° and B ≈ 180° − 45.585° = 134.415°. Both leave room for angle C (A + B < 180°), so two different triangles fit the same measurements. This is the ambiguous case.
Check the law of sines on the 6–8–60° triangle: calculate a ÷ sin A and c ÷ sin C. What do you notice?
a ÷ sin A = 6 ÷ sin 46.102° ≈ 8.327 and c ÷ sin C = 7.211 ÷ sin 60° ≈ 8.327. The ratios are equal, as the law of sines says (b ÷ sin B gives 8.327 too).