Law of Sines & Law of Cosines Worksheet
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.
What to do
- Open the triangle solver and choose “Two sides + angle (SAS)”. Enter “Side a” = 6, “Side b” = 8 and “Included angle C (°)” = 60, then press “Solve triangle”. Record side c (shown in the steps below the diagram), angles A and B, and the area.
- Change “Included angle C (°)” to 90, then to 120. Press “Solve triangle” each time and record the results.
- Choose “Three sides (SSS)”, enter 7, 8 and 9 as sides a, b and c, and solve. Record the results.
- Choose “Two sides + opposite angle (SSA)” and enter “Side a” = 7, “Side b” = 10 and “Angle A (°)” = 30. Press “Solve triangle”: the solver finds two triangles. Record both, using the solution buttons that appear below the button.
Record your results
| Setting | Side c | Angle A | Angle B | Area |
|---|---|---|---|---|
| SAS: a = 6, b = 8, C = 60° | ||||
| SAS: a = 6, b = 8, C = 90° | ||||
| SAS: a = 6, b = 8, C = 120° | ||||
| SSS: a = 7, b = 8, c = 9 | ||||
| SSA: a = 7, b = 10, A = 30° (triangle 1) | ||||
| SSA: a = 7, b = 10, A = 30° (triangle 2) |
Questions
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?
What happens to the law of cosines when C = 90°? Which famous theorem do you get?
As angle C grew from 60° to 90° to 120°, how did side c change? Use the sign of cos C to explain why.
In the 7–8–9 triangle, which angle is the largest, and why? Find it with the law of cosines.
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.
Check the law of sines on the 6–8–60° triangle: calculate a ÷ sin A and c ÷ sin C. What do you notice?
Free from Geometry Lab · geometrylab.net · HSG-SRT.D.10: Prove the laws of sines and cosines and use them to solve problems. HSG-SRT.D.11: Understand and apply the law of sines and the law of cosines to find unknown measurements in right and non-right triangles.
Teaching with this? Find the answer key and more activities on the Geometry Lab for teachers page.