Law of Sines vs Law of Cosines: Which One to Use and Examples
Know when to use the Law of Sines or the Law of Cosines for any triangle, with both formulas and two worked examples.
The two laws
- sin A / a = sin B / b = sin C / c
- c² = a² + b² − 2ab cos C
Both work on any triangle, not just right triangles.
How to pick
- Know an angle and the side opposite it (a matching pair)? Sines.
- Know SAS or SSS (no matching pair)? Cosines.
Cosines first, then sines finishes the rest.
Example 1
A = 40°, B = 60°, a = 10. Find b.
- A and a are a matching pair, so use sines
- sin 40° / 10 = sin 60° / b
- b = 10 sin 60° / sin 40°
b ≈ 13.5
Example 2
a = 5, b = 7, C = 60°. Find c.
- Two sides and the angle between them (SAS): cosines
- c² = 25 + 49 − 2(5)(7)cos 60°
- c² = 74 − 35 = 39
c ≈ 6.24
Common mistake
SSA always gives one triangle
SSA can give two triangles, one, or none (the ambiguous case).
Check whether the sine you find is above 1, and whether the supplementary angle also fits.
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