Law of Sines
The Law of Sines relates sides and angles of any triangle: a/sin(A) = b/sin(B) = c/sin(C).
Solve triangles with step-by-step examples.
The Formula
a / sin(A) = b / sin(B) = c / sin(C)
The Law of Sines states that in any triangle, the ratio of a side to the sine of its opposite angle is constant.
This works for all triangles — not just right triangles.
Variables
| Symbol | Meaning |
|---|---|
| a, b, c | The three sides of the triangle |
| A, B, C | The angles opposite to sides a, b, c respectively |
| sin | The sine function |
Example 1
In triangle ABC, A = 40°, B = 60°, and a = 10. Find b.
Using: a / sin(A) = b / sin(B)
10 / sin(40°) = b / sin(60°)
10 / 0.6428 = b / 0.8660
15.557 = b / 0.8660
b = 15.557 × 0.8660
b ≈ 13.47
Example 2
In triangle ABC, a = 8, A = 30°, and b = 12. Find angle B.
Using: a / sin(A) = b / sin(B)
8 / sin(30°) = 12 / sin(B)
8 / 0.5 = 12 / sin(B)
16 = 12 / sin(B)
sin(B) = 12 / 16 = 0.75
B = arcsin(0.75) ≈ 48.59°
When to Use It
Use the Law of Sines when:
- You know two angles and one side (AAS or ASA)
- You know two sides and an angle opposite one of them (SSA — watch for the ambiguous case)
- The triangle is not a right triangle
- You need to find a missing side or angle in a non-right triangle