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
Key Notes
- Formula: a/sinA = b/sinB = c/sinC = 2R: Each ratio equals the diameter of the circumscribed circle (2R). Given any two angle-side pairs, the third can be found. This is the primary tool for AAS and ASA triangle configurations.
- The ambiguous case (SSA): Given two sides and a non-included angle, there may be 0, 1, or 2 valid triangles. When the given angle is acute and the side opposite it is shorter than the other given side, two different triangles satisfy the conditions. Always check for this.
- When to use Law of Sines vs Law of Cosines: Use Law of Sines for AAS, ASA, and SSA (check for ambiguity). Use Law of Cosines for SAS and SSS (or SSA when you want to avoid the ambiguous case by solving for an angle directly).
- Finding missing angles — watch for supplementary solutions: arcsin always returns a value between −90° and 90°. The supplement (180° − θ) is also a valid sine solution. When solving for angles with the Law of Sines, check whether the supplement produces a valid triangle (positive angle sum).
- Applications: The Law of Sines is used in surveying (triangulation from two known angles and one measured distance), navigation (cross-bearing fixes), astronomy (parallax calculations), and engineering whenever a triangle's dimensions must be determined from angle measurements.