Britannica Sources used in this answer

When should the law of sines be used to solve a triangle?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

In trigonometry, the law of sines establishes that the lengths of the sides of a triangle are proportional to the sines of their opposite angles. This principle is expressed by the formula:12

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

You should use the law of sines to solve a triangle in the following scenarios:

Two angles and one side are known: When you have two angles and the length of one side (either opposite one of the known angles or between them), you can use the law of sines to find the remaining side lengths or the third angle.23
Two sides and an angle opposite one of them are known: This configuration allows you to calculate the other unknown angle opposite the second known side.23

When these specific conditions are not met—such as when you are given three sides or two sides and an included angle—the law of cosines is the appropriate tool for solving the triangle.3