law of sines

The law of sines is a trigonometric relationship between the sides and angles of any triangle. It is especially useful for solving non‑right (oblique) triangles.

Statement

For a triangle with side lengths a,b,ca,b,ca,b,c opposite angles A,B,CA,B,CA,B,C respectively, the law of sines states
asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}sinAa​=sinBb​=sinCc​.

This common value also equals 2R2R2R, where RRR is the radius of the circumcircle of the triangle.

When it is used

The law of sines is used to find unknown sides or angles when certain combinations are known (such as two angles and one side, or two sides and an angle opposite one of them).

It applies to any plane triangle, not just right triangles, making it a central tool in solving oblique triangles.

Basic example setup

If AAA, BBB, and side aaa are known, first find C=180∘−A−BC=180^\circ -A-BC=180∘−A−B, then use
asin⁡A=bsin⁡B\dfrac{a}{\sin A}=\dfrac{b}{\sin B}sinAa​=sinBb​ to solve for bbb and
asin⁡A=csin⁡C\dfrac{a}{\sin A}=\dfrac{c}{\sin C}sinAa​=sinCc​ to solve for ccc.

Rearranging a proportion like asin⁡A=bsin⁡B\dfrac{a}{\sin A}=\dfrac{b}{\sin B}sinAa​=sinBb​ gives a=sin⁡Asin⁡B ba=\dfrac{\sin A}{\sin B},ba=sinBsinA​b, which is how the missing side is computed.

If you want, a follow-up can walk step‑by‑step through a concrete numerical example.