Law of Sines: Formula, Proof, and Examples

The law of sines relates every side of a triangle to the sine of its opposite angle. It works for acute, right and obtuse triangles. The relation is especially useful when one complete opposite side-angle pair is known, as in angle-side-angle or angle-angle-side data. With two sides and a non-included angle, it also reveals whether the information determines zero, one or two triangles.

Match each side with the angle across from it

Use lowercase $a$, $b$ and $c$ for the sides opposite uppercase angles $A$, $B$ and $C$. Pairing a side with an adjacent angle breaks the proportion. Before substituting, sketch the triangle and place each label. The angles must total 180°, and the largest side must lie opposite the largest angle.

The common ratio also equals twice the circumradius $R$. This provides both a geometric interpretation and a check: $\frac{a}{\sin A}$, $\frac{b}{\sin B}$ and $\frac{c}{\sin C}$ should agree to the retained precision.

Why the circumcircle produces the same ratio

Every nondegenerate triangle has a circumscribed circle through its three vertices. Draw a diameter through one vertex and join its other endpoint to a second vertex. The angle subtended by a diameter is a right angle. In the resulting right triangle, the sine of the inscribed angle is the opposite chord divided by the diameter $2R$. Repeating the argument for the other sides gives the same ratio.

This proof also handles an obtuse original angle because supplementary inscribed angles have the same sine. The relation depends on the triangle being nondegenerate; three collinear points do not determine a finite circumcircle in the same way.

Worked example with two angles and one side

Suppose $A=42°$, $B=68°$ and $a=10$ cm. First find $C=180°-42°-68°=70°$. Then use the known pair $a,A$ to calculate the other sides. Keep the trigonometric values unrounded until the final step.

The result is approximately $b=13.86$ cm and $c=14.04$ cm. Since $C$ is the largest angle, $c$ should be the largest side, which it is. Substitution into the three ratios provides a numerical check.

Why SSA data can describe two different triangles

If $A=30°$, $a=7$ and $b=10$, the law gives $\sin B=\frac{5}{7}$. Sine has the same positive value for an acute angle and its supplement. Thus $B$ can be about 45.58° or 134.42°. Both leave a positive third angle, so both triangles are valid. In other data sets, the supplementary option may make the angle sum reach or exceed 180°, leaving only one triangle.

This is the ambiguous side-side-angle case. A calculator that returns only the principal inverse-sine value can silently miss the second triangle. Check the supplement and the angle sum every time.

Choose the law that matches the given information

Use the law of sines first for ASA, AAS and many SSA problems. Use the law of cosines for side-angle-side data or three known sides because those cases do not begin with an opposite pair. In a right triangle, $\sin 90°=1$, so the side opposite the right angle is the diameter of the circumcircle and the formula reduces consistently to familiar right-triangle ratios. For the function values themselves, see sine, cosine and tangent.