Law of Sines

The law of sines relates every side of a triangle to the sine of its opposite angle.

Side and angle information determine which triangle relationship applies. A non-right triangle has three labelled sides and angles, plus a dropped altitude. A right-angle marker and side-length labels connect trigonometric ratios, area, and Pythagorean reasoning.

Match each side with the angle directly opposite it; that pairing is the entire structure of the law.

The law

For any triangle with sides $a,b,c$ opposite angles $A,B,C$,

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

The reciprocal form is equally valid. Use one consistent orientation and match opposite pairs carefully.

Worked example: AAS

The larger angle $68^\circ$ is opposite the longer side, which supports the result.

Finding an angle

If sides $a,b$ and angle $A$ are known,

\sin B=\frac{b\sin A}{a}.

Inverse sine gives a principal angle, but another angle $180^\circ-B$ has the same sine. This creates the ambiguous SSA case.

Area formula

With two sides and included angle,

K=\frac12ab\sin C.

This follows by using $b\sin C$ as a perpendicular height. It can calculate area without first finding every side.

Common mistakes

Pairing a side with an adjacent angle. Match opposites.

Using sine law with no known opposite pair. Start with cosine law when appropriate.

Keeping only the inverse-sine principal result in SSA. Check its supplement.

Accepting angles whose sum reaches or exceeds $180^\circ$. No triangle remains.

Rounding an angle early. It affects later side calculations.

Quick self-check

Are sides labelled opposite matching capital-letter angles?
Is there a known side-angle opposite pair?
Does triangle angle sum provide another angle first?
If solving SSA, has the supplementary angle been tested?
Are larger angles opposite larger sides?
Are units and final precision appropriate?