Law of Sines

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 $A,B,C$,

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

When to use it

The sine law is especially useful for:

• ASA: two angles and included side;
• AAS: two angles and a non-included side;
• SSA: two sides and a non-included angle, which may be ambiguous.

At least one known side-opposite-angle pair is needed to start directly.

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,

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

The ambiguous case

For SSA information, there may be zero, one, or two triangles. After finding a candidate $B_1$:

1. calculate $B_2=180^\circ-B_1$;
2. check whether $A+B_2<180^\circ$;
3. if so, a second triangle exists;
4. solve each triangle separately.

Also reject a computed sine value outside $[-1,1]$, which means no triangle fits.

Why ambiguity occurs

An angle and a non-included side can allow the remaining side to swing into two positions, producing an acute or obtuse angle with the same sine value.

A diagram helps reveal both geometric possibilities instead of relying only on a calculator screen.

Area formula

With two sides and included angle,

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

Choosing sine or cosine law

Use sine law when a side-opposite-angle pair is known. Use cosine law for SSS or SAS information without an opposite pair.

Sometimes cosine law finds one side or angle first, then sine law completes the triangle.

Applications

Oblique triangles model surveying, navigation, forces, property boundaries, and inaccessible distances. Draw orientation, label units, and distinguish bearings or angles measured from different reference directions.

Round only after all required values are found.

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?

Related topics

• Law of Cosines
• Trigonometry
• Right Triangle Trigonometry
• Triangle Area