20.6 The sine rule — Learn

Prerequisites: sine ratios, angle sum in a triangle, and calculator degree mode.

Objectives: use the sine rule to find a missing side or angle in a non-right-angled triangle.

A side and its opposite angle form a pair. The sine rule is $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$, where lower-case letters are opposite their matching capital angles.

Example 1. In a triangle, $A=30^\circ$, $B=45^\circ$, and $a=8$ cm. Find $b$.

Example 2. Find angle $A$ if $a=6$ cm, $b=10$ cm and $B=90^\circ$.

Watch out. Pair each side with its opposite angle. A wrong pairing changes the equation; mark opposite pairs on the diagram first.

Progress check

For the garden diagonal, the right-triangle ratios still work. The sine rule is useful when the triangle has no right angle.

You should now be able to

• match each side to its opposite angle before using the sine rule;
• use the sine rule to find a missing side or angle;
• check whether a second angle is possible by using the triangle's angle sum.