Sine Rule
The sine rule is used to find missing sides or angles in non-right-angled triangles when you know an opposite side-angle pair. Match each side with the angle opposite it, then use \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\) to find a missing side. When two sides and one angle are given, flip the sine rule to \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b}\) to find a missing angle.
Common mistake: Always pair a side with its opposite angle. If side \(c\) is opposite angle \(C\), use \(\dfrac{c}{\sin C}\), not \(\dfrac{c}{\sin B}\).
Worked example:
In the triangle, \(A=35^\circ\), \(B=70^\circ\), and \(a=9\text{ cm}\). Find \(b\), to 1 decimal place.
- Use the sine rule.\[\dfrac{b}{\sin70^\circ}=\dfrac{9}{\sin35^\circ}\]\[b=\dfrac{9\sin70^\circ}{\sin35^\circ}=14.7\]
Answer:
\(b = 14.7\text{ cm}\).
Worked example:
In the triangle, \(A=48^\circ\), \(a=10\text{ cm}\), and \(b=8\text{ cm}\). Find \(B\), to 1 decimal place.
- Use the sine rule to find \(\sin B\).\[\dfrac{\sin B}{8}=\dfrac{\sin48^\circ}{10}\]\[\sin B=\dfrac{8\sin48^\circ}{10}\]\[B=\sin^{-1}\left(\dfrac{8\sin48^\circ}{10}\right)=36.5^\circ\]
Answer:
\(B = 36.5^\circ\).