Solving problems using the sine and cosine rule - Higher

To solve problems involving non-right-angled triangles, the correct rule must first be chosen.

The cosine rule can be used in any triangle to calculate:

  • a side when two sides and the angle in between them are known
  • an angle when three sides are known

The sine rule can be used in any triangle to calculate:

  • a side when two angles and an opposite side are known
  • an angle when two sides and an opposite angle are known

Example

Ship A leaves port P and travels on a bearing of 200°. A second ship B leaves the same port P and travels on a bearing of 165°. After half an hour ship A has travelled 5.2 km and ship B has travelled 5.8 km. How far apart are the ships after half an hour? Give the answer to three significant figures.

Diagram showing positions of 2 yachts compared to a lighthouse

Remember a bearing is an angle measured clockwise from north. The angle APB = 200 - 165 = 35^\circ.

Two sides and the angle in between are known. Calculate the length AB.

Use the cosine rule.

a^2 = b^2 + c^2 - 2bc \cos{A}

\text{AB}^2 = 5.2^2 + 5.8^2-2 \times 5.2 \times 5.8 \cos{35}

\text{AB}^2 = 11.26874869

AB = 3.36 km