Using the sine and cosine rules together (AQA GCSE Further Maths): Revision Notes

Worked Example: Navigation Problem

Given information:

• Bentham is 8km from Aldbury on a bearing of 037°
• Chorton is 9km from Bentham on a bearing of 150°

We need to find:

• The size of angle ABC
• The distance from Chorton to Aldbury
• The bearing of Chorton from Aldbury

Step 1: Find angle ABC

From the bearings, we can work out that:

• Angle ABS = 37° (alternate angles)
• Angle SBC = 30° (adjacent angles on a straight line, since the bearing is 150°)
• Therefore angle ABC = 37° + 30° = 67°

Step 2: Find the distance AC using the cosine rule

We have two sides (AB = 8km, BC = 9km) and the included angle (67°), so we use the cosine rule:

$b^2 = a^2 + c^2 - 2ac \cos B$

$b^2 = 9^2 + 8^2 - 2 \times 9 \times 8 \times \cos 67°$

$b^2 = 81 + 64 - 144 \times \cos 67°$

$b^2 = 145 - 144 \times 0.3907$

$b^2 = 88.7347$

$b = \text{:success[9.42km]} \text{ (to 2 decimal places)}$

So Chorton is 9.42km from Aldbury.

Step 3: Find the bearing using the sine rule

Now we use the sine rule to find angle BAC:

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

$\frac{\sin A}{9} = \frac{\sin 67°}{9.42}$

$\sin A = \frac{9 \times \sin 67°}{9.42}$

$\sin A = 0.879$

$A = 61.57°$

The bearing of Chorton from Aldbury is 099° (037° + 62°, where 62° is the rounded value of angle A).

Worked Example: Finding Area of Triangular Land

Given: A triangular plot with sides of 70m, 80m, and 95m. Find: The area in hectares.

Step 1: Find an angle using the cosine rule

Since we have all three sides, we use the cosine rule to find an angle first:

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

$\cos A = \frac{70^2 + 80^2 - 95^2}{2 \times 70 \times 80}$

$\cos A = \frac{4900 + 6400 - 9025}{11200}$

$\cos A = \frac{2275}{11200}$

$\cos A = 0.203$

$A = 78.28°$

Step 2: Calculate the area

Now we can use the area formula:

Area = $\frac{1}{2}bc \sin A$

Area = $\frac{1}{2} \times 70 \times 80 \times \sin 78.28°$

Area = 2741m²

Area = 0.27 hectares (to 2 decimal places)