Draw a sketch, it should form a triangle.

bearings-hq3

Find the bearing of A from B by adding 180° to the bearing of B from A.

$105 + 180 = 285$

Find the angle in the triangle at B by subtracting the two bearings.

$285 - 230 = 55$

bearings-hq3-2

[1]

To find the distance between A and C you need to know the angle opposite it in the triangle. You know an angle and the length of the side opposite to it, you also know another length so you can therefore find the angle opposite it by using the sine rule.

The formula for the sine rule is $\frac{\sin A}{a} = \frac{\sin B}{b}$ , where a and b are the lengths of the two sides that are opposite the angles A and B respectively.
Substitute a = 95, b = 180, A = y and B = 55 into the formula.

$\frac{\sin (y)}{95} = \frac{\sin (55)}{180}$

[1]

Multiply both sides by 95 and then find the angle by using the inverse sin function.

$\begin{array}{rcl} \sin (y) & = & \frac{\sin (55)}{180} \times 95 \\ y & = & \sin^{- 1} (\frac{\sin (55)}{180} \times 95) \\ y & = & 25.615534 . . . \end{array}$

[1]

The interior angles of the triangle should add up to 180°. Use the answer button on your calculator to find the third angle.

$\begin{array}{rcl} Angle A C B & = & 180 - 55 - 25.615534 . . . \\ = & 99.384465 . . . \end{array}$

You now have enough information to find the side LN using either the sine rule or the cosine rule.

Bearings, Constructions & Scale Drawings (CIE IGCSE Maths: Extended)

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