Surveying Problems

Definition 1. The bearing of A from B relative to a North, N, is the (possibly reflex) clockwise angle formed between NA and AB. If the angle is smaller than 100°, we still denote it using 3 digits, as exemplified below.

A coastguard station on a 300-metre cliff at O monitors two vessels: Ship A and Ship B. Ship A is located at a bearing of 040° from the North, and Ship B is located at a bearing of 115° from the North.

Problem 1. Calculate the distance between Ship A and Ship B. Hence, calculate the bearing of Ship B from Ship A.

(Click for Solution)

Solution. Firstly, $\angle AOB = 115^\circ - 40^\circ = 75^\circ$ . By the law of cosines,

$\begin{aligned} AB^2 &= 8^2 + 11^2 - 2 \cdot 8 \cdot 11 \cdot \cos 75^\circ \\ &= 185 - 176 \cos 75^\circ \\ &= 139.44784, \\ AB &= \sqrt{139.44784} \\ &= 11.808 \\ &\approx 11.8\, \text{km}. \end{aligned}$

By the law of sines,

$\begin{aligned} \frac{\sin \angle OAB}{11} &= \frac{\sin 75^\circ}{11.808} \\ \sin \angle OAB &= 0.89982 \\ \angle OAB &= 64.135^\circ. \end{aligned}$

Draw an extra north $\mathrm N_2$ at $A$ as follows.

Using internal angles, since $O\mathrm N \parallel A \mathrm N_2$ ,

$\angle OA\mathrm N_2 = 180^\circ - 40^\circ = 140^\circ.$

Since angles at a point sum to $360^\circ$ , the required bearing is given by

$\displaystyle 360^\circ - 140^\circ - 64.135^\circ = 155.865^\circ \approx 155.9^\circ.$

Definition 2. The angle of elevation of a point T from G is defined by $\alpha$ as follows. Similar, the angle of depression of G from T is defined by $\beta$ as follows.