Elevation, depression and bearings

Example From a point 40 m from the foot of a tower, the angle of elevation of the top is 32°. Find the height.

opp and adj: tan.

1. \(h=40\tan32^\circ=24.99\ldots\)

Answer 25.0 m

Example From the top of a 60 m cliff, the angle of depression of a boat is 25°. How far is the boat from the foot of the cliff?

Move the 25° down to the boat. Height 60 is opposite, distance is adjacent.

1. \(\tan25^\circ=\dfrac{60}{d}\)
2. \(d=\dfrac{60}{\tan25^\circ}=128.7\)

Answer 129 m (3 s.f.)

Example A building is 30 m tall. From point P the elevation of the top is 20°; from Q, nearer, it is 35°. Find PQ.

Two ground distances, then subtract.

1. \(\dfrac{30}{\tan20^\circ}=82.42\)
2. \(\dfrac{30}{\tan35^\circ}=42.84\)
3. \(PQ=39.58\)

Answer 39.6 m

Example B is 8 km east and 6 km north of A. Find the distance AB and the bearing of B from A.

The angle θ at A is measured from north, so the east side is opposite.

1. \(AB=\sqrt{8^{2}+6^{2}}=10\)
2. \(\theta=\tan^{-1}\dfrac86=53.1^\circ\)

Answer 10 km on a bearing of 053.1°

Example A ship sails 12 km on a bearing of 060°. How far east and how far north is it from its start?

Angle from north 60°: east is opposite, north is adjacent.

1. east \(=12\sin60^\circ=10.39\)
2. north \(=12\cos60^\circ=6\)

Answer 10.39 km east, 6 km north

Example AC = 20 m and angle CAB = 35°. Find the shortest distance from C to the line AB.

CX is perpendicular, so triangle ACX has a right angle at X. CX is opposite 35°.

1. \(CX=20\sin35^\circ=11.47\)

Answer 11.5 m