The idea, step by step
Elevation and depression use a horizontal line
An angle of elevation is measured upward from the observer’s horizontal; depression is measured downward from that horizontal. With parallel horizontals, a depression angle equals the corresponding elevation angle at the other end.
The vertical rise found by trigonometry begins at eye or instrument level. Add the eye height to obtain a height above the ground. Do not add eye height to a sloping distance or assume a line-of-sight distance is horizontal.
A bearing is measured from north
Write a bearing as a clockwise angle from north, normally with three digits. East is 090°, south is 180° and west is 270°. Draw a north line at the departure point.
A reverse bearing differs by 180°, adjusted into 000°–359°. For a distance d at bearing β, its eastward and northward components are d sinβ and d cosβ; a negative value indicates west or south.
A common trap
Using an angle of depression as the angle between the line of sight and the vertical.
Instead: Draw the observer’s horizontal. The given angle is measured from that horizontal.
See the thinking
Worked example 1
An observer’s eyes are 1.6 m above level ground. A tower is 20 m horizontally away and its top has elevation 32°. Find the tower height.
- The rise above eye level is 20tan32°.
- This rise is 12.4974 m.
- Add 1.6 m because the instrument is above the ground.
- H≈14.1 m.
Worked example 2
A camp is 12 km east and 16 km north of a starting point. Find its distance and bearing from the start.
- Distance=√(12²+16²)=20 km.
- The bearing is measured from north, so tanβ=12/16.
- β=36.8699…°, giving bearing 037°.
Check your understanding
What is the reverse bearing of 248°?
Show answer
068°
248°−180°=68°, written as 068°.
Your turn
6 questions · 13 marks · Work at your own pace.
Read the questions online
- A vertical tower is 16 m horizontally from an observer at ground level. Its top has elevation 35°. Find its height to one decimal place.2 marks
- Repeat the tower calculation if the observing instrument is 1.5 m above the ground.2 marks
- From an 8 m high platform, a boat is seen at a depression of 25°. Find its horizontal distance from the platform base.2 marks
- A point is 14 km east and 20 km north of a station. Find its bearing to the nearest degree.2 marks
- Find the reverse bearing of 032°.2 marks
- A boat travels 24 km on bearing 210°. Find its southward and westward distances, each to one decimal place.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A vertical tower is 16 m horizontally from an observer at ground level. Its top has elevation 35°. Find its height to one decimal place.2 marks
h=16tan35°.
h≈11.2 m.
11.2 m - Repeat the tower calculation if the observing instrument is 1.5 m above the ground.2 marks
The trigonometric rise is still 16tan35°.
Add the instrument height 1.5 m.
H≈12.7 m.
12.7 m - From an 8 m high platform, a boat is seen at a depression of 25°. Find its horizontal distance from the platform base.2 marks
The boat’s elevation angle to the platform is 25°.
tan25°=8/d, so d=8/tan25°.
d≈17.2 m.
17.2 m - A point is 14 km east and 20 km north of a station. Find its bearing to the nearest degree.2 marks
β=tan⁻¹(14/20)=34.992…°.
It lies in the north-east quadrant, so the bearing is 035°.
035° - Find the reverse bearing of 032°.2 marks
032°+180°=212°.
212° - A boat travels 24 km on bearing 210°. Find its southward and westward distances, each to one decimal place.3 marks
E=24sin210°=−12 km.
N=24cos210°≈−20.7846 km.
The negative signs mean west and south respectively.
20.8 km south; 12.0 km west
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: practical trigonometry, including similarity, right triangles, triangle area, sine/cosine rules and contextual interpretation.
Official curriculum source ↗