Angles of elevation and depression
Measure from a horizontal reference and account for an observer's height.
Section Paper 2 · Extended · Non-calculator
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Use tangent and inverse trigonometric ratios.
- Recognise parallel horizontal lines.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
An angle of depression is measured from which reference?
What you will learn
- Model elevation and depression with a right triangle.
- Distinguish horizontal distance, line of sight and total height.
The reference is horizontal
An angle of elevation is measured upward from horizontal to a line of sight; an angle of depression is measured downward from horizontal. If an observer looks down to a point on level ground, the depression angle equals the elevation angle seen from that point because the horizontal reference lines are parallel.
For a point 10 m horizontally from the foot of a pole and angle of elevation 45° from ground level, height is 10 tan 45° = 10 m. The horizontal distance is adjacent to the angle; the line of sight is the hypotenuse. Read which distance is supplied before choosing tangent or sine.
Account for the eye level
If an observer's eye is 1.6 m above ground, a calculated vertical rise from eye to the top is only part of the object's total height. At horizontal distance 12 m and elevation 30°, the rise is 12 tan 30° = 4√3 m, so total height is 1.6 + 4√3 ≈ 8.53 m.
A depression problem from a bridge can ask for horizontal distance: for bridge eye level 8 m above water and depression 30°, distance is 8/tan 30° = 8√3 ≈ 13.9 m. Draw the right angle at the perpendicular foot, not at the observer. Real measurements can carry uncertainty; give the model result to the requested accuracy without claiming an exact survey.
Pause and explain
From ground level, elevation to a pole top is 45° at horizontal distance 10 m. Find height.
Worked example
An observer's eye is 1.6 m above level ground, 12 m horizontally from a tower. Elevation to the top is 30°. Find total tower height.
Show the worked solution
- The rise above eye level is 12 tan 30° = 4√3 m.
- Add the 1.6 m eye height.
- Total is 1.6 + 4√3 ≈ 8.53 m to three significant figures.
Answer 1.6 + 4√3 m ≈ 8.53 m.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
From ground level, a tower is 15 m horizontally away and its top has elevation 30°. Find its height.
Give me a hint
Use tangent with the horizontal adjacent leg.
Compare my reasoning
- h = 15 tan 30°.
- This gives 5√3 m.
- To three significant figures, height is 8.66 m.
5√3 m ≈ 8.66 m.
Look for these in your work
- Used a horizontal reference.
- Distinguished rise from line of sight.
An observer's eye is 1.5 m high, 20 m horizontally from a tree. Elevation to its top is 35°. Find tree height to three significant figures.
Give me a hint
Add eye height after calculating the rise.
Compare my reasoning
- Rise is 20 tan 35° ≈ 14.0042 m.
- Total height is about 15.5042 m.
- Round the final result to 15.5 m.
15.5 m.
Look for these in your work
- Included eye height.
- Rounded only at the end.
A point on a bridge is 8 m above water and observes a boat at depression 30°. Find horizontal distance to the boat.
Give me a hint
The water-level triangle has elevation thirty degrees.
Compare my reasoning
- tan 30° = 8/d.
- d = 8/tan 30° = 8√3 m.
- The horizontal distance is approximately 13.9 m.
8√3 m ≈ 13.9 m.
Look for these in your work
- Matched depression with the parallel-line elevation angle.
- Solved for the denominator length.
Common mistakes
- Measuring elevation from vertical.
- Treating line of sight as horizontal distance.
- Forgetting eye height.
What can you explain now?
When is eye height added to a trigonometric rise?
Compare with the explanation
When the rise is measured from an eye above the object's ground level.
The right-triangle calculation measures vertical separation from eye level, so the ground-to-eye distance completes total height.
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Make it stick
Tomorrow, sketch an elevation and a depression problem and mark both horizontal references.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.