What this chapter is about
This chapter shows how the trigonometric ratios you learnt earlier can solve real-world problems involving heights and distances. When direct measurement is impossible — finding the height of a tall building, the width of a river, or the distance of a ship from shore — trigonometry provides a practical method using angles and one known length.
You will work with two key angles: the angle of elevation (looking upward from horizontal) and the angle of depression (looking downward from horizontal). These angles, combined with right-triangle trigonometry, let you calculate unknown lengths in situations where climbing or crossing is not feasible.
After studying this chapter, you should be able to draw correct diagrams from word problems, identify the right triangle involved, choose the appropriate trigonometric ratio, and compute heights or distances accurately.
Key ideas
- The angle of elevation is the angle between the horizontal line of sight and the line of sight directed upward to an object above the observer.
- The angle of depression is the angle between the horizontal line of sight and the line of sight directed downward to an object below the observer.
- When an observer looks down at an object, the angle of depression from the observer equals the angle of elevation from the object to the observer (alternate angles with a horizontal line).
- In any right triangle, tan θ = opposite side / adjacent side, sin θ = opposite side / hypotenuse, and cos θ = adjacent side / hypotenuse.
- Drawing a clear, labelled diagram is essential: mark the right angle, the known angle, the known length, and the unknown length.
- Most height-and-distance problems reduce to finding one side of a right triangle when another side and an acute angle are known.
- Problems may involve two positions of observation (moving closer or farther) to set up two equations and eliminate an unknown.
Formulas and facts to remember
tan θ = perpendicular / base — use when height and horizontal distance are involved.
sin θ = perpendicular / hypotenuse — use when height and slant distance are involved.
cos θ = base / hypotenuse — use when horizontal and slant distances are involved.
Standard values: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
Standard values: sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2.
Standard values: cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2.
Angle of depression from A to B = Angle of elevation from B to A (when horizontal lines through A and B are parallel).
Worked examples
Example 1: Finding the height of a temple tower
A person standing 40 metres away from the base of a temple tower observes the top at an angle of elevation of 60°. Find the height of the tower.
Solution: Let the height of the tower be h metres. The horizontal distance from the observer to the base is 40 m. In the right triangle formed: tan 60° = h / 40 √3 = h / 40 h = 40 × √3 h = 40 × 1.732 = 69.28 m (approximately)
The height of the temple tower is approximately 69.28 metres.
Example 2: Finding horizontal distance using angle of depression
From the top of a 25-metre-high lighthouse, the angle of depression to a fishing boat is 30°. How far is the boat from the base of the lighthouse?
Solution: Let the horizontal distance from the lighthouse base to the boat be d metres. The angle of depression from the top equals the angle of elevation from the boat, which is 30°. In the right triangle: tan 30° = 25 / d 1/√3 = 25 / d d = 25 × √3 d = 25 × 1.732 = 43.3 m (approximately)
The boat is approximately 43.3 metres from the lighthouse base.
Example 3: Two angles of elevation from different points
A statue stands on a pillar. From a point on the ground, the angles of elevation to the top of the pillar and the top of the statue are 45° and 60° respectively. If the pillar is 20 metres high, find the height of the statue.
Solution: Let the horizontal distance from the observation point to the pillar base be d metres. Let the height of the statue be s metres.
For the pillar top: tan 45° = 20 / d 1 = 20 / d d = 20 m
For the statue top (total height = 20 + s): tan 60° = (20 + s) / d √3 = (20 + s) / 20 20√3 = 20 + s s = 20√3 − 20 s = 20(√3 − 1) s = 20 × 0.732 = 14.64 m (approximately)
The height of the statue is approximately 14.64 metres.
Common mistakes
Confusing angle of elevation with angle of depression → Remember elevation is looking up, depression is looking down; always draw the horizontal reference line first.
Using the wrong trigonometric ratio → Identify which sides you have (opposite, adjacent, hypotenuse relative to the given angle) before choosing sin, cos or tan.
Forgetting to add the observer's height when standing above ground level → If a person of height 1.5 m observes a tower, the calculated height is from eye level; add 1.5 m for total tower height.
Placing the right angle incorrectly in the diagram → The right angle is at the base of the vertical object (tower, building), not at the observer's position.
Rounding too early in calculations → Keep √3 or √2 in your working until the final step to avoid accumulating errors.
Quick revision
Angle of elevation: observer looks upward; angle of depression: observer looks downward.
For most problems, tan θ is your primary tool since you deal with height and horizontal distance.
Always draw the diagram first — label all known and unknown quantities.
Use tan 30° = 1/√3, tan 45° = 1, tan 60° = √3 for quick calculations.
When two observation points are given, form two equations and solve simultaneously.
The observer's own height must be added to find the full height of tall objects.