Line of sight, horizontal level
The line of sight is the straight line from the observer's eye to the object. The horizontal level is the flat line through the eye, parallel to the ground. In Class 10 questions, the ground is flat and towers, poles and buildings stand vertical, so they meet the ground at 90°.
Angle of elevation
When the object is above the eye, you raise your head. The angle between the line of sight and the horizontal is the angle of elevation. As you walk towards a tower, this angle becomes larger.
Angle of depression
When the object is below the eye, you lower your head. The angle between the horizontal and the line of sight is the angle of depression. It is measured from the horizontal line at the top, never from the vertical tower.
The horizontal line at the top is parallel to the ground, so the angle of depression and the angle of elevation are alternate angles and are equal. This lets you mark the angle at the bottom of the diagram.
Solving with one right triangle
- Draw a rough diagram. Mark the right angle.
- Mark what you know (distance or height) and the angle.
- Choose the ratio that links the known side and the unknown side. Height and ground distance → tan. Height and slant length (ladder, string) → sin. Ground and slant → cos.
- Put in values from the table and solve.
Use √3 ≈ 1.732 and √2 ≈ 1.414 when a decimal answer is asked.
Problems with two right triangles
Many board questions use two triangles:
- An observer at two points on the same line (angles 30° and 60°) → one unknown height h and one unknown distance x.
- A flagstaff on a building: two angles from the same point → two heights sharing the same ground distance.
- Two buildings or a building and a tower facing each other.
Write one tan equation per triangle. Solve the two equations together, usually by expressing x from one and putting it in the other. These questions carry 3 to 5 marks.
Key formulas and definitions
- tan θ = height / horizontal distance
- sin θ = height / slant length, cos θ = horizontal distance / slant length
- angle of depression = angle of elevation (alternate angles)
- tan 30° = 1/√3, tan 45° = 1, tan 60° = √3
- √3 ≈ 1.732, √2 ≈ 1.414
Worked examples
1. From a point 30 m from the foot of a tower, the angle of elevation of its top is 45°. Find the height.
tan 45° = h/30, so h = 30 × 1 = 30 m.
2. A kite string is 60 m long and makes 60° with the ground. Find the height of the kite (string straight).
sin 60° = h/60, so h = 60 × √3/2 = 30√3 ≈ 51.96 m.
3. A pole casts a shadow 10√3 m long when the sun's elevation is 30°. Find the pole's height.
tan 30° = h/(10√3), so h = 10√3 × 1/√3 = 10 m.
4. From the top of a 75 m lighthouse, the angle of depression of a boat is 30°. How far is the boat from the lighthouse?
Angle of elevation from the boat = 30°. tan 30° = 75/d, so d = 75√3 ≈ 129.9 m.
5. A ladder leans on a wall making 60° with the ground. Its foot is 2.5 m from the wall. Find the ladder's length.
cos 60° = 2.5/L, so L = 2.5 ÷ 1/2 = 5 m.
6. The angle of elevation of a tower's top changes from 30° to 60° when you walk 20 m towards it. Find the height.
Let height h and near distance x. tan 60° = h/x → h = √3 x. tan 30° = h/(x + 20) → √3 h = x + 20. Put x = h/√3: √3 h = h/√3 + 20 → 3h = h + 20√3 → h = 10√3 ≈ 17.32 m.
7. From a point on the ground, the angles of elevation of the bottom and top of a 10 m flagstaff on a building are 45° and 60°. Find the height of the building.
Let building height h and distance d. tan 45° = h/d → d = h. tan 60° = (h + 10)/d → √3 h = h + 10 → h = 10/(√3 − 1) = 5(√3 + 1) ≈ 13.66 m.
Common mistakes
- Measuring the angle of depression from the vertical tower instead of the horizontal line.
- Using sin when you know the height and ground distance. Height with ground → tan.
- Forgetting to add the observer's height when the question gives eye height (e.g. a 1.5 m tall boy).
- In two-triangle problems, using the full distance for the near triangle. Each triangle has its own base.