Worked example 1
Question: A point 20 m from a tower sees its top at 45°. Find the tower height, ignoring eye height.
Solution: tan45°=h/20. Since tan45°=1, h=20 m.
What this chapter covers
Model heights and horizontal distances with right triangles using angles of elevation and depression, usually 30°, 45° or 60°.
Angles of elevation and depression
The angle of elevation is measured upward from the horizontal; the angle of depression is measured downward. Draw a labelled right-triangle diagram before selecting a ratio. Keep the observer’s eye height and ground distance distinct.
Choosing the ratio
Use tanθ when the known horizontal side and unknown vertical side form opposite/adjacent. Use sine or cosine when a sloping distance is involved. If a problem has two right triangles, solve one relationship at a time.
Key formulae: tanθ = height ÷ horizontal distance; sinθ = height ÷ line of sight; cosθ = horizontal distance ÷ line of sight.
Worked examples: CBSE question patterns
These are original practice questions modeled on common board-exam patterns; they are not reproduced past-paper questions.
Worked example 2
Question: A kite string 40 m long makes a 30° angle with the ground. Find the kite’s vertical height above the holder’s hand.
Solution: sin30°=h/40, so h=40×1/2=20 m.
Practical activity / investigation
With teacher supervision, measure a flagpole’s shadow and use a clinometer made from a protractor, straw and thread to estimate its angle of elevation. Calculate height using tanθ, then compare with a safe direct measurement if available.
Record: Write the aim, materials, steps, observations and conclusion in your notebook. Follow teacher supervision for any classroom activity.
Quick practice with answers
Try each question before opening its solution.
Practice 1: A tree casts a 10√3 m shadow when the Sun’s elevation is 30°. Find its height.
tan30°=h/(10√3)=1/√3, so h=10 m.
Practice 2: A 10 m ladder reaches a wall at 60° to the ground. How high does it reach?
h=10sin60°=5√3 m.
Practice 3: A 15 m building is viewed at 45°. Find the horizontal distance.
tan45°=15/d, so d=15 m.
Revision checklist
- Can I explain the key idea without memorising a worked solution?
- Have I selected the right formula and written units where needed?
- Can I show each algebraic or geometric step clearly?
- Can I check my answer by substitution, estimation or a diagram?
Aligned to the Class X Mathematics curriculum for 2026–27 and the NCERT Mathematics textbook. Always confirm any school-specific internal assessment with your teacher.