CBSE Class 10 Maths: Heights & Distances — Notes 2026

Angle of Elevation and Depression

• Angle of elevation: angle between horizontal and line of sight to object above horizontal
• Angle of depression: angle between horizontal and line of sight to object below horizontal
• Key: angle of elevation from A to B = angle of depression from B to A (alternate interior angles)

Using Trigonometric Ratios

• tan θ = opposite/adjacent → use when height and horizontal distance involved
• sin θ and cos θ: use when slant distance (hypotenuse) is given or required
• Standard values: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3; sin 30° = ½, cos 60° = ½

Single Observation Problems

• Tower of height h observed from point P at angle of elevation α: tan α = h/d → d = h/cot α
• Example: pole 20 m high; angle of elevation 60°: d = 20/tan 60° = 20/√3 = 20√3/3 m
• Draw neat diagram; label angle, height (opposite), distance (adjacent); identify which trig ratio to use

Two Observation Problems

• Observer at base of building; two poles observed at different angles of elevation
• Two people observing same tower from opposite sides at angles α and β: height = d×tan α×tan β / (tan α + tan β)
• Standard CBSE problem: from top of hill, angles of depression to two objects on ground

Problem with Height Change

• Nearer the object, greater the angle of elevation; moving towards tower increases angle
• Moving problem: man walks d metres towards tower; angle changes from α to β; find height: h = d/(cot α − cot β)
• CBSE frequently asks: from top of lighthouse, angles of depression to two ships; find distance between ships

Common Exact Values to Memorize

• sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3
• sin 45° = cos 45° = 1/√2, tan 45° = 1
• sin 60° = √3/2, cos 60° = ½, tan 60° = √3; sin 90° = 1, cos 90° = 0

CBSE Exam Strategy

• Heights & Distances: 4–6 marks in CBSE; usually 1 long question
• Always draw labelled diagram first — diagrams earn marks even if calculation has errors
• Express answers in simplified surd form: 20√3/3, not 11.55; rationalize denominators