Heights & Distances: JEE Main Trigonometry Guide

Heights and distances is one of the lighter chapters in JEE Main Mathematics, contributing one question per session. It is almost entirely applied trigonometry — setting up a triangle, identifying the known and unknown, and solving using the right trigonometric ratio. The key skill is translating the word problem into a clear diagram and identifying the right triangle(s) to work with.

Key Terminology and Setup

Angle of elevation: the angle from the horizontal line of sight upward to the object (looking up). Angle of depression: the angle from the horizontal line of sight downward to the object (looking down). Both are measured from the horizontal. Since horizontal lines are parallel, the angle of depression from a height equals the angle of elevation to that height from the base (alternate interior angles). JEE Main uses this alternate angle equality in problems involving a tower on top of a hill or an observer at a height looking down at an object.

Basic trigonometric ratios: sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent. For heights and distances, tanθ is used most frequently: tan(elevation) = height/horizontal distance. For the full trigonometry foundation, see our Trigonometry Guide.

Standard Single-Triangle Problems

Tower problem: a tower of height h casts a shadow of length l. Find the elevation angle: tanθ = h/l. If the elevation angle is θ = 60°: tan60° = √3, so h = l√3. JEE Main gives you two of the three quantities (height, shadow, angle) and asks for the third. Substitute tanθ for the given angle immediately and solve.

Observer and tower: from a point 100 m from the base of a tower, the angle of elevation of the top is 30°. Height of tower: h = 100 × tan30° = 100/√3 = 100√3/3 m. JEE Main also asks for the angle of elevation at a different distance — set up a new right triangle with the new horizontal distance and the same height. Take a free mock test on trigonometry applications including heights and distances.

Two-Triangle Problems

Most JEE Main heights and distances problems involve two angles and require setting up two right triangle equations. The observer moves between two positions and angles of elevation change. Standard setup: from point A, elevation to tower top is α; from point B (closer, on the same line), elevation is β (β > α). Let tower height = h, distance from B to base = x. Then tanβ = h/x and tanα = h/(x + AB). Divide or subtract to eliminate x and solve for h in terms of AB, α, β.

Result formula: h = AB × tanα × tanβ / (tanβ − tanα). This formula covers the two-position tower problem completely. Memorise it — it is faster than re-deriving under exam pressure. Verify with numbers: if AB = 50, α = 30°, β = 60°: h = 50 × (1/√3) × √3 / (√3 − 1/√3) = 50 × 1 / (2/√3) = 50√3/2 = 25√3 m.

Elevation and Depression from a Height

An observer at the top of a tower looks down at objects at the base and at a distance. Angle of depression = arctan(height/horizontal distance). JEE Main sets up a scenario: from the top of a cliff 80 m high, the angle of depression to two boats is 30° and 45°. On the same side: boat A (30°, farther) at distance 80/tan30° = 80√3 m, boat B (45°, closer) at 80/tan45° = 80 m from the cliff base. Distance between boats: 80√3 − 80 = 80(√3 − 1) m.

Problems with Bearings and Non-Collinear Observers

Advanced heights and distances (JEE Main integer-type): observers at two positions not in a straight line with the tower base. Use the sine rule or cosine rule in the triangle formed by the two observer positions and the tower base. JEE Main occasionally sets this up — draw the plan view (top view), identify the triangle, and apply the appropriate rule. For the sine and cosine rule reference, see our Properties of Triangles Guide.

Exam Strategy

Heights and distances is a diagram-first chapter. Draw the figure carefully — label all known angles, heights, and distances before writing any equation. The most common errors are: (1) confusing angle of elevation and depression; (2) using the wrong triangle; (3) algebraic errors when solving two simultaneous equations for height. The two-triangle formula h = AB × tanα × tanβ / (tanβ − tanα) covers the single most common question type — memorise it. Upgrade for ₹149/month for 100+ heights and distances problems at all levels.