Trigonometry & Height and Distances

Module 18 — Lesson 0018 · In Depth · 40 min

Every height-and-distance problem is one or more right triangles. Identify P (opposite), B (adjacent), H (hypotenuse) relative to the observer's angle — then pick the ratio that connects what you know to what you need. tan θ is king because it links height directly to ground distance.

1. The Foundation: The Right-Angled Triangle

All height-and-distance problems reduce to right triangles. Label sides relative to the angle of observation θ:

Side Also Called Role
Perpendicular (P) Opposite Facing θ — usually the height (tower, building)
Base (B) Adjacent Touching θ — usually ground distance
Hypotenuse (H) — Opposite 90°, longest side — ladder, kite string, line of sight

Pythagoras: H² = P² + B² — use when two sides are known and you need the third, without any trig.

Right triangle labeling relative to observer angle θ θ P (height) B (ground) H (line of sight) Top Observer Base (90°)
All H&D problems are this triangle — label P/B/H relative to θ, then pick sin/cos/tan.

2. Trigonometric Ratios — SOH CAH TOA

Ratio Formula Typical Use in H&D
sin θ P / H Ladder / kite string problems (you know slant length)
cos θ B / H Horizontal distance from slant length
tan θ P / B Most common — height ↔ ground distance

Why tan dominates: towers are vertical (P) and ground is horizontal (B) — you almost always know or want one of those and the angle between them. tan directly connects them without needing the hypotenuse.

Mnemonic: SOH (Sin = Opp/Hyp), CAH (Cos = Adj/Hyp), TOA (Tan = Opp/Adj). Also: sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ.

Standard Angles — Memorize

30°, 45°, 60° cover 90% of aptitude problems. Also know complements (θ and 90°−θ swap sin↔cos).

θ sin θ cos θ tan θ
30° 1/2 √3/2 1/√3
45° 1/√2 1/√2 1
60° √3/2 1/2 √3

Values: √3 ≈ 1.732, √2 ≈ 1.414, 1/√2 ≈ 0.707, 1/√3 ≈ 0.577. Pattern: sin increases 1/2 → 1/√2 → √3/2 as angle grows; cos is the reverse.

3. Key Terminology

Crucial rule: Angle of depression = angle of elevation (alternate interior angles with the two horizontal lines). So a "depression 30° from a 50m cliff" is mathematically identical to "elevation 30° from boat to cliff top" — just flip the triangle. Draw the horizontals and the transversal (line of sight) — the Z-shape proves equality.
Angle of depression equals angle of elevation — alternate interior angles horizontal horizontal Eye Boat/base θ = depression θ = elevation Z
Depression and elevation are the same angle — alternate interior angles of the “Z” formed by two horizontals and the line of sight.

4. Common Problem Scenarios & Methods

Case 1: The Broken Tree

A tree breaks and the top touches the ground — forming a right triangle.

Broken tree forming a right triangle ground break point tip touches ground P standing H broken (slant) B Total height = P + H
Broken tree: standing = P, broken = H, ground = B → H² = P² + B² and total = P + H.
Worked: 15m tree breaks, top touches ground 9m from base. Let standing = x, broken = 15−x = H. Then H² = x² + 9² → (15−x)² = x² + 81 → 225 −30x + x² = x² +81 → 30x=144 → x=4.8m standing, 10.2m broken.

Case 2: Shadow Problems

Compare height to shadow length via tan θ = height / shadow:

Condition tan θ θ
Height = Shadow 1 45°
Height = √3 × Shadow √3 60°
Shadow = √3 × Height 1/√3 30°

Intuition: taller shadow → smaller angle. At noon shadow is short (high sun, 60°), at evening shadow is long (low sun, 30°).

Case 3: Two Objects on the Same Side

Observer sees two points in a straight line from the base of a tower. Angles α (nearer, larger) and β (farther, smaller), height h.

Distance between points = h (cot β − cot α) — because each distance from base is h × cot(angle).

Derivation: Let distances from base be d₁ (near) and d₂ (far). tan α = h/d₁ → d₁ = h·cot α, similarly d₂ = h·cot β. Separation = d₂ − d₁ = h(cot β − cot α). Since α>β, cot β > cot α, so positive.

Case 4: Two Objects on Opposite Sides

Observer (or tower) between two objects.

Distance between points = h (cot α + cot β) — sum, not difference, because distances are on opposite sides of the base.

Same side vs opposite sides of a tower Same side separation = h(cot β − cot α) top base h near far α (larger) β (smaller) to find Opposite sides separation = h(cot α + cot β) top h left right α β to find
Same side → subtract ground distances; opposite sides → add. Both distances are h·cot(angle).

5. Advanced Concepts

a) Complementary Angles

If angles of elevation from two points at distances a and b are complementary (sum 90°):

Height h = √(a × b)

Why: Let angles be θ and 90°−θ. Then tan θ = h/a and tan(90°−θ) = cot θ = h/b → (h/a) × (h/b) = tan θ × cot θ = 1 → h² = ab. So height is the geometric mean of the two distances.

b) Reflection in a Lake (Cloud Problem)

Cloud at height H above lake, reflection at depth H below. Observer at height h above lake, elevation to cloud = α, depression to reflection = β.

H = h × (tan β + tan α) / (tan β − tan α)

Setup: Cloud height above observer = H−h, reflection depth below observer = H+h. Then tan α = (H−h)/d, tan β = (H+h)/d where d is horizontal distance. Solve two equations for H.

c) Movement and Speed

If a car/boat moves toward a tower and the angle changes from α to β in time t:

  1. Compute distance traveled d = h(cot α − cot β) (if moving toward) using Case 3.
  2. Speed = d / t.

Same formula works away from tower with sign flipped.

6. Summary of Variables

Symbol Meaning
h, H Height of object (tower, hill)
d, x Horizontal distance
θ, α, β Angles of elevation/depression
L Length (ladder, string)

In a right triangle, tan θ = ?

tan 45° = ?

Height = Shadow. Angle of elevation?

Angle of depression from a cliff equals?

Broken tree: standing part, broken part, total height?

Two points on same side of tower, angles α>β, separation = ?

Two points on opposite sides of tower, separation = ?

Two complementary elevation angles from distances a and b: height h = ?

Notes

Right Triangle :-

P ⇒ opposite θ (height) ; B ⇒ adjacent θ (ground) ; H ⇒ hypotenuse (slant)

H² = P² + B²

Ratios :- SOH CAH TOA

sin ⇒ P/H ; cos ⇒ B/H ; tan ⇒ P/B → most used (height/ground)

tan = sin/cos ; sin²+cos²=1

30° ⇒ sin 1/2, cos √3/2, tan 1/√3 ; 45° ⇒ 1/√2, 1/√2, 1 ; 60° ⇒ √3/2, 1/2, √3

Terminology :-

Elevation ⇒ up from horizontal ; Depression ⇒ down from horizontal

Depression = Elevation (alternate interior, Z-shape) × equal

Cases :-

a) Broken tree ⇒ standing P + broken H = total ; H²=P²+B²

b) Shadow ⇒ tan = h/shadow ; h=shadow→45° ; h=√3·shadow→60° ; shadow=√3·h→30°

c) Same side ⇒ separation = h(cot β − cot α) (α>β)

d) Opposite sides ⇒ separation = h(cot α + cot β)

Advanced :-

Complementary (sum 90°) ⇒ h = √(a×b) (GM of distances)

Reflection (cloud) ⇒ H = h·(tanβ+tanα)/(tanβ−tanα)

Movement ⇒ distance h(cotα−cotβ) → speed = d/t

Variables :-

h,H ⇒ height ; d,x ⇒ horizontal distance ; θ,α,β ⇒ angles ; L ⇒ slant length

Primary source: none pinned yet — drop your preferred video/book resource into RESOURCES.md and it will be linked here. Ask me anything that's unclear.

← Prev: Mensuration 2D Next: Progressions →

Questions? Ask your agent — you can follow up on any concept, quiz answer, or get extra practice problems tuned to this module.