Some Applications of Trigonometry

Heights & Distances · Mathematics Ch.9 · Class X CBSE

📐 Line of Sight, Elevation & Depression

Adjust the sliders to see how angle of elevation changes the line of sight.
Definition

Angle of Elevation

When we look upward at an object, the angle between our horizontal line of sight and the line to the object is the angle of elevation.

Definition

Angle of Depression

When we look downward at an object, the angle between our horizontal line of sight and the line to the object is the angle of depression.

Key Property

Alternate Angles

Angle of depression from point A to B = Angle of elevation from B to A (by alternate interior angles with a horizontal transversal).

💡 The angle is always measured from the horizontal, never from the vertical. This is the most common mistake in exams!

🏗️ Finding Heights

Height = distance × tan(angle)
Formula
height = distance × tan θ
Where θ = angle of elevation from the point on the ground to the top of the object
Example

Tower Height

A tower's angle of elevation from a point 20 m away is 60°.

h = 20 × tan 60° = 20√3 ≈ 34.64 m

Example

Pole with Shadow

A 6 m pole casts a 2√3 m shadow. Find the sun's elevation angle.

tan θ = 6/(2√3) = √3 → θ = 60°

Remember

Observer's Height

If the observer has height (e.g., 1.5 m), add it to the calculated perpendicular distance to get the total height of the object.

📏 Finding Distances

Distance = height / tan(angle of depression)
Formula
distance = height / tan θ = height × cot θ
Where θ = angle of depression from the top of the structure to the object below
Example

Ship from Lighthouse

From a 150 m lighthouse, angle of depression to a ship = 30°.

d = 150/tan 30° = 150√3 ≈ 259.8 m

Example

Car from Building

From a 60 m building top, angle of depression to a car = 45°.

d = 60/tan 45° = 60/1 = 60 m

💡 When angle = 45°, height = distance (since tan 45° = 1). When angle = 30°, distance = √3 × height. When angle = 60°, distance = height/√3.

🔺 Two-Triangle Problems (HOTS)

Set up two tan equations with the common height h, then eliminate the extra variable.
Method — Two angles from two points (same side)
h = d × (tan α · tan β) / (tan β − tan α)
Where d = distance between the two observation points, α < β
5-Mark Pattern

Step-by-Step

  • Let height = h, base to near point = x
  • tan β = h/x → x = h/tan β
  • tan α = h/(x + d) → x + d = h/tan α
  • Subtract: d = h/tan α − h/tan β
  • Solve for h
Example

Classic CBSE Problem

Angles of elevation of a tower from two points 40 m apart are 30° and 60°.

h = 40 × tan30°·tan60° / (tan60°−tan30°)

= 40 × (1/√3)(√3) / (√3−1/√3) = 40/(2/√3) = 20√3 m

Caution

Same Side vs Opposite

If two points are on same side: subtract distances.
If on opposite sides: add distances.
Read the question carefully!

🌍 Real-World Applications

Surveying

Measuring Mountain Heights

Surveyors use theodolites to measure angles from known baselines. The Great Trigonometric Survey of India used this to calculate the height of Mt. Everest (8848 m) in the 1850s!

Navigation

Ship & Aircraft Positioning

Lighthouses, radar, and pilots use angles of depression/elevation to determine distances. Air traffic controllers calculate descent paths using trigonometry.

Architecture

Building Design

Roof slopes, ramp angles, staircase dimensions, and shadow analysis for sunlight planning all use heights and distances principles.

Astronomy

Stellar Distances

Parallax method uses trigonometry to find distances to stars. Earth's orbit serves as the baseline, and tiny angle shifts give stellar distances.

Military

Range Finding

Artillery uses angles of elevation to calculate projectile range. Spotters use angles of depression to estimate enemy positions.

Everyday

Satellite Dishes

Satellite dish angle depends on your latitude. The dish must be tilted at the correct angle of elevation to point at the geostationary satellite.

Key Takeaway
Trigonometry lets us measure what we cannot reach
Heights of mountains, distances across rivers, depths of valleys — all computed from a safe observation point using angles and a single known measurement.