Some Applications of Trigonometry

Play with it

Measure a tower without climbing

Set how far you stand (d) and the angle of elevation θ to the top. The height comes straight from h = d · tan θ — watch it update.

height h = d · tan θdrag d and θ

distance d 30 m

angle of elevation θ 35°

Try:

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The three ideas in this chapter

The line of sight is the straight line from your eye to the object you're looking at. The angle of elevation is the angle this line makes with the horizontal when the object is above you.

This sets up a right triangle: the height is the side opposite the angle, the ground distance is adjacent, so:

tan θ = height / distance → height = d · tan θ

Watch out: if you measure from eye level, the object's full height = (eye height) + d·tan θ. Many problems take the observer at ground level to keep it simple.

When you look down at something below you (a boat from a cliff, the ground from a plane), the angle below the horizontal is the angle of depression.

The handy trick

The angle of depression from the top equals the angle of elevation from the bottom — they're equal alternate angles between two parallel horizontals. So you can solve the triangle from whichever end is easier.

Common mistake: measuring the angle of depression from the vertical. It's always measured down from the horizontal, just like elevation is measured up from it.

Draw the triangle, mark what you know and what you want, then pick the ratio that links them: tan for height-vs-distance, sin/cos when the hypotenuse (a rope, ladder or line of sight) is involved.

Worked example · height of a tower

From a point 30 m from the foot of a tower, the angle of elevation of its top is 60°. Find the height.

Known: d = 30 m (adjacent), θ = 60°. Want: height h (opposite).
tan 60° = h/30 → h = 30 · tan 60° = 30√3.
h = 30√3 ≈ 51.96 m.

Worked example · ladder length

A ladder makes 60° with the ground and reaches 6 m up a wall. Find its length.

Here 6 m is opposite, the ladder L is the hypotenuse → use sin.
sin 60° = 6/L → L = 6 / sin 60° = 6 / (√3/2) = 12/√3.
L = 4√3 ≈ 6.93 m.

Why this matters

Where you'll actually use this

This is the chapter that built the world's maps. One angle and one distance let surveyors, pilots and astronomers measure things far too big or too far to reach with a tape.

Aviation & landing

A pilot at a known altitude reads the angle of depression to the runway and instantly knows the horizontal distance still to fly (d = h / tan θ). Air-traffic control and autopilot glide-slopes are built on exactly this triangle.

Angle of depression

Surveying & mapping

Surveyors point a theodolite (an angle-measuring telescope) at a peak from a measured baseline and compute its height with tan θ. This is how mountains were measured and how every construction site checks its levels today.

Heights & distances

🚢 Lighthouses & ships

Keepers find a ship's distance from the angle of depression to it.

🔭 Astronomy

The heights of lunar mountains and distances to stars were first found with angle measurements.

🏗️ Cranes & construction

Operators use elevation angles to position loads at the right height and reach.

🔒 More real-world applications

Lighthouses, astronomy, construction and more — each explained with a diagram. Free to unlock.

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Check yourself

Competency quiz

Modelled on CBSE's competency pattern — MCQ, assertion–reason and case-study items.

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