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§9Class 10, Chapter 9

Some Applications of Trigonometry: Heights and Distances You Can't Measure Directly

This companion covers how a single known angle and a single known side unlock the height of a tower, the width of a river, or the length of a shadow — including trickier two-triangle problems with flagstaffs, statues, and pairs of buildings, with tools, flashcards and a CBSE-style quiz to test yourself.

angle of elevationangle of depressiontan θ = opp/adjline of sight

👁️9.1 Line of Sight, Elevation, and Depression

In the last chapter you learned the six trigonometric ratios. This chapter puts them to work on a genuinely practical problem: finding heights and distances you can't directly measure — the height of a tower you can't climb, the width of a river you can't wade across, the altitude of a balloon drifting overhead.

θ
Angle of Elevation
Looking UP at an object above the horizontal
θ
Angle of Depression
Looking DOWN at an object below the horizontal

Picture a student looking up at the top of a minar. The line drawn from the student's eye straight to the top of the minar is called the line of sight. The angle this line makes with the horizontal is called the angle of elevation — it's the angle formed whenever you raise your head to look at something above you.

👁️Line of Sight. The line of sight is the line drawn from the eye of an observer to the point on the object being viewed.
⬆️Angle of Elevation. The angle of elevation of a point viewed is the angle formed by the line of sight with the horizontal, when the point is above the horizontal level (i.e., when you raise your head to look at it).
⬇️Angle of Depression. The angle of depression of a point viewed is the angle formed by the line of sight with the horizontal, when the point is below the horizontal level (i.e., when you lower your head to look at it) — like a girl on a balcony looking down at a flower pot across a river.
🙂
Think about your own chin.. Elevation is the angle your chin tilts UP to see something high up. Depression is the angle your chin tilts DOWN to see something below you. Both angles are always measured from the same reference: the horizontal line straight out from your eye.

🗺️The General Roadmap

To find the height CD of a minar without measuring it, you'd need exactly three pieces of information: the distance you're standing from its foot, the angle of elevation of its top, and your own eye-level height (since your line of sight starts from your eyes, not the ground).

SPLIT THE HEIGHTThe total height CD = CB (the part above your eye level) + BD (your own eye height, since BD = AE).
CHOOSE A RATIOIn the right triangle formed, side BC is opposite the known angle A, and AB is the known horizontal distance. Since these are the two sides you have and need, use tan A = BC/AB (or its reciprocal, cot A).
SOLVE FOR THE UNKNOWN SIDERearranging gives BC — the extra height above your eye level.
ADD BACK THE OBSERVER'S HEIGHTAdd AE (your eye height) to BC to get the full height CD of the object.
⚠️Don't forget the observer's own height.. It's easy to solve the triangle correctly and then report BC as "the height" — but if the angle was measured from eye level rather than ground level, you must add the observer's height back in to get the true, ground-to-top height.

🗼9.1 Worked Examples — One Right Triangle at a Time

Many height-and-distance problems boil down to a single right triangle: one known angle, one known side, one thing to find. The only real decision is which trigonometric ratio connects what you know to what you want.

A tower stands vertically on the ground. From a point 15 m from its foot, the angle of elevation of its top is 60°. Find the height of the tower.

SET UPLet AB be the tower (height to find), and CB = 15 m be the known horizontal distance, with ∠ACB = 60°.
CHOOSE THE RATIOtan 60° connects AB (opposite) and CB (adjacent): tan 60° = AB/BC
SOLVE√3 = AB/15 ⟹ AB = 15√3 m ≈ 25.98 m
🪤The most common mistake in this whole chapter.. Students frequently solve for the triangle's side correctly, then forget to add (or subtract) the observer's height, ladder-target offset, or similar fixed adjustment mentioned in the problem — always re-read the question to check what final quantity is actually being asked for.

🏙️9.1 Worked Examples — Two Right Triangles Together

Some problems give you two different angles of elevation or depression to two different points on the same vertical line (or from two different observing points). These need two separate right triangles and two equations solved together — but the strategy is still the same trigonometry, just applied twice.

From a point P, the angle of elevation of the top of a 10 m building is 30°, and of the top of a flagstaff on the building is 45°. Find the flagstaff's length and the distance of the building from P. (√3 = 1.732)

TRIANGLE 1 — JUST THE BUILDINGtan 30° = AB/AP = 10/AP ⟹ AP = 10√3 ≈ 17.32 m (this is the distance of the building from P).
TRIANGLE 2 — BUILDING PLUS FLAGSTAFFLet DB = x (flagstaff length). tan 45° = AD/AP = (10+x)/(10√3)
SOLVE1 = (10+x)/(10√3) ⟹ x = 10(√3−1) ≈ 7.32 m — the flagstaff's length.

📝9.2 Chapter Summary

  1. Line of sight: the line from the observer's eye to the point being viewed.
  2. Angle of elevation: the angle between the line of sight and the horizontal, when looking UP at something above the horizontal level.
  3. Angle of depression: the angle between the line of sight and the horizontal, when looking DOWN at something below the horizontal level.
  4. The core technique: heights, lengths, and distances of otherwise-inaccessible objects can all be found using trigonometric ratios, given one known side and one known angle of a right triangle.
  5. Two-angle problems (flagstaffs, statues on pedestals, shadows, two buildings, river widths from bridges) need two right triangles solved together, often via the alternate-angles relationship between elevation and depression.
🛰️
One chapter, endless real applications.. This exact technique, using known angles and distances to find inaccessible heights, is the working basis of surveying, navigation, astronomy, and architecture.