What is the angle of elevation?

The angle of elevation is the angle formed by the line of sight with the horizontal when looking up at an object above the horizontal level.

What is the angle of depression?

The angle of depression is the angle formed by the line of sight with the horizontal when looking down at an object below the horizontal level.

How do I choose which trigonometric ratio to use in a height-and-distance problem?

If both the known and unknown quantities are legs of the right triangle (height and horizontal distance), use tan θ. If one of them is the hypotenuse (like a ladder's length or a string's length), use sin θ or cos θ instead.

How do I solve a problem with two right triangles, like a flagstaff on a building?

Set up one trigonometric equation per triangle. If they share a common side (like the horizontal distance from the observer), solve the simpler triangle first and substitute that value into the second equation.

How do I solve a problem with two separate measurements of the same height, like a shadow at two different times?

Express the same unknown height using each triangle's own equation, then set the two expressions equal to each other. This eliminates the height and leaves one solvable equation in the remaining unknown.

Chapter 9 · CBSE Class 10 Maths

Some Applications of Trigonometry

Two ways to learn — a crisp formula reference, or a story that makes two-triangle problems feel obvious.

Angle of elevationAngle of depressionChoosing the right ratioTwo-triangle problems

Also useful:Chapter 8 — Introduction to Trigonometry Chapter 10 — Circles NCERT Solutions — Class 10 Maths 1-on-1 CBSE Class 10 Tuition

CBSEClass 10MathematicsChapter 9⏱ 8 min crisp · 11 min story

Four scenes that build the ideas from scratch, step by step.

How this works

Four scenes, one thread: every height-and-distance problem is a right triangle in disguise. Draw the line of sight, pick the ratio that connects what you know to what you want, and solve — except when the picture secretly hides two triangles, which needs two equations instead of one. Read the story, then tap "The Maths" for the formal result.

Looking Up, Looking Down

A student looks up at the top of a tall minar. The line from her eye to the minar's tip is called the line of sight, and the angle it makes with the horizontal — since she's looking UP — is the angle of elevation.

Teacher

Now flip the scene: a girl on a balcony looks down at a flower pot on a lower step. Same idea, but since she's looking DOWN, that angle has a different name.

Looking UP at something above eye level → angle of ELEVATION

Looking DOWN at something below eye level → angle of DEPRESSION

To find the minar's height without measuring it directly, three things are needed: the distance to its foot, the angle of elevation, and the observer's own eye-height (since the triangle's calculation only reaches up to eye level — the observer's height has to be added back on top).

The Tower and the Ladder

A tower's angle of elevation is 60° from a point 15 m away. Height and distance are both legs of the right triangle — so tan connects them directly.

Student

tan 60° = height/15, so height = 15 × √3. That's straightforward when both quantities are legs. What if the unknown is the SLANTED side instead — like a ladder?

Tower: tan 60° = height/15 → height = 15√3 ≈ 25.98 m

An electrician needs her ladder, inclined at 60°, to reach a point 3.7 m up a pole. The ladder itself IS the hypotenuse now — height and hypotenuse call for sin, not tan.

sin 60° = 3.7/ladder length

ladder length = 3.7 / (√3/2) = 7.4/√3 ≈ 4.28 m

The choice of ratio always comes down to: what do you already know, and what do you want? If both are legs (height & horizontal distance), use tan. If one of them is the hypotenuse, use sin or cos instead.

Two Triangles, One Picture

A 10 m building has a flagstaff on its roof. From point P on the ground, the building's top has elevation 30°, and the flagstaff's top (from the same P) has elevation 45°. Find the flagstaff's length.

Student

There are two different heights here (the building, and building+flagstaff) but only one horizontal distance AP. Is one equation enough?

No — two unknowns (AP, and the flagstaff length) need two equations, one from EACH right triangle sharing that same base AP.

Triangle 1 (to building top): tan 30° = 10/AP → AP = 10√3

Triangle 2 (to flagstaff top): tan 45° = (10+x)/AP → 10+x = AP = 10√3

x = 10√3 − 10 ≈ 7.32 m

The first triangle's result (AP) feeds directly into the second equation — solving them in sequence, not simultaneously from scratch, since AP is shared.

The Shrinking Shadow

A tower's shadow is 40 m longer when the sun's angle is 30° than when it's 60°. Neither the tower's height nor either shadow length is known upfront — genuinely two unknowns, needing two full equations this time (not one feeding into the other).

Teacher

Let the height be h and the SHORTER shadow (at 60°) be x. The longer shadow is then x+40. Write one tan equation for each sun angle.

At 60°: tan 60° = h/x → h = √3·x

At 30°: tan 30° = h/(x+40) → h = (x+40)/√3

Equate: √3·x = (x+40)/√3 → 3x = x+40 → x = 20

Substituting back, h = √3 × 20 = 20√3 m. Both unknowns fall out together because the SAME height h was expressed two different ways — once per triangle — and setting those two expressions equal eliminates h, leaving one equation in x alone.

This "express the same quantity two ways, then equate" move is the general technique whenever a problem gives two separate angle/distance measurements of the same object — whether it's a shrinking shadow, two buildings, or a bridge over a river.

The Big Picture

Every idea in this chapter branches from one habit: draw the line of sight, identify the right triangle, and pick the ratio connecting what's known to what's wanted. The only extra skill is spotting when a picture secretly holds TWO triangles — sharing a base, or sharing a height — which calls for two equations instead of one.

Looking up vs looking down?

Elevation (up) vs depression (down)

Unknown is a leg (height/distance)?

Use tan θ = opposite/adjacent

Unknown is the hypotenuse?

Use sin θ or cos θ

Two triangles sharing a base?

Solve one, substitute into the other

Two triangles, same unknown height?

Express h two ways, set them equal

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