Is it correct to say: Measuring the angle of depression from the vertical?

Both elevation and depression are measured from the HORIZONTAL line through the observer's eye.

Is it correct to say: Placing the angle of depression at the observer's corner of the triangle?

Transfer it to the other end using alternate angles: depression at the top = elevation at the bottom point.

Is it correct to say: Ignoring the observer's height?

If the angle is measured from the eyes of a 1.5 m tall person, the calculated height is above eye level. Add 1.5 m.

Is it correct to say: Using sin when the horizontal distance is given?

Height with horizontal distance → tan. Use sin or cos only with a slant length such as a string or ladder.

Is it correct to say: Rounding √3 too early?

Work in surd form and substitute √3 = 1.732 only at the end, otherwise rounding errors grow.

Is it correct to say: Mixing up which angle belongs to the nearer point?

The nearer point always sees the LARGER angle of elevation. If you get a negative distance, the angles are swapped.

📄 Formula sheet (PDF)

Some Applications of Trigonometry

Line of sight, angles of elevation and depression, and heights-and-distances problems using trigonometric ratios — NCERT Class 10 Maths Ch 9

📐 12 formulas✏️ 4 examples🎯 6 practice⚖️ Unit V (Trigonometry): 12 marks (with Ch 8)🏫 CBSE📚 Class 10✓ 2025–26 syllabus

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Board Exam Tips

→Draw a neat labelled figure first: mark the horizontal line, the vertical object, the angle and every known length. Most errors start with a wrong figure.
→The CBSE syllabus limits this chapter to angles of 30°, 45° and 60° and to problems with at most two right triangles — know tan, sin and cos of these angles cold.
→If the height and the horizontal distance are involved, use tan. Use sin or cos only when a slant length (string, ladder, line of sight) appears.
→Angle of depression at the top equals angle of elevation at the bottom (alternate angles). Mark it inside the right triangle before writing any ratio.
→Keep answers in surd form (20√3 m) until the last step; use √3 = 1.732 only if the question asks for a decimal value.

📐 Formulas(12)

Angle of Elevation★ Board fav

tan θ = \frac{h}{d}

Symbol	Meaning
$θ$	Angle of elevation (degrees)
$h$	Height of the object above eye level (m)
$d$	Horizontal distance from the observer to the foot of the object (m)

Height and Distance from tan

h = d tan θ, d = \frac{h}{t a n θ} = h cot θ

Slant Length: sin and cos

h = l sin θ, d = l cos θ

Symbol	Meaning
$l$	Slant length — hypotenuse of the right triangle (m)
$h$	Vertical height (m)
$d$	Horizontal distance (m)

Angle of Depression★ Board fav

θ_{depression} = θ_{elevation} (alternate angles)

Allowing for the Observer's Height★ Board fav

H = h_{o} + d tan θ

Symbol	Meaning
$H$	Total height of the object (m)
$h_{o}$	Height of the observer's eyes above the ground (m)
$d$	Horizontal distance from the observer to the object (m)

tan Values Used in This Chapter

tan 3 0^{\circ} = \frac{1}{3}, tan 4 5^{\circ} = 1, tan 6 0^{\circ} = 3

sin and cos Values Used in This Chapter

sin 3 0^{\circ} = cos 6 0^{\circ} = \frac{1}{2}, sin 6 0^{\circ} = cos 3 0^{\circ} = \frac{3}{2}, sin 4 5^{\circ} = cos 4 5^{\circ} = \frac{1}{2}

Shadow Length

s = h cot θ = \frac{h}{t a n θ}

Symbol	Meaning
$s$	Length of the shadow (m)
$h$	Height of the vertical object (m)

Two Points on the Same Side★ Board fav

x = h (cot α - cot β) \Rightarrow h = \frac{x}{c o t α - c o t β}

Symbol	Meaning
$x$	Distance between the two observation points (m)
$h$	Height of the object (m)
$α, β$	Angles of elevation from the farther and nearer points

Special Case: 30° and 60°

α = 3 0^{\circ}, β = 6 0^{\circ} \Rightarrow h = \frac{3}{2} x, B D = \frac{x}{2}

Two Points on Opposite Sides

x = h (cot α + cot β)

Building and Tower

d = h cot β, H = h + d tan α

Symbol	Meaning
$h$	Height of the building (m)
$H$	Height of the tower (m)
$d$	Horizontal distance between building and tower (m)

✏️ Solved Examples

1Solved Exampleeasy3 steps

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

Height and horizontal distance are involved, so use tan.

tan 6 0^{\circ} = \frac{h}{30}

2Solved Exampleboard4 steps

From the top of a building 60 m high, the angle of depression of a car on the road is 30°. How far is the car from the foot of the building?

The angle of depression from the top equals the angle of elevation of the top seen from the car (alternate angles). So, at the car, the angle is 30°.

3Solved Exampleboard4 steps

The angle of elevation of the top of a tower from a point on the ground is 30°. After walking 40 m towards the tower, the angle of elevation becomes 60°. Find the height of the tower and the distance of the second point from its foot.

Let the height be h and the distance from the second point to the foot be y. At the second point (60°):

tan 6 0^{\circ} = \frac{h}{y} \Rightarrow h = 3 y (1)

4Solved ExampleHOTS5 steps

From the top of a 20 m high building, the angle of elevation of the top of a tower is 30° and the angle of depression of its foot is 45°. Find the height of the tower.

Draw a horizontal line from the building's top to the tower. It meets the tower at a point 20 m above the ground. Let the horizontal distance between them be d.

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Measuring the angle of depression from the vertical
✓Both elevation and depression are measured from the HORIZONTAL line through the observer's eye.
2
Placing the angle of depression at the observer's corner of the triangle
✓Transfer it to the other end using alternate angles: depression at the top = elevation at the bottom point.
3
Ignoring the observer's height
✓If the angle is measured from the eyes of a 1.5 m tall person, the calculated height is above eye level. Add 1.5 m.
4
Using sin when the horizontal distance is given
✓Height with horizontal distance → tan. Use sin or cos only with a slant length such as a string or ladder.
5
Rounding √3 too early
✓Work in surd form and substitute √3 = 1.732 only at the end, otherwise rounding errors grow.
6
Mixing up which angle belongs to the nearer point
✓The nearer point always sees the LARGER angle of elevation. If you get a negative distance, the angles are swapped.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
A kite is flying at a height of 75 m above the ground. Its string, assumed straight, makes an angle of 60° with the ground. Find the length of the string.
Q2
A 6 m ladder leans against a vertical wall and makes an angle of 60° with the ground. How high up the wall does it reach?
Q3
A boy whose eyes are 1.5 m above the ground stands 30 m from a building. The angle of elevation of the top of the building from his eyes is 60°. Find the height of the building.
Q4
Find how much longer the shadow of an 18 m pole is when the sun's altitude is 30° than when it is 60°.
Q5
Two poles of equal height stand on either side of a road 100 m wide. From a point on the road between them, the angles of elevation of their tops are 60° and 30°. Find the height of the poles and the distances of the point from them.
Q6
From the top of a lighthouse 90 m high, the angles of depression of two ships on the same side are 30° and 45°. Find the distance between the ships.

📝 Notes

Some Applications of Trigonometry

This chapter applies the ratios from Chapter 8 to measure heights and distances you cannot reach: towers, buildings, kites, ships and shadows.

Key terms

Line of sight: the line from the observer's eye to the object.
Angle of elevation: the angle between the line of sight and the horizontal when the object is above eye level.
Angle of depression: the angle between the line of sight and the horizontal when the object is below eye level.

Because horizontal lines are parallel, the angle of depression seen from A equals the angle of elevation seen from B. Use this to move the angle into a right triangle.

Which ratio to use

Known and wanted	Ratio
Height and horizontal distance	tan θ = h/d
Height and slant length	sin θ = h/l
Horizontal distance and slant length	cos θ = d/l

Problems with two triangles

When two angles are given, set up one tan equation per triangle and link them through the shared length (the height of the tower, or the common horizontal distance). Typical patterns:

Walking towards a tower: the two angles share the height h.
Two ships or two cars seen from a lighthouse: both triangles share the height.
Building and tower: the horizontal line from the building's top splits the tower into two parts.
Two poles across a road: the two distances add up to the road's width.

Answer style

Give the answer in surd form first (for example 20√3 m), then use √3 = 1.732 if a decimal is asked. Always attach the unit. A quick sanity check: at 45° the height equals the horizontal distance, and the nearer point always sees the larger angle.