Class 10 Mathematics – Some Applications of Trigonometry

📝 Section 1 — Classroom Notes

🌎 Real-World Connection: Imagine standing far from a tall building and wanting to know its height without climbing it. By measuring the distance from the base and the angle at which you look up to the top, trigonometry gives you the exact answer. Surveyors measure mountain heights, engineers design mobile towers, and sailors navigate at sea — all using these same ideas of heights and distances.

📐 Topic 1: What Are Heights and Distances

We use trigonometry to find heights.
We can measure tall objects without climbing.
We need an angle and one distance.
A right triangle is always formed.
This chapter uses sin, cos, tan ratios.

📐 Topic 2: Line of Sight

The line of sight goes from eye to object.
It is a straight line always.
It connects observer to the point seen.
It forms an angle with the horizontal.

📐 Topic 3: Angle of Elevation

When we look up, it is elevation.
The angle is above the horizontal line.
Used to find height of tall things.
Example: looking up at a kite.
The angle is always between 0° and 90°.

📐 Topic 4: Angle of Depression

When we look down, it is depression.
The angle is below the horizontal line.
Used from a high place looking down.
Example: looking down from a cliff.
Depression angle = elevation angle (alternate angles).

📐 Topic 5: How to Draw the Figure

Always draw a right triangle first.
The vertical line is the height.
The horizontal line is the distance.
Mark the angle at the observer point.
Label all known and unknown sides.

📐 Topic 6: Using tan for Heights

tan θ = opposite / adjacent.
Use tan when you know distance on ground.
tan θ = height / distance.
Most height problems use tan.

📐 Topic 7: Using sin and cos

sin θ = opposite / hypotenuse.
cos θ = adjacent / hypotenuse.
Use sin when a rope or wire is given.
Use cos to find the ground distance.
Pick the ratio that has one unknown.

📐 Topic 8: Two-Position Problems

Observer sees from two different spots.
Two angles of elevation are given.
Both triangles share the same height.
Write two equations using tan.
Solve them together to find the answer.

📐 Topic 9: Shadow Problems

The sun makes shadows on the ground.
Shadow length depends on the sun angle.
tan (sun angle) = height / shadow.
Higher sun = shorter shadow.

📐 Topic 10: Tips for Word Problems

Read the problem twice, slowly.
Draw the figure with all labels.
Mark the right angle clearly.
Choose sin, cos, or tan wisely.
Always write the final answer with units.

🎨 Section 2 — Easy Illustrations

Picture 1 — Elevation is looking up; Depression is looking down

Picture 2 — Use tan to find the height of a tower from distance

Picture 3 — The sun angle decides how long the shadow is

Picture 4 — Two positions give two angles and two equations

Picture 5 — Four simple steps solve every problem

✏️ Section 3 — Worked Examples

✏ Example 1

A kite string is 80 m long. The string makes a 45° angle with the ground. Find the height of the kite.

1The string is the hypotenuse = 80 m.

2We need the height (opposite side).

3Use sin because we have hypotenuse and need opposite.

4sin 45° = height / 80

5√2 / 2 = height / 80

6height = 80 × √2 / 2 = 40√2

🌟 Height of kite = 40√2 ≈ 56.6 m

✏ Example 2

A 12 m ladder leans on a wall. It makes a 30° angle with the ground. How high up the wall does it reach?

1The ladder is the hypotenuse = 12 m.

2We need the wall height (opposite side).

3Use sin because we have hypotenuse and need opposite.

4sin 30° = height / 12

51/2 = height / 12

6height = 12 × 1/2 = 6

🌟 The ladder reaches 6 m up the wall

✏ Example 3

From the top of a 45 m building, the angle of depression to a car is 45°. How far is the car from the base?

1Building height = 45 m (opposite side).

2We need the distance from base (adjacent side).

3Angle of depression = 45° = angle of elevation.

4Use tan because we have opposite and need adjacent.

5tan 45° = 45 / distance

61 = 45 / distance, so distance = 45

🌟 The car is 45 m from the base

📊 Section 4 — Cheat Sheet

🔎 Standard Trigonometric Values

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	Not defined

🔎 Which Ratio to Use?

You Know	You Need	Use
Distance + Angle	Height	tan
Hypotenuse + Angle	Height	sin
Hypotenuse + Angle	Distance	cos
Height + Angle	Distance	tan
Height + Distance	Angle	tan⁻¹

🔎 Common Problem Types

Problem Type	What to Do
Height of a tower	Use tan with distance
Kite / wire / rope	Use sin (hypotenuse given)
Shadow problem	Use tan with shadow length
Two-position problem	Write two tan equations
Depression from top	Depression angle = elevation angle

📄 Section 5 — Take-Home Handout

📚 Part 1: Key Definitions

Line of sight — straight line from eye to object.
Angle of elevation — angle above horizontal, looking up.
Angle of depression — angle below horizontal, looking down.
Horizontal — flat line at eye level.

📚 Part 2: Important Formulas

tan θ = opposite / adjacent = height / distance
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
Angle of depression = angle of elevation (alternate angles).

📚 Part 3: Memory Tricks

SOH — Sin = Opposite / Hypotenuse
CAH — Cos = Adjacent / Hypotenuse
TOA — Tan = Opposite / Adjacent
Remember: Some Old Horses Can Always Hear Their Owner Approaching

📚 Part 4: Common Mistakes

Forgetting to draw the right angle.
Mixing up elevation and depression.
Using the wrong ratio (sin vs tan).
Not writing the answer with units (m, cm).

📚 Part 5: Practice Questions

A pole is 20 m tall. Shadow is 20 m. Find sun angle.
A rope of 50 m at 30° angle. Find height.
From a 30 m tower, depression to a dog is 60°. Find distance.

✍ Write your answers here

✅ Detailed Solutions

Solution to Q1: Finding the Sun’s Angle of Elevation

Question: A vertical pole 20 m tall casts a shadow 20 m long on the ground. Find the angle of elevation of the sun.

1 Let the angle of elevation of the sun be θ.

2 In the right triangle: tan θ = height ÷ shadow length.

3 tan θ = 20 ÷ 20 = 1.

4 tan 45° = 1, therefore θ = 45°.

∴ Angle of elevation of the sun = 45°

Solution to Q2: Vertical Height Reached by a Rope

Question: A rope 50 m long is tied to the top of a pole and makes an angle of 30° with the ground. Find the height of the pole.

1 The rope is the hypotenuse = 50 m. The height is the opposite side.

2 sin θ = opposite ÷ hypotenuse.

3 sin 30° = 1/2.

4 Height = 50 × (1/2) = 25 m.

∴ Height of the pole = 25 m

Solution to Q3: Horizontal Distance from a Tower

Question: From the top of a 30 m tower, the angle of depression to a point on the ground is 60°. Find the horizontal distance from the base of the tower to that point.

1 The angle of depression from the top = angle of elevation from the ground = 60°.

2 tan 60° = height ÷ horizontal distance (d).

3 √3 = 30 ÷ d ⇒ d = 30 ÷ √3.

4 Rationalise: d = 30 × √3 ÷ 3 = 10√3 ≈ 17.32 m.

∴ Horizontal distance = 10√3 ≈ 17.32 m

🃏 Section 6 — Revision Flashcards

Click each card to see the answer

👁

What is the line of sight?

A straight line drawn from the observer's eye to the object being looked at.

⬆

What is the angle of elevation?

The angle formed between the horizontal and the line of sight when looking upward.

⬇

What is the angle of depression?

The angle formed between the horizontal and the line of sight when looking downward.

🔢

What does tan θ equal?

tan θ = Opposite / Adjacent = Height / Distance on ground.

🌊

How are depression and elevation related?

They are equal because they are alternate interior angles formed by a transversal cutting parallel lines.

✍

When do we use sin?

When the hypotenuse (rope, wire, ladder) is known and we need the opposite side (height).

⛺

What is SOH CAH TOA?

Sin = O/H, Cos = A/H, Tan = O/A. A memory trick for trigonometric ratios.

💡

What is the first step in solving a word problem?

Draw the figure with a right triangle, label all known sides and angles, then pick the correct ratio.

Class 10 Mathematics — Some Applications of Trigonometry

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📐 Class 10 Mathematics — Some Applications of Trigonometry Interactive Visual Revision Notes Aligned with the Class 10 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

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