NOTES

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📘 Some Applications of Trigonometry Class 10 Notes (Chapter 9)

🔷 Introduction

Trigonometry helps us solve many real-life problems. In simple words, it helps us find height and distance without measuring them directly. For example, we can find the height of a tower, a tree, or a building. Therefore, this chapter connects maths with everyday life.

Moreover, these concepts help students understand how engineers and surveyors work.

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🔷 Line of Sight

When you look at an object, your eye creates a straight line toward that object. We call this line the line of sight.

For instance, when you look at the top of a tower, your eyes form a line between you and the tower. Thus, every height and distance problem starts with the line of sight.

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🔷 Horizontal Line

A horizontal line runs parallel to the ground. Your eye level creates this line naturally.

In addition, this line helps us measure angles correctly. Therefore, it plays an important role in trigonometry diagrams.

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🔷 Angle of Elevation

When you look upward at an object, your line of sight forms an angle with the horizontal line. We call this angle the angle of elevation.

For example, you create an angle of elevation when you look at:

• A tall building
• A mountain
• A flying kite
• The top of a tower

Hence, the object always stays above your eye level.

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🔷 Angle of Depression

Sometimes you look downward at an object. In that case, your line of sight forms an angle below the horizontal line. We call this angle the angle of depression.

For example, you create this angle when you look:

• From a bridge to a road
• From a hill to a car
• From a building to the ground

As a result, the object stays below your eye level.

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🔷 Important Trigonometric Ratios

We use trigonometric ratios to solve height and distance questions. Most importantly, tangent helps in most problems.

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Tangent Ratio

$$\tan\theta= \frac{Perpendicular}{Base}$$tanθ=BasePerpendicular​

Therefore, we often choose tangent first.

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Sine Ratio

$$\sin\theta= \frac{Perpendicular}{Hypotenuse}$$sinθ=HypotenusePerpendicular​

Similarly, sine helps when hypotenuse is known.

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Cosine Ratio

$$\cos\theta= \frac{Base}{Hypotenuse}$$cosθ=HypotenuseBase​

Likewise, cosine helps when the base is important.

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🔷 Height and Distance Formula

In most questions:$$\tan\theta= \frac{Height}{Distance}$$tanθ=DistanceHeight​

Therefore, we get:

Height Formula

$$Height=Distance\times\tan\theta$$Height=Distance×tanθ

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Distance Formula

$$Distance= \frac{Height}{\tan\theta}$$Distance=tanθHeight​

Thus, these formulas solve most chapter problems.

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🔷 Solved Example

Example

A tree is 20 m high. The angle of elevation from a point on the ground is 45°. Find the distance between the tree and the point.

Solution

Given:

Height = 20 m

Angle:$$\theta=45^\circ$$θ=45∘

Now we use:$$\tan45^\circ= \frac{Height}{Distance}$$tan45∘=DistanceHeight​

We know:$$\tan45^\circ=1$$tan45∘=1

So:$$1= \frac{20}{Distance}$$1=Distance20​

Therefore:$$Distance=20m$$Distance=20m

Answer:

Distance = 20 m

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🔷 Important Points

• Angle of elevation always points upward.
• Angle of depression always points downward.
• Horizontal lines stay parallel.
• Equal alternate angles help in diagrams.

Most importantly, always draw a neat figure before solving.

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🔷 Real-Life Uses

People use trigonometry in many fields. For example, it helps in:

• Construction
• Aviation
• Navigation
• Surveying
• Architecture
• Mountain measurement

Therefore, this chapter has many practical uses.