NOTES

📘 Introduction to Trigonometry Class 10 Notes (Chapter 8)

🔷 Introduction

Trigonometry is an important branch of mathematics. In simple words, trigonometry studies the relationship between angles and sides of a triangle.

The word trigonometry comes from Greek words meaning “triangle measurement.” Therefore, this chapter helps students solve problems related to height, distance, and angles.

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🔷 What is Trigonometry?

Trigonometry mainly deals with right-angled triangles.

A right-angled triangle has:

• One angle equal to 90°
• One longest side called hypotenuse
• Two other sides called base and perpendicular

Thus, understanding the sides is the first step in trigonometry.

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🔷 Parts of a Right Triangle

Consider △ABC where:$$\angle B =90^\circ$$∠B=90∘

✔ Hypotenuse

The side opposite to the right angle is called hypotenuse.

It is always the longest side.

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✔ Perpendicular

The side opposite to the given angle is called perpendicular.

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✔ Base

The side adjacent to the given angle is called base.

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🔷 Trigonometric Ratios

There are six trigonometric ratios.

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1. Sine (sin)

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

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2. Cosine (cos)

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

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3. Tangent (tan)

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

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4. Cosecant (cosec)

$$\cosec \theta= \frac{Hypotenuse}{Perpendicular}$$cosecθ=PerpendicularHypotenuse​

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5. Secant (sec)

$$\sec \theta= \frac{Hypotenuse}{Base}$$secθ=BaseHypotenuse​

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6. Cotangent (cot)

$$\cot \theta= \frac{Base}{Perpendicular}$$cotθ=PerpendicularBase​

Therefore, these ratios are used to solve triangle problems.

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🔷 Important Relationship Between Ratios

Reciprocal Relations

$$\sin\theta=\frac1{\cosec\theta}$$sinθ=cosecθ1​ $$\cos\theta=\frac1{\sec\theta}$$cosθ=secθ1​ $$\tan\theta=\frac1{\cot\theta}$$tanθ=cotθ1​

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Division Relation

$$\tan\theta= \frac{\sin\theta}{\cos\theta}$$tanθ=cosθsinθ​ $$\cot\theta= \frac{\cos\theta}{\sin\theta}$$cotθ=sinθcosθ​

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🔷 Trigonometric Identities

Identity 1

$$\sin^2\theta+\cos^2\theta=1$$sin2θ+cos2θ=1

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Identity 2

$$1+\tan^2\theta=\sec^2\theta$$1+tan2θ=sec2θ

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Identity 3

$$1+\cot^2\theta=\cosec^2\theta$$1+cot2θ=cosec2θ

Hence, these identities help in solving equations quickly.

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🔷 Values of Trigonometric Ratios

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

As a result, remembering these values saves time in exams.

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🔷 Applications of Trigonometry

Trigonometry is used in:

• Architecture
• Navigation
• Engineering
• Astronomy
• Construction
• Computer graphics

Thus, trigonometry has many real-life applications.