Mathematics Class 10 - Introduction To-Trigonometry Notes

Introduction to Trigonometry

Welcome to the chapter on Introduction to Trigonometry for Class 10. In this chapter, you will learn about the basics of trigonometry, trigonometric ratios, and how to use them to solve problems involving right-angled triangles. By the end of this chapter, you will be able to find unknown sides and angles in triangles using trigonometric formulas.

Key Concepts

• Trigonometry: The branch of mathematics that deals with the relationships between the sides and angles of triangles.
• Right-angled Triangle: A triangle with one angle equal to 90°.
• Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
• Perpendicular: The side opposite to the angle being considered.
• Base: The side adjacent to the angle being considered.

Trigonometric Ratios

The six trigonometric ratios for an acute angle θ in a right-angled triangle are:

• Sine (sin θ): Perpendicular / Hypotenuse
• Cosine (cos θ): Base / Hypotenuse
• Tangent (tan θ): Perpendicular / Base
• Cosecant (csc θ): Hypotenuse / Perpendicular
• Secant (sec θ): Hypotenuse / Base
• Cotangent (cot θ): Base / Perpendicular

Values of Trigonometric Ratios

The values of sin, cos, and tan for standard angles (0°, 30°, 45°, 60°, 90°) are important to remember.

θ	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

Trigonometric Identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

Applications of Trigonometry

• Finding heights and distances
• Solving problems involving right-angled triangles
• Used in navigation, engineering, and astronomy

Practice Questions

1. In a right-angled triangle, if the perpendicular is 3 cm and the hypotenuse is 5 cm, find sin θ.
2. If cos θ = 4/5, find sin θ using the identity sin²θ + cos²θ = 1.
3. Find tan 45°.
4. If tan θ = 1/√3, what is the value of θ?
5. State the six trigonometric ratios for angle θ.

Challenge Yourself

• Prove that sin²θ + cos²θ = 1 for θ = 30°.
• A ladder 10 m long leans against a wall making an angle of 60° with the ground. Find the height at which the ladder touches the wall.

Did You Know?

• Trigonometry comes from the Greek words "trigonon" (triangle) and "metron" (measure).
• Trigonometry is used in designing buildings, bridges, and even in music!

Glossary

• Trigonometric Ratio: A ratio of the lengths of two sides of a right-angled triangle.
• Hypotenuse: The longest side of a right-angled triangle.
• Perpendicular: The side opposite the angle being considered.
• Base: The side adjacent to the angle being considered.

Answers to Practice Questions

1. sin θ = Perpendicular / Hypotenuse = 3/5
2. sin²θ = 1 - (4/5)² = 1 - 16/25 = 9/25 ⇒ sin θ = 3/5
3. tan 45° = 1
4. θ = 30° (since tan 30° = 1/√3)
5. sin θ, cos θ, tan θ, csc θ, sec θ, cot θ

---

Practice trigonometry to solve real-world problems and master geometry!