# NCERT Class 10 Mathematics Chapter 8 Introduction to Trigonometry: Summary, Concepts, and Revision Notes | SwaVid

This chapter introduces the fundamental concepts of trigonometry, a branch of mathematics dealing with the relationships between the sides and angles of...

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# Introduction to Trigonometry

## Concepts in this chapter

## The ideas this chapter keeps returning to

## Where this chapter usually trips people up

## Find the gaps before this chapter finds them for you.

## Related chapters

### Trigonometric Ratios of an Acute Angle

Chapter 8 · Class 10 Mathematics

This chapter introduces the fundamental concepts of trigonometry, a branch of mathematics dealing with the relationships between the sides and angles of triangles. Specifically, it focuses on right-angled triangles and defines six trigonometric ratios for acute angles, laying the groundwork for more advanced applications.

Your study route

Key topics

Trigonometric ratios are defined for acute angles in a right-angled triangle. For an acute angle A, sine (sin A) is the ratio of the side opposite to angle A to the hypotenuse. Cosine (cos A) is the ratio of the side adjacent to angle A to the hypotenuse. Tangent (tan A) is the ratio of the side opposite to angle A to the side adjacent to angle A. The other three ratios (cosecant, secant, cotangent) are reciprocals of sine, cosine, and tangent, respectively.

Example

Example 1 on page 179 illustrates how to find all trigonometric ratios of angle A when tan A is given as 4/3. Exercise 8.1, Question 1, asks to find sin C, cos C, sin A, cos A for a given right triangle.

Watch out

Students often confuse which side is 'opposite' and which is 'adjacent' relative to the chosen acute angle. They might also mix up the definitions of sin, cos, and tan.

Trigonometric ratios are defined for acute angles in a right-angled triangle. For an acute angle A, sine (sin A) is the ratio of the side opposite to angle A to the hypotenuse. Cosine (cos A) is the ratio of the side adjacent to angle A to the hypotenuse. Tangent (tan A) is the ratio of the side opposite to angle A to the side adjacent to angle A. The other three ratios (cosecant, secant, cotangent) are reciprocals of sine, cosine, and tangent, respectively.

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Example

Example 1 on page 179 illustrates how to find all trigonometric ratios of angle A when tan A is given as 4/3. Exercise 8.1, Question 1, asks to find sin C, cos C, sin A, cos A for a given right triangle.

Why it matters

These ratios form the basis of trigonometry, allowing us to relate angles and side lengths in right-angled triangles. They are fundamental for solving problems in geometry, physics, and engineering.

Watch out

Students often confuse which side is 'opposite' and which is 'adjacent' relative to the chosen acute angle. They might also mix up the definitions of sin, cos, and tan.

Ask at home

Ask your child to draw a right-angled triangle, label its vertices and an acute angle, and then correctly identify the opposite side, adjacent side, and hypotenuse relative to that angle. Then, ask them to write down the definitions of sin, cos, and tan for that angle using the side labels.

Chapter 8, "Introduction to Trigonometry", establishes the core principles of trigonometry by defining six trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent. These ratios relate the sides of a right-angled triangle to its acute angles. The chapter then explores the values of these ratios for specific angles like 0°, 30°, 45°, 60°, and 90°, which are crucial for solving various problems. Furthermore, it introduces the concept of trigonometric ratios of complementary angles, showing how ratios of (90° - A) relate to ratios of A. Finally, the chapter covers fundamental trigonometric identities, particularly the Pythagorean identities, which are essential for simplifying expressions and proving other trigonometric relationships. Mastering these foundational concepts is vital for understanding subsequent applications of trigonometry.

Chapter summary

Chapter 8, "Introduction to Trigonometry", establishes the core principles of trigonometry by defining six trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent. These ratios relate the sides of a right-angled triangle to its acute angles. The chapter then explores the values of these ratios for specific angles like 0°, 30°, 45°, 60°, and 90°, which are crucial for solving various problems. Furthermore, it introduces the concept of trigonometric ratios of complementary angles, showing how ratios of (90° - A) relate to ratios of A. Finally, the chapter covers fundamental trigonometric identities, particularly the Pythagorean identities, which are essential for simplifying expressions and proving other trigonometric relationships. Mastering these foundational concepts is vital for understanding subsequent applications of trigonometry.

What you should learn

Keep these close

SOH CAH TOA: sin = Opposite/Hypotenuse, cos = Adjacent/Hypotenuse, tan = Opposite/Adjacent.

Cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent respectively.

tan A = sin A / cos A and cot A = cos A / sin A.

Values for 0°, 30°, 45°, 60°, 90° must be memorized for sin, cos, and tan.

Complementary angle identities: sin(90°-A) = cos A, cos(90°-A) = sin A, tan(90°-A) = cot A, etc.

Pythagorean Identities: sin²A + cos²A = 1, 1 + tan²A = sec²A, 1 + cot²A = cosec²A.

Trigonometric ratios are unitless as they are ratios of lengths.

The value of sin A and cos A always lies between 0 and 1 (inclusive for acute angles).

The value of tan A can be any positive real number for acute angles.

The ratios depend only on the angle, not on the size of the right-angled triangle.

Always identify the hypotenuse, opposite, and adjacent sides correctly relative to the angle.

Common confusions

It is easy to think

Confusing the 'opposite' and 'adjacent' sides when the reference angle changes, or relative to the wrong angle.

The clearer idea

Always identify the angle first. The 'opposite' side is directly across from the angle, and the 'adjacent' side is next to the angle but not the hypotenuse. The hypotenuse is always opposite the right angle.

It is easy to think

Treating 'sin A' as 'sin multiplied by A' instead of 'sine of angle A'.

The clearer idea

sin, cos, tan are functions of an angle. 'sin' is not a separate entity that multiplies 'A'. It represents the ratio associated with angle A.

It is easy to think

Incorrectly applying trigonometric identities, especially the Pythagorean identities, by forgetting the squares or mixing up the terms (e.g., sin A + cos A = 1).

The clearer idea

Remember the exact forms: sin²A + cos²A = 1, 1 + tan²A = sec²A, 1 + cot²A = cosec²A. Practice deriving them from the Pythagorean theorem to solidify understanding.

It is easy to think

Assuming trigonometric ratios apply to any triangle, not just right-angled triangles, without additional rules.

The clearer idea

The definitions of sin, cos, tan (Opposite/Hypotenuse, etc.) are strictly for right-angled triangles. For general triangles, different laws (like Sine Rule, Cosine Rule) are used, which are not part of Class 10 NCERT.

It is easy to think

Mixing up the values of trigonometric ratios for specific angles (e.g., sin 30° = √3/2 instead of 1/2).

The clearer idea

Create a mnemonic or a table and practice filling it out repeatedly. Understand the derivation from special triangles (30-60-90 and 45-45-90) to help recall.

Before you push ahead

Most stuck chapters trace back to one earlier idea. Check the prerequisites first, or let SwaVid adapt this chapter to the way your child learns.

Keep exploring Class 10

Source

- Define and calculate the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent) for acute angles in a right-angled triangle.
- Recall and apply the trigonometric ratio values for specific angles: 0°, 30°, 45°, 60°, and 90°.
- Understand and utilize the relationships between trigonometric ratios of complementary angles.
- Prove and apply the fundamental trigonometric identities (Pythagorean identities) to simplify expressions.
- Solve problems involving unknown sides or angles of right-angled triangles using trigonometric ratios.
- Define and calculate the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent) for acute angles in a right-angled triangle.
- Recall and apply the trigonometric ratio values for specific angles: 0°, 30°, 45°, 60°, and 90°.
- Understand and utilize the relationships between trigonometric ratios of complementary angles.
- Prove and apply the fundamental trigonometric identities (Pythagorean identities) to simplify expressions.
- Solve problems involving unknown sides or angles of right-angled triangles using trigonometric ratios.
- NCERT Class 10 Mathematics textbook: Mathematics : Chapter 8: Introduction to Trigonometry

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