CBSE Class 10 Mathematics Chapter 8: Introduction to Trigonometry NCERT Solutions

Reviewed by NCERT Help Editorial Team · Updated on 29 Jul 2026

NCERT Solutions PDF Class 10 PDF

This chapter introduces students to the fundamental concepts of trigonometry, a branch of mathematics dealing with the relationships between the sides and angles of triangles. The NCERT Solutions for Class 10 Mathematics, Chapter 8, cover essential topics like trigonometric ratios, trigonometric identities, and their applications. These solutions provide a clear, step-by-step approach to solving problems, ensuring students understand the underlying principles. They are designed to help students build a strong foundation in trigonometry, which is crucial for higher studies in mathematics and science. By working through these detailed explanations, students can effectively revise the chapter and prepare for their board examinations, gaining confidence in tackling complex problems.

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Board	CBSE
Class	Class 10
Subject	Mathematics
Session	2026
Language	English
Type	NCERT Solutions
Chapter	Chapter 8: Introduction to Trigonometry

Chapter summary

Chapter 8, Introduction to Trigonometry, for Class 10 Maths NCERT Solutions focuses on defining and applying trigonometric ratios (sine, cosine, tangent, etc.) in right-angled triangles. It covers calculating these ratios using side lengths and understanding their relationships. The solutions also address problems involving trigonometric identities and their use in simplifying expressions and solving equations. This chapter is foundational for understanding more advanced trigonometric concepts.

Learning outcomes

Understand the definitions of trigonometric ratios in a right-angled triangle.
Calculate trigonometric ratios for specific angles.
Apply the Pythagorean theorem to find unknown side lengths in right-angled triangles.
Solve problems involving the difference between trigonometric ratios like tan P and cot R.
Determine the values of trigonometric ratios for angles A and C.
Apply trigonometric concepts to solve practical geometry problems.

Topics covered

Paper topics

Trigonometric Ratios
Ratios of Specific Angles
Trigonometric Identities
Right-angled Triangles
Pythagorean Theorem
Sine, Cosine, Tangent
Cotangent, Secant, Cosecant
Angles in a Triangle
Problem Solving with Trigonometry
Geometric Applications of Trigonometry

Important topics

Definition of Trigonometric Ratios
Calculating Ratios using Side Lengths
Pythagorean Theorem in Trigonometry
Evaluating tan P - cot R
Finding sin A, cos A, sin C, cos C

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Questions and Solutions

Exercise 8.1, Question 1

In a triangle ABC, which is right-angled at vertex B, the length of side AB is 24 cm and the length of side BC is 7 cm. Determine the values of:

sin A and cos A
sin C and cos C

Solution:

First, we need to find the length of the hypotenuse AC using the Pythagorean theorem in the right-angled triangle ABC.

$AC^2 = AB^2 + BC^2$

Substitute the given values of AB and BC:

$AC^2 = (24 \text{ cm})^2 + (7 \text{ cm})^2$

$AC^2 = 576 \text{ cm}^2 + 49 \text{ cm}^2$

$AC^2 = 625 \text{ cm}^2$

Now, take the square root to find the length of AC:

$AC = \sqrt{625 \text{ cm}^2} = 25 \text{ cm}$

So, the hypotenuse AC is 25 cm.

Now we can determine the trigonometric ratios:

(i) For angle A:

The side opposite to angle A is BC, and the side adjacent to angle A is AB. The hypotenuse is AC.

The sine of angle A is the ratio of the opposite side to the hypotenuse:

$\sin A = \frac{\text{Side opposite to } \angle A}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{7 \text{ cm}}{25 \text{ cm}} = \frac{7}{25}$

The cosine of angle A is the ratio of the adjacent side to the hypotenuse:

$\cos A = \frac{\text{Side adjacent to } \angle A}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{24 \text{ cm}}{25 \text{ cm}} = \frac{24}{25}$

(ii) For angle C:

The side opposite to angle C is AB, and the side adjacent to angle C is BC. The hypotenuse is AC.

The sine of angle C is the ratio of the opposite side to the hypotenuse:

$\sin C = \frac{\text{Side opposite to } \angle C}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{24 \text{ cm}}{25 \text{ cm}} = \frac{24}{25}$

The cosine of angle C is the ratio of the adjacent side to the hypotenuse:

$\cos C = \frac{\text{Side adjacent to } \angle C}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{7 \text{ cm}}{25 \text{ cm}} = \frac{7}{25}$

Exercise 8.1, Question 2

In the given figure, find the value of tan P - cot R.

Right-angled triangle PQR with PQ=12cm, PR=13cm

Solution:

The given triangle PQR is right-angled at Q. We are given the lengths of the hypotenuse PR = 13 cm and one side PQ = 12 cm. We need to find the length of the third side QR using the Pythagorean theorem.

$PR^2 = PQ^2 + QR^2$

Substitute the given values:

$(13 \text{ cm})^2 = (12 \text{ cm})^2 + QR^2$

$169 \text{ cm}^2 = 144 \text{ cm}^2 + QR^2$

Rearrange the equation to solve for QR^2:

$QR^2 = 169 \text{ cm}^2 - 144 \text{ cm}^2$

$QR^2 = 25 \text{ cm}^2$

Take the square root to find the length of QR:

$QR = \sqrt{25 \text{ cm}^2} = 5 \text{ cm}$

Now we have all three sides of the triangle: PQ = 12 cm, QR = 5 cm, and PR = 13 cm.

We need to find the value of tan P - cot R.

For angle P:

The side opposite to P is QR, and the side adjacent to P is PQ.

$\tan P = \frac{\text{Side opposite to } \angle P}{\text{Side adjacent to } \angle P} = \frac{QR}{PQ} = \frac{5 \text{ cm}}{12 \text{ cm}} = \frac{5}{12}$

For angle R:

The side opposite to R is PQ, and the side adjacent to R is QR.

$\cot R = \frac{\text{Side adjacent to } \angle R}{\text{Side opposite to } \angle R} = \frac{QR}{PQ} = \frac{5 \text{ cm}}{12 \text{ cm}} = \frac{5}{12}$

Now, calculate tan P - cot R:

$\tan P - \cot R = \frac{5}{12} - \frac{5}{12} = 0$

Therefore, the value of tan P - cot R is 0.

Common mistakes

Confusing the opposite and adjacent sides relative to the angle.
Incorrectly applying the Pythagorean theorem.
Errors in calculating ratios, especially with fractions.
Mixing up the definitions of different trigonometric ratios (e.g., sin vs. cos).
Not correctly identifying the hypotenuse in a right-angled triangle.

Revision tips

Memorize the definitions of the six trigonometric ratios (sin, cos, tan, cosec, sec, cot).
Practice drawing right-angled triangles and labeling sides (opposite, adjacent, hypotenuse) correctly for each angle.
Work through all examples and exercises to solidify understanding of the Pythagorean theorem's application.
Focus on problems that require calculating differences between ratios, like tan P - cot R.
Review the relationship between ratios of complementary angles.

Practice MCQs

Q1. In a right-angled triangle ABC, right-angled at B, if AB = 24 cm and BC = 7 cm, what is the length of the hypotenuse AC?

Q2. For angle A in triangle ABC (right-angled at B), what is sin A?

Q3. In a right-angled triangle PQR, right-angled at Q, if PQ = 12 cm and PR = 13 cm, what is the length of QR?

Q4. What is the value of tan P - cot R in the given figure (where PQ=12, PR=13, QR=5)?

Q5. In triangle ABC, right-angled at B, if BC = 7 cm and AC = 25 cm, what is cos C?

Frequently asked questions

What is the main focus of Chapter 8: Introduction to Trigonometry for Class 10?

This chapter introduces the basic trigonometric ratios (like sine, cosine, tangent) in right-angled triangles and how to calculate them using the sides of the triangle. It also covers problems involving these ratios.

How are the Pythagorean theorem and trigonometry related in these solutions?

The Pythagorean theorem is essential for finding the length of an unknown side of a right-angled triangle when two sides are known. This allows us to calculate all trigonometric ratios for the angles in the triangle.

What does it mean to find sin A and cos A in a triangle?

Finding sin A and cos A means calculating the ratio of the side opposite to angle A to the hypotenuse (for sin A) and the ratio of the side adjacent to angle A to the hypotenuse (for cos A).

How do these solutions help in preparing for exams?

These solutions provide clear, step-by-step explanations for each problem, helping students understand the methods and concepts. This detailed approach aids in revision and builds confidence for exam questions.

What is the significance of calculating tan P - cot R?

Calculating expressions like tan P - cot R helps students practice applying the definitions of tangent and cotangent to different angles within the same triangle and understanding how these ratios relate.

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