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CBSE · Class 10 · Mathematics · Chapter 8

Introduction to Trigonometry — Class 10 Mathematics (CBSE) explained

  • Board exam 2026-27
  • 4 lessons
  • 35 scenes
  • About 26 min
  • Updated

Quick answer

Introduction to Trigonometry is Chapter 8 of CBSE Class 10 Mathematics. It covers right triangles in surroundings, optional inverse-notation distinction, recovering a ratio geometrically, forty-five-degree values, zero-degree ratios and ninety-degree ratios. AI Lessons teaches it in 4 narrated lessons (about 26 minutes) with an AI tutor and board-pattern practice.

What you’ll learn

  1. Lesson 1: Right Triangles and Well-Defined Ratios

    6 min
    • Right Triangles Connect Heights, Distances and Angles
    • Opposite and Adjacent Depend on the Reference Angle
    • Define the Three Basic Trigonometric Ratios
    • Connect All Six Ratios
    • Enrichment: How Sine and Cosine Get Their NamesNot assessed
    • Similarity Makes the Ratios Well Defined
    • Read Ratio Names and Squares Correctly
    • Enrichment: A Reciprocal Is Not Sine Inverse
    • A Known Ratio Determines a Missing Side Ratio

  2. Lesson 2: Using Ratios and Building Standard-Angle Triangles

    6 min
    • Example 1: Find the Other Five Ratios
    • Example 2: Equal Sines Give Equal Acute Angles
    • Example 3: Evaluate a Sum and Difference of Squares
    • Example 4: Verify the Given Product
    • Example 5: Use a Difference of Side Lengths
    • Exercise 8.1: Interpret Before Calculating
    • Derive the Six Ratios for 45°
    • An Equilateral Triangle Produces 30° and 60°

  3. Lesson 3: Standard Angles and Their Applications

    6 min
    • Read the Ratios for 30° and 60°
    • Motivate the Ratios at 0°
    • Motivate the Ratios at 90°
    • Interpret Table 8.1
    • Example 6: Select Ratios for Missing Sides
    • Example 7: Determine the Acute Angles
    • Example 8: Solve for A and B
    • Exercise 8.2: Substitute and Justify

  4. Lesson 4: Trigonometric Identities and Practice

    8 min
    • Prove the Sine–Cosine Identity
    • Derive the Tangent–Secant Identity
    • Derive the Cotangent–Cosecant Identity
    • Find Ratios Using Identities
    • Example 9: Express Ratios in Terms of Sine
    • Example 10: Convert to Sine and Cosine
    • Example 11: Factor Before Cancelling
    • Enrichment: Follow the Longer Proof in Example 12
    • Exercise 8.3: Choose a Short Identity Route
    • Recap: Definitions, Values, Identities and Practice

Key concepts

Right triangles in surroundings
Trigonometry studies relationships between the sides and angles of a triangle. Its name comes from Greek words meaning three, sides and measure.
Optional inverse-notation distinction
The expression called sine inverse A is different. It is not another way to write cosecant A.
Recovering a ratio geometrically
A side length is positive, so choose the positive value.
Forty-five-degree values
Cosine forty five degrees has the same value.
Zero-degree ratios
Cotangent and cosecant zero degrees are not defined because their reciprocal expressions divide by zero. The endpoint values are definitions motivated by the changing triangle.
Ninety-degree ratios
At ninety degrees, both require division by zero, so neither is defined.
Fundamental trigonometric identity
An identity is an equation true for all values under consideration.
Cotangent–cosecant identity
A valid identity must use expressions that are defined.

Exam focus

Board exam
CBSE assesses this chapter in the 2026-27 year-end board examination.
Syllabus
INTRODUCTION TO TRIGONOMETRY: 1. Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined). 2. Motivate the ratios whichever are defined at 0° and 90°. Values of the trigonometric ratios of 30°, 45° and 60°. 3. Relationships between the ratios. TRIGONOMETRIC IDENTITIES: 1. Proof and applications of the identity sin^2 A + cos^2 A = 1. 2. Only simple identities to be given. Competencies: understands the definitions of the basic trigonometric functions (including the introduction of the sine and cosine functions); uses trigonometric identities to solve problems. Learning outcomes: evaluates trigonometric ratios; describes trigonometric ratios of standard angles and solving related expressions; proves trigonometric identities using sin^2 A + cos^2 A = 1 and other identities.
Not assessed
These parts go beyond the 2026-27 syllabus and are taught for understanding only: Enrichment: How Sine and Cosine Get Their Names.
Exam note
This one textbook chapter covers two syllabus topics: Introduction to Trigonometry and Trigonometric Identities (8.4).
Exam note
Identities: "Only simple identities to be given" - keep identity proofs short and simple.
Exam note
Trigonometric ratios of complementary angles are not in the syllabus (the textbook mentions complementary angles only in its historical note).
Paper pattern
Mathematics Standard: 80 marks, 3 hours, 38 questions. Section A: 18 MCQs + 2 assertion-reason (1 mark each); B: 5 VSA (2 marks); C: 6 SA (3 marks); D: 4 LA (5 marks); E: 3 case-study questions (4 marks). No calculators.
Question style
About 50% of board questions are competency-focused.
Practice in the app
118 board-pattern questions with marking schemes: 33 multiple choice (1 mark), 15 assertion–reason (1 mark), 26 very short answer (2 marks), 25 short answer (3 marks), 6 long answer (5 marks) and 13 case-based (4 marks).

Sample questions with marking points

Board-pattern questions from the AI Lessons practice set for this chapter. The marking points show what an examiner looks for; full model answers are in the app.

  1. Very short answer2 marks

    Given sin A = 1/3, a student calculates sin² A as 2/3. Identify the error and find the correct value.

    Marking points

    • Identifies that sin² A means (sin A)², not 2 sin A.1
    • Correctly squares 1/3 to obtain 1/9.1

  2. Short answer3 marks

    A student writes: 'Since tan A = 4/3, sin A = 4/3 and cos A = 3/4.' Explain the errors and determine the correct values of sin A, cos A and cot A.

    Marking points

    • Explains that sine and cosine require the hypotenuse, while 3/4 is the reciprocal of tangent.0.5
    • Uses proportional legs 4k and 3k and Pythagoras to obtain hypotenuse 5k.1
    • Obtains sin A = 4/5.0.5
    • Obtains cos A = 3/5.0.5
    • Obtains cot A = 3/4.0.5

  3. Long answer5 marks

    For an acute angle A, sec A = 13/12. A student claims that (1 + tan² A)/(1 + cot² A) = 144/25 and (1 - tan A)/(1 - cot A) = 5/12. Check both claims using identities and reciprocal relationships. Give the correct values and explain why the second quotient is defined for this angle.

    Marking points

    • Uses tan² A = sec² A - 1 to obtain 25/144.1
    • Chooses the positive value tan A = 5/12.0.5
    • Uses the reciprocal relationship to obtain cot A = 12/5.0.5
    • Simplifies the first quotient to tan² A = 25/144 using identities, or obtains the same value by correct substitution.1
    • Shows that the second denominator is -7/5 ≠ 0.0.5
    • Substitutes correctly into the second quotient to obtain (7/12)/(-7/5).0.5
    • Obtains the second quotient as -5/12.0.5
    • Explicitly rejects both proposed values.0.5

Frequently asked questions

What does Introduction to Trigonometry cover in Class 10 Mathematics?

It covers right triangles in surroundings, optional inverse-notation distinction, recovering a ratio geometrically, forty-five-degree values, zero-degree ratios and ninety-degree ratios. AI Lessons teaches it in 4 lessons: Right Triangles and Well-Defined Ratios, Using Ratios and Building Standard-Angle Triangles, Standard Angles and Their Applications and Trigonometric Identities and Practice.

Is Introduction to Trigonometry in the CBSE Class 10 board exam?

Yes. CBSE assesses Introduction to Trigonometry in the 2026-27 year-end board examination. 1 short part goes beyond the syllabus and is marked as not assessed.

For an acute angle A, which expression equals cosec A?

1/(sin A). Cosecant is the reciprocal of sine, so cosec A = 1/(sin A).

What is a ratio name?

A ratio name is an abbreviation, not a product.

Which ratio is defined and has the value 1?

Sec 0°. Since cos 0° = 1, sec 0° = 1/(cos 0°) = 1.

What is an identity?

An identity is an equation true for all values under consideration.

For an acute angle A, which expression gives sec A entirely in terms of sin A?

1/√(1 - sin² A). For an acute angle, cos A = √(1 - sin² A), and sec A is its reciprocal.

How does AI Lessons teach Introduction to Trigonometry?

As 4 narrated, animated lessons (35 scenes, about 26 minutes in all), with an AI tutor that knows the step you are on and board-pattern practice with marking schemes. The full lessons open after sign-in on an AI Lessons account.

Watch the full lesson

Introduction to Trigonometry, as a narrated, animated lesson

4 lessons · 35 animated scenes · about 26 minutes, with an AI tutor that knows the step you are on and 118 board-pattern questions with marking schemes.

Lessons open after sign-in. A parent or guardian holds the account and adds each child.

Get early accessHow it works

Last updated . Written by the AI Lessons team from our own lesson scripts and practice questions, aligned to the Central Board of Secondary Education syllabus for 2026-27. Always check your school’s latest syllabus.