Login / Register
Follow Us
0
0
Register Free!
Register Free!
Online Mock Tests & Exam Preparation | Target Exams
Select Medium / माध्यम चुनें:
Eng (English) Hindi (हिन्दी)
CBSE • Class X • Mathematics • Ch 8
Estimated Time: 45 Mins
Study Progress: In Progress

Introduction to Trigonometry

In Class 10 Mathematics, "Introduction to Trigonometry" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

Take Chapter CBT Test View All Chapters
Overview Core Concepts Summary Self Check Take MCQ Test

📐 Have You Ever Wondered?

How do astronomers calculate the distance to stars or architects design suspension bridge cables? Trigonometry (measuring triangles) establishes fixed...

How do astronomers calculate the distance to stars or architects design suspension bridge cables? Trigonometry (measuring triangles) establishes fixed ratios between angles and sides of right triangles.

Why This Chapter Matters

In Class 10 Mathematics, "Introduction to Trigonometry" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

Before You Begin (Prerequisites)

  • Right-angled triangles and Pythagoras theorem.
  • Ratios and fractions.
  • Algebraic identities.

What You Will Learn (Core Objectives)

  • Define the six Trigonometric Ratios: $\sin, \cos, \tan, \cot, \sec, \csc$ relative to an acute angle $\theta$.
  • Calculate exact trigonometric values for standard angles: $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$.
  • Prove the fundamental Pythagorean trigonometric identities: $\sin^2\theta + \cos^2\theta = 1, 1 + \tan^2\theta = \sec^2\theta, 1 + \cot^2\theta = \csc^2\theta$.
  • Simplify trigonometric expressions using identities.
  • Prove complex board examination trigonometric identities.

Chapter Roadmap & Progression

1 1. The Six Ratios of a Right Triang...
2 2. Table of Standard Angles
3 3. The Three Master Trigonometric I...

Complete Concept Guide (100% Curriculum Coverage)

1. The Six Ratios of a Right Triangle

In a right triangle relative to acute angle $\theta$:
• $\mathbf{\sin\theta = \frac{\text{Opposite (Perpendicular)}}{\text{Hypotenuse}}}$ • $\mathbf{\cos\theta = \frac{\text{Adjacent (Base)}}{\text{Hypotenuse}}}$ • $\mathbf{\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{\text{Perpendicular}}{\text{Base}}}$
• Reciprocals: $\csc\theta = \frac{1}{\sin\theta}$, $\sec\theta = \frac{1}{\cos\theta}$, $\cot\theta = \frac{1}{\tan\theta}$.

2. Table of Standard Angles

$\theta$$0^\circ$$30^\circ$$45^\circ$$60^\circ$$90^\circ$
$\sin\theta$$0$$1/2$$1/\sqrt{2}$$\sqrt{3}/2$$1$
$\cos\theta$$1$$\sqrt{3}/2$$1/\sqrt{2}$$1/2$$0$
$\tan\theta$$0$$1/\sqrt{3}$$1$$\sqrt{3}$Not Defined

3. The Three Master Trigonometric Identities

  • $\mathbf{\sin^2\theta + \cos^2\theta = 1}$
  • $\mathbf{1 + \tan^2\theta = \sec^2\theta} \implies \sec^2\theta - \tan^2\theta = 1$
  • $\mathbf{1 + \cot^2\theta = \csc^2\theta} \implies \csc^2\theta - \cot^2\theta = 1$

Visual Learning & Conceptual Map

Introduction to Trigonometry Master Matrix

Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture

1. The Six Ratios of a Right Triangle • 2. Table of Standard Angles

Chapter Summary & 10 Key Takeaways

Takeaway 1
T-Ratios: SOH-CAH-TOA (Sine, Cosine, Tangent).
Takeaway 2
Standard Values: $\sin 30^\circ = 1/2$, $\cos 60^\circ = 1/2$, $\tan 45^\circ = 1$.
Takeaway 3
Identity I: $\sin^2\theta + \cos^2\theta = 1$.
Takeaway 4
Identity II: $\sec^2\theta - \tan^2\theta = 1$.
Takeaway 5
Identity III: $\csc^2\theta - \cot^2\theta = 1$.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
If $\tan A = \frac{4}{3}$, find all other trigonometric ratios of $\angle A$.
Reveal Answer & Explanation
Answer: Opposite = 4k, Adjacent = 3k. Hypotenuse $= \sqrt{16+9} = 5k$. $\sin A = \frac{4}{5}, \cos A = \frac{3}{5}, \cot A = \frac{3}{4}, \sec A = \frac{5}{3}, \csc A = \frac{5}{4}$.
Ratios from 3-4-5 triangle.
2
Evaluate: $\sin^2 30^\circ + \cos^2 30^\circ$
Reveal Answer & Explanation
Answer: By Identity I, $\sin^2\theta + \cos^2\theta = 1$. Alternatively: $(1/2)^2 + (\sqrt{3}/2)^2 = 1/4 + 3/4 = 1$.
Value is 1.
3
Prove that: $\frac{\sin\theta}{1 + \cos\theta} + \frac{1 + \cos\theta}{\sin\theta} = 2\csc\theta$
Reveal Answer & Explanation
Answer: LHS $= \frac{\sin^2\theta + (1 + \cos\theta)^2}{\sin\theta(1 + \cos\theta)} = \frac{\sin^2\theta + 1 + 2\cos\theta + \cos^2\theta}{\sin\theta(1 + \cos\theta)} = \frac{2 + 2\cos\theta}{\sin\theta(1 + \cos\theta)} = \frac{2(1+\cos\theta)}{\sin\theta(1+\cos\theta)} = \frac{2}{\sin\theta} = 2\csc\theta$.
Standard proof.
4
If $\sin(A - B) = \frac{1}{2}$ and $\cos(A + B) = \frac{1}{2}$, find $A$ and $B$.
Reveal Answer & Explanation
Answer: $A - B = 30^\circ$ and $A + B = 60^\circ$. Adding: $2A = 90^\circ \implies A = 45^\circ, B = 15^\circ$.
A = 45°, B = 15°.
5
Simplify: $(\sec\theta + \tan\theta)(1 - \sin\theta)$
Reveal Answer & Explanation
Answer: $(\frac{1}{\cos\theta} + \frac{\sin\theta}{\cos\theta})(1 - \sin\theta) = \frac{1+\sin\theta}{\cos\theta}(1 - \sin\theta) = \frac{1 - \sin^2\theta}{\cos\theta} = \frac{\cos^2\theta}{\cos\theta} = \cos\theta$.
Simplifies to cos theta.
Finished Studying This Chapter?
READY TO PRACTICE?

Timed CBT Practice Tests (Exam Simulator)

Put your concepts to the test with official curriculum-aligned Foundation and Advanced practice tests. Get instant accuracy scores, time metrics, and step-by-step verified explanations.

Foundation Test
15 MCQs • Core Concepts
Advanced Test
15 MCQs • HOTS & Traps

All Class 10 Mathematics Chapters

AI Study Friend

Instant Concept Tutor

Have a doubt in Introduction to Trigonometry? Ask our AI study tutor for rapid explanations or customized quizzes.