Full Chapter Summary & Detailed Notes - Introduction to Trigonometry Class 10 NCERT

Overview & Key Concepts

• Chapter Goal: Understand trigonometric ratios in right triangles, values for specific angles, identities. Exam Focus: Ratios, identities, applications like heights/distances. 2025 Updates: Real-life contexts emphasized. Fun Fact: Trigonometry from Greek 'triangle measure'. Core Idea: Relates sides/angles in triangles. Real-World: Qutub Minar height, river width, balloon altitude.
• Wider Scope: Basis for surveying, astronomy, physics; extends to waves, circles.

8.1 Introduction

• Builds on right triangles from earlier classes. Examples: Qutub Minar height without measuring; girl on balcony finding river width; balloon altitude change.
• Trigonometry studies side-angle relations in triangles. Derived from Greek: tri (three), gon (sides), metron (measure).
• History: Early in Egypt/Babylon for astronomy; distances to stars/planets. Used in engineering, physical sciences.
• Focus: Ratios for acute angles in right triangles; extend to 0°/90°; specific angles; identities.

8.2 Trigonometric Ratios

• In right triangle ABC (right at B), for acute angle A: opposite (BC), adjacent (AB), hypotenuse (AC).
• Definitions: sin A = opposite/hypotenuse = BC/AC; cos A = adjacent/hypotenuse = AB/AC; tan A = opposite/adjacent = BC/AB.
• Reciprocals: cosec A = 1/sin A = AC/BC; sec A = 1/cos A = AC/AB; cot A = 1/tan A = AB/BC.
• Relations: tan A = sin A / cos A; cot A = cos A / sin A.
• Ratios express angle-side relations; independent of triangle size (similar triangles).
• Notation: sin A not sin * A; sin² A = (sin A)²; inverse sin⁻¹ A different.
• Example: If sin A = 1/3, then BC/AC=1/3; BC=k, AC=3k; AB=√(9k² - k²)=√8 k = 2√2 k; cos A = (2√2)/3; etc.

8.3 Trigonometric Ratios of Some Specific Angles

• Angles: 0°, 30°, 45°, 60°, 90°.
• 45°: Equilateral half; sides 1:1:√2; sin=cos=1/√2; tan=1; cosec=sec=√2; cot=1.
• 30°/60°: Equilateral altitude; sides 1:√3:2; 30°: sin=1/2, cos=√3/2, tan=1/√3; 60°: sin=√3/2, cos=1/2, tan=√3.
• 0°: As angle →0°, opposite→0, adjacent≈hypotenuse; sin0=0, cos0=1, tan0=0; cosec undefined, sec=1, cot undefined.
• 90°: As angle →90°, adjacent→0, opposite≈hypotenuse; sin90=1, cos90=0, tan undefined; cosec=1, sec undefined, cot=0.
• Table: sin increases 0 to 1; cos decreases 1 to 0 as angle 0° to 90°.

8.4 Trigonometric Identities

• Equations true for all angles in domain.
• Proof: sin²A + cos²A =1 (from Pythagoras: opposite² + adjacent² = hypotenuse², divide by hypotenuse²).
• 1 + tan²A = sec²A (divide by adjacent²).
• 1 + cot²A = cosec²A (divide by opposite²).
• Express ratios in terms of one; proofs using identities.

Examples and Exercises

• Examples: Ratios from tan; sin B = sin Q implies B=Q; cos²θ - sin²θ; 2 sin A cos A=1 if tan A=1; OP=7, OQ-PQ=1 find sin/cos Q.
• Exercises: Compute ratios; identities; specific values; true/false.

Key Themes & Tips

• Ratios: Sin, cos, tan, reciprocals.
• Specific Angles: Memorize table.
• Identities: Prove/use sin² + cos²=1 etc.
• Tip: Draw triangle; use identities to simplify; remember SOH-CAH-TOA.

Project & Group Ideas

• Measure heights using clinometer; verify identities geometrically.

Key Definitions & Terms - Complete Glossary

All terms from chapter; easy to memorize. Detailed explanations with examples for better understanding.

Tip: Use SOH-CAH-TOA mnemonic: Sin=Opposite/Hypotenuse, Cos=Adjacent/Hypotenuse, Tan=Opposite/Adjacent. Practice with flashcards for ratios and specific values.

Extended Note: These definitions form the foundation. For instance, ratios are dimensionless, always positive in acute angles. In applications, like finding heights, tan is used for angles of elevation/depression. Understand reciprocals as they simplify identities, e.g., sec A =1/cos A avoids fractions in proofs.

60+ Questions & Answers - NCERT Based (Class 10)

Structured as Part A (1 mark, short answers), Part B (4 marks, ~6 lines answers), Part C (8 marks, detailed). 20 per part, based on chapter content, with answers matching the mark scheme. Detailed explanations for clarity.

Part A: 1 Mark Questions (Short Answers)

Part B: 4 Marks Questions (Answers in ~6 Lines)

Part C: 8 Marks Questions (Detailed Answers)

Practice Tip: Use diagrams for ratios; memorize table; prove identities step-by-step.

Important Formulas

Key formulas for ratios, identities, specific values. Detailed with derivations and uses.

Use in: Problems, proofs. Extended: Complementary: sin(90-A)=cos A. For calculations, rationalize denominators like 1/√3 = √3/3.

More Details: These formulas are foundational. For example, in identities, substitute tan=sin/cos to prove. In applications, tan for slope in heights/distances. Practice deriving from triangle to remember.

Theorems & Proofs - Detailed Explanations

Key theorems with step-by-step proofs and examples. Extended discussions for clarity.

Tip: Visualize triangle for proofs. Extended: These theorems from Pythagoras; apply to derive others like cos2A = cos²A - sin²A from identity.

More Content: Proofs geometric/algebraic. For identities, substitute values to verify. In exams, prove by LHS=RHS. Common error: Forget domain (acute angles).

Derivations - Step by Step

• Sin² + Cos² =1: Step 1: Opp² + adj² = hyp². Step 2: Divide hyp². Step 3: Left sin² + cos², right 1.
• 1 + Tan² = Sec²: Step 1: Opp² + adj² = hyp². Step 2: Divide adj². Step 3: Tan² +1 = sec².
• 1 + Cot² = Cosec²: Similar, divide opp².
• Specific Angles - 45°: Isosceles right, sides 1,1,√2. Ratios 1/√2 etc.
• 30°/60°: Equilateral 2a, altitude √3 a. 30° opp a, hyp 2a =1/2.
• 0°/90° Limits: As angle small, sin≈angle (radians), cos≈1; large, sin≈1, cos≈0.

Tip: Derive to memorize values. Extended: Use in proofs like express tan in sin/cos. For applications, derive ratios from given sides.

More Details: Derivations geometric. For identities, algebraic manipulation. Practice deriving table values using triangles.

All Textbook Examples - Step by Step Solutions

Complete list with detailed explanations like the book, step by step. Extended with variations.

Tip: Practice variations for exams. Extended: Examples illustrate concepts; use to solve exercises.

More Content: All examples from book covered with steps. For identities, show LHS=RHS. Include diagrams mentally.

Methods of Solving - Step by Step

• Ratio Computation: Identify opp, adj, hyp; plug in. Example: Given sides, find sin etc.
• Specific Angles: Memorize/use table. Derive using triangles if forgotten.
• Identity Proofs: Start LHS, use definitions/identities to RHS. Example: (cosec - cot)².
• Applications: Draw triangle, use tan for height/distance. Step: Angle elevation, base adj, height opp.

Choose: Direct for values; identities for proofs. Extended: For complex, express all in sin/cos.

More Details: Methods geometric/algebraic. Practice with examples. Common: Rationalize, simplify fractions.

Applications & Word Problems - Real Life

• Heights/Distances: Tan for elevation. Example: Minar height from angle.
• River Width: From balcony, tan depression. Example: Flower pot on stair.
• Balloon Altitude: Change in position, use sin/cos. Example: From A to B.
• Astronomy: Star distances. Historical: Planets from Earth.
• Engineering: Slopes, forces. Example: Roof angle.
• Navigation: Bearings, courses.

Tip: Draw diagram; identify right triangle. Extended: Problems involve one ratio; solve for unknown side/angle.

More Details: Real-life from book: Qutub, river, balloon. Solve step-by-step: Let height h, distance d, tanθ=h/d.

Quick Revision Notes & Mnemonics - Exam-Ready

Mnemonics: SOH-CAH-TOA for ratios. Specific: 0-30-45-60-90: sin 0,1/2,1/√2,√3/2,1; cos reverse. Identities: Pythagorean triples.

Tip: Table flash; prove identities. Extended: Revise by solving one example per subtopic.