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I. Chapter Summary

This chapter introduces students to the fundamental concepts of trigonometry—a branch of mathematics that deals with relationships between the sides and angles of right-angled triangles. The chapter focuses on defining trigonometric ratios, calculating them for standard angles, understanding complementary angles, and deriving identities that form the basis for advanced trigonometric applications in later classes and real-life fields like engineering, astronomy, and architecture.

II. Key Concepts Covered

Concept	Explanation
Right-Angled Triangle	A triangle with one angle equal to 90°.
Trigonometric Ratios	Ratios of the sides of a right-angled triangle w.r.t. one acute angle:
$-sin theta = frac{text{Opposite}}{text{Hypotenuse}}$
$-cos theta = frac{text{Adjacent}}{text{Hypotenuse}}$
$-tan theta = frac{text{Opposite}}{text{Adjacent}}$
$-csc theta = frac{1}{sin theta},quad sec theta = frac{1}{cos theta},quad cot theta = frac{1}{tan theta}$
Trigonometric Ratios of Standard Angles	Values for angles: 0°, 30°, 45°, 60°, 90° (to be memorized)
Trigonometric Identities
1. $sin^2 theta + cos^2 theta = 1$
2. $1 + tan^2 theta = sec^2 theta$
3. $1 + cot^2 theta = csc^2 theta$
Complementary Angles
$sin(90^circ – theta) = cos theta$
$tan(90^circ – theta) = cot theta$

III. Important Questions

(A) Multiple Choice Questions (1 Mark)

1. What is the value of sin 90°?
    a) 0
    b) 1 ✅ (PYQ 2020)
    c) Undefined
    d) 0.5
2. If cos A = 3/5, then tan A = ?
    a) $frac{3}{4}$✅
    b) $frac{4}{3}$
    c) $frac{5}{3}$
    d) 1.67
3. Which of the following is equal to cot A?
    a) $frac{1}{tan A}$✅
    b) sin A
    c) tan A
    d) $frac{1}{sin A}$
4. If $sin A = frac{1}{2}, quad cos A = frac{sqrt{3}}{2}$, then A =
    a) 30° ✅
    b) 45°
    c) 60°
    d) 90°

(B) Short Answer Questions (2/3 Marks)

1. Prove that: $sin^2 A + cos^2 A = 1$
   
2. Find the value of $frac{tan 45^circ + cot 45^circ}{sec 30^circ + csc 30^circ}$​.
3. If sin $A = frac{4}{5}$ and A is acute, find the values of all other trigonometric ratios of angle A.
4. Use complementary angles to evaluate: $cos 30^circ times sin 60^circ + sin 30^circ times cos 60^circ$.

(C) Long Answer Questions (5 Marks)

1. Prove the identity:
    $frac{1 + tan^2 A}{1 + cot^2 A} = tan^2 A cot^2 A$
   
2. If sin $A = frac{3}{5}$, use identity to find:
    a) cos A
    b) tan A
    c) sec A
    d) cosec A
    e) cot A
3. Find the values of all trigonometric ratios for angle 60° using an equilateral triangle method.
4. Verify: $frac{sin^2 60^circ + cos^2 30^circ}{tan 45^circ + cot 45^circ} = 1$
   

(D) HOTS (Higher Order Thinking Skills)

1. If $tan A = frac{3}{4}$​, find an expression for $sin A + cos A$ in terms of a single fraction.
2. Given that an angle A satisfies $tan A + cot A = 2$ find the value of $sin^2 A + cos^2 A$
   

IV. Key Formulas/Concepts

Formula	Use
$sin theta = frac{text{Opposite}}{text{Hypotenuse}}​$	Triangle sides
$cos theta = frac{text{Adjacent}}{text{Hypotenuse}}$	Triangle sides
$tan theta = frac{text{Opposite}}{text{Adjacent}}​$	Triangle sides
$sin^2 theta + cos^2 theta = 1$	Identity
$1 + tan^2 theta = sec^2 theta$	Identity
$1 + cot^2 theta = csc^2 theta$	Identity
Standard Values Table
	Angle
	——-
	0°
	30°
	45°
	60°
	90°

V. Deleted Portions (CBSE 2025–2026)

 No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)

Unit/Chapter	Estimated Marks	Type of Questions Typically Asked
Chapter 8: Introduction to Trigonometry	6–8 Marks	1 Long Answer, 2 Short Answers, 1–2 MCQs

VII. Previous Year Questions (PYQs)

Year	Marks	Question
2023	1	Evaluate $sin 30^circ + cos 60^circ$
2022	3	If $sin A = frac{3}{5}$, find all other ratios
2021	5	Prove identity-based expressions
2020	2	Find trigonometric ratios using identity

VIII. Real-World Application Examples

Application	Explanation
Surveying & Mapping	Trigonometry helps in calculating distances and heights.
Navigation & Aviation	Used to find directions, angles of elevation and depression.
Architecture	Designing ramps, sloped roofs, and bridges involves trigonometry.
Physics	Motion on inclined planes uses trigonometric principles.

IX. Student Tips & Strategies for Success

Time Management

• Make a trigonometric value chart and revise daily.
• Practice identities 10 minutes a day.

Exam Preparation

• Use diagrams for every triangle-based question.
• Memorize standard ratios with mnemonic tricks like:
  “Some People Have Curly Brown Hair Till Painted Black”
  $sin A = frac{text{Perpendicular}}{text{Hypotenuse}}, quad
  cos A = frac{text{Base}}{text{Hypotenuse}}, quad
  tan A = frac{text{Perpendicular}}{text{Base}}$
  
  

Stress Management

• Practice identities through games and flashcards.
• Create visual aids and triangle drawings to recall values better.

X. Career Guidance & Exploration (Class-Specific)

For Classes 9–10

Streams

• Science: Engineering, Architecture, Astronomy
• Commerce: Finance (Statistics), Actuarial Sciences
• Arts: Surveying, Geography

Competitive Exams

• NTSE, RMO, CUET, JEE Foundation, Olympiads

XI. Important Notes

• Always show steps in trigonometric identity proofs.
• Practice with diagrams. Label clearly.
• Use unit triangle or reference triangle method to memorize standard values.
• For the latest syllabus, check CBSEacademic.nic.in and NCERT.nic.in.