Trigonometric Functions – NCERT Class 11 Mathematics Chapter 3 – Angles, Radian Measure, Graphs, and Identities

Explores angle measurement in degrees and radians, definitions and properties of trigonometric functions, values of trigonometric ratios, graphs, signs in different quadrants, domain and range, trigonometric identities for sum, difference, double and triple angles, and numerous illustrative examples and exercises.

Updated: 4 days ago

Categories: NCERT, Class XI, Mathematics, Trigonometry, Trigonometric Functions, Angle Measurement, Chapter 3

Tags: Trigonometric Functions, Radian Measure, Degree Measure, Angle Conversion, Sine, Cosine, Tangent, Identities, Quadrants, Graphs, Domain and Range, Sum and Difference Identities, Double Angle, Triple Angle, Example Problems, NCERT Class 11, Mathematics, Chapter 3

Trigonometric Functions: Class 11 NCERT Chapter 3 - Ultimate Study Guide, Notes, Questions, Quiz 2025

Full Chapter Summary & Detailed Notes - Trigonometric Functions Class 11 NCERT

Overview & Key Concepts

Chapter Goal: Generalize trig ratios to functions using unit circle; study angles in degrees/radians, properties of sin/cos etc. Exam Focus: Conversions, arc lengths, trig values at quadrantal angles. 2025 Updates: Emphasis on radian measure applications, identities like $$ \sin^2 x + \cos^2 x = 1 $$. Fun Fact: Derived from Greek 'trigon' (triangle) + 'metron' (measure); used by Arya Bhatt. Core Idea: Angle as rotation; trig functions from unit circle coordinates. Real-World: Navigation, seismology, tides. Ties: Builds on Class 10 heights/distances; leads to Ch4 identities. Expanded: Examples from PDF, conversion table, unit circle diagram.
Wider Scope: From right-triangle ratios to any angle via radians.
Expanded Content: Positive/negative angles, degree/radian relations, trig signs in quadrants.

3.1 Introduction

Trigonometry measures triangle sides; evolved for navigation/surveying. Now in physics (atoms), music (tones), oceanography (tides). Extends acute ratios to functions for any angle.

3.2 Angles

Definition: Rotation from initial to terminal side at vertex; anticlockwise positive, clockwise negative (Fig 3.1).
Units: Revolution (full 360°), degree (1/360 rev), radian (arc length = radius in unit circle).

Box 1: Common Angle Measures (Simple Way: Quick Conversion Table)

Degree	Radian	Approx.
0°	0	0
30°	$$ \pi/6 $$	0.52
45°	$$ \pi/4 $$	0.79
60°	$$ \pi/3 $$	1.05
90°	$$ \pi/2 $$	1.57
180°	$$ \pi $$	3.14
270°	$$ 3\pi/2 $$	4.71
360°	$$ 2\pi $$	6.28

Simple Way: $$ \pi $$ rad = 180°; multiply degrees by $$ \pi/180 $$ for radians.

3.2.1 Degree Measure

1° = 1/360 revolution; 1° = 60', 1' = 60''. Examples: 360°, -30° (Fig 3.3).

3.2.2 Radian Measure

Arc length = radius in unit circle (Fig 3.4); full circle 2$$ \pi $$ rad. General: $$ \theta = l / r $$ (Fig 3.5).

3.2.3 Relation: Degree & Radian

$$ \pi $$ rad = 180°; 1 rad ≈ 57°17'. Radian = ($$ \pi/180 $$) × Degree.

3.3 Trigonometric Functions

Unit Circle: Center origin; point P(a,b), arc AP = x rad → cos x = a, sin x = b (Fig 3.6).
Identity: $$ \cos^2 x + \sin^2 x = 1 $$; extends to tan, cot etc.
Periodicity: sin/cos repeat every 2$$ \pi $$: $$ \sin(2n\pi + x) = \sin x $$.
Quadrantal Angles: 0, $$ \pi/2 $$, $$ \pi $$, $$ 3\pi/2 $$, 2$$ \pi $$ values (table on p.51).
Other Functions: csc x = 1/sin x, sec x = 1/cos x, tan x = sin/cos, cot x = cos/sin.
Identities: 1 + tan²x = sec²x; 1 + cot²x = csc²x.

Simple Example 1: Conversion (Step-by-Step)

40°20' to radian. Step 1: 20' = 1/3 ° → 40 + 1/3 = 121/3 °. Step 2: × $$ \pi/180 $$ = $$ 121\pi / 540 $$. Simple Way: Minutes to fraction, multiply $$ \pi/180 $$

Summary

Angles: Rotation measure; degrees/radians via unit circle. Trig: Ratios to functions; key identity $$ \sin^2 x + \cos^2 x = 1 $$; signs via quadrants (Fig 3.7).
Applications: Arc length l = rθ; wheel revolutions.

Why This Guide Stands Out

Math-focused: Conversions, unit circle, identities with steps. Free 2025 with MathJax.

Key Themes & Tips

Aspects: Measures, functions, properties.
Tip: Memorize $$ \pi $$ = 180°; practice conversions.

Exam Case Studies

Convert 25° to radian, find arc length, trig values at $$ \pi/2 $$

Project & Group Ideas

Measure angles in real rotations (clock hands).
Plot unit circle trig values.

Key Definitions & Terms - Complete Glossary

All terms from chapter; detailed with examples, relevance. Expanded: 15+ terms with depth.

Angle

Rotation measure from initial to terminal side. Relevance: Basis trig. Ex: Positive anticlockwise. Depth: Vertex fixed.

Initial Side

Starting ray position. Relevance: Reference. Ex: OA in Fig 3.1. Depth: Fixed for measure.

Terminal Side

Ending ray after rotation. Relevance: Defines angle. Ex: OB. Depth: Can coincide after 2$$ \pi $$

Vertex

Rotation point. Relevance: Common origin. Ex: O. Depth: Fixed.

Positive Angle

Anticlockwise rotation. Relevance: Standard. Ex: +90°. Depth: >0.

Negative Angle

Clockwise rotation. Relevance: Direction. Ex: -30°. Depth: <0.

Degree Measure

1/360 revolution. Relevance: Common unit. Ex: 1° = 60'. Depth: Minutes/seconds subdivide.

Radian Measure

Arc/radius in unit circle. Relevance: Natural for calc. Ex: 1 rad ≈57°. Depth: 2$$ \pi $$ full circle.

Unit Circle

Radius 1, center origin. Relevance: Defines trig. Ex: P(a,b) on circle. Depth: a² + b² =1.

Sine Function

sin x = b (y-coord). Relevance: Opposite/hyp. Ex: sin $$ \pi/2 $$ =1. Depth: -1 to 1 range.

Cosine Function

cos x = a (x-coord). Relevance: Adjacent/hyp. Ex: cos 0 =1. Depth: Even: cos(-x)=cos x.

Tangent Function

tan x = sin x / cos x. Relevance: Opp/adj. Ex: tan $$ \pi/4 $$ =1. Depth: Undefined at odd $$ \pi/2 $$

Quadrantal Angle

Multiple of $$ \pi/2 $$. Relevance: Standard values. Ex: 0, $$ \pi $$. Depth: Axes points.

Periodicity

Repeats every 2$$ \pi $$. Relevance: Functions cycle. Ex: sin(x+2$$ \pi $$)=sin x. Depth: n integer.

Arc Length

l = r θ (radians). Relevance: Applications. Ex: Wheel distance. Depth: θ in rad.

Tip: Radian for calc; degree for everyday. Depth: Signs: sin odd, cos even. Errors: Forget radian unit. Historical: Arya Bhatt. Interlinks: Ch4 identities. Advanced: Principal values. Real-Life: Pendulum swings. Graphs: Unit circle. Coherent: Angles → Measures → Functions.

Additional: Notational: θ° for degrees, θ for radians. Pitfalls: Mix units in formulas.

60+ Questions & Answers - NCERT Based (Class 11) - Expanded from PDF Exercise 3.1

Expanded based on full NCERT Ex 3.1 (7 questions) + variations. Part A: 20 (1 mark short), Part B: 20 (4 marks medium with steps), Part C: 20 (8 marks long with detailed steps). All from PDF; answers point-wise, numerical stepwise with MathJax. Added more: Full Ex 3.1 solved, variations for pendulum, wheel, chord etc.

Part A: 1 Mark Questions (20 Qs - Short from Ex 3.1 & Variations)

1. What is the radian measure of 180°?

1 Mark Answer:

$$ \pi $$ radian.

2. Convert 90° to radians (approx).

1 Mark Answer:

$$ \pi/2 $$ rad.

3. What is 1 radian in degrees (approx)?

1 Mark Answer:

57°.

4. sin 0° = ?

1 Mark Answer:

5. cos $$ \pi/2 $$ = ?

1 Mark Answer:

6. tan $$ \pi $$ = ?

1 Mark Answer:

7. Is sin(-x) = -sin x?

1 Mark Answer:

Yes (odd function).

8. Full circle in radians?

1 Mark Answer:

2$$ \pi $$.

9. Arc length formula?

1 Mark Answer:

l = r θ (θ rad).

10. sin $$ \pi $$ = ?

1 Mark Answer:

11. cos 0 = ?

1 Mark Answer:

12. 1° in radians (approx)?

1 Mark Answer:

0.0175.

13. Positive angle direction?

1 Mark Answer:

Anticlockwise.

14. sin $$ 3\pi/2 $$ = ?

1 Mark Answer:

15. Key identity?

1 Mark Answer:

$$ \sin^2 x + \cos^2 x = 1 $$

16. Radian measure of 360°?

1 Mark Answer:

$$ 2\pi $$ rad.

17. 1 minute = ? degrees

1 Mark Answer:

1/60 °.

18. cos $$ \pi $$ = ?

1 Mark Answer:

19. Domain for tan x undefined?

1 Mark Answer:

$$ (2n+1)\pi/2 $$.

20. 1 second = ? minutes

1 Mark Answer:

1/60 '.

Part B: 4 Marks Questions (20 Qs - Medium from Ex 3.1 Full + Variations)

1. Find radian measures for: (i) 25°

4 Marks Answer (Step-by-Step):

Step 1: $$ \theta = 25 \times \pi / 180 $$
Step 2: Simplify: $$ 5\pi / 36 $$ rad.
Verify: Approx 0.436 rad.
Relevance: Basic conversion.

16. Wheel makes 360 rev/min; radians in 1 sec? (Ex 3.1 Q3)

4 Marks Answer (Step-by-Step):

Step 1: 360 rev/min = 6 rev/sec.
Step 2: 1 rev = $$ 2\pi $$ rad → $$ 12\pi $$ rad/sec.
Verify: Rate calculation.
Relevance: Rotation.

17. Degree measure for 22 cm arc, r=100 cm ($$ \pi=22/7 $$) (Ex 3.1 Q4)

4 Marks Answer (Step-by-Step):

Step 1: $$ \theta $$ rad = 22/100 = 0.22.
Step 2: Deg = 0.22 × 180 ×7/22 = 12.6°.
Verify: Approx 12°36'.
Relevance: Arc angle.

18. Diameter 40 cm, chord 20 cm; minor arc length (Ex 3.1 Q5)

4 Marks Answer (Step-by-Step):

Step 1: r=20 cm; angle = 60° (cos rule).
Step 2: Arc = $$ (60/360) \times 2\pi \times 20 = 20\pi/3 $$ cm.
Verify: Equilateral triangle.
Relevance: Chord-arc.

19. Same arc lengths subtend 60° & 75°; radii ratio (Ex 3.1 Q6)

4 Marks Answer (Step-by-Step):

Step 1: $$ l = r_1 \theta_1 = r_2 \theta_2 $$; $$ \theta_1 = \pi/3 $$, $$ \theta_2 = 5\pi/12 $$ rad.
Step 2: $$ r_1 / r_2 = \theta_2 / \theta_1 = 5/4 $$.
Ratio: 5:4 (inverse since l same).
Relevance: Proportional radii.

20. Pendulum l=75 cm, arc=10 cm; radian angle (Ex 3.1 Q7 i)

4 Marks Answer (Step-by-Step):

Step 1: $$ \theta = 10/75 = 2/15 $$ rad.
Step 2: Approx 0.133 rad or 7.64°.
Verify: Small angle.
Relevance: Swing angle.