NCERT Solutions for Class 11 Maths Chapter 3 – Trigonometric Functions
Chapter 3 is, without question, the most formula-dense chapter in Class 11 Mathematics. And yet, it is also one of the most rewarding — once you understand the unit circle and the way trigonometric functions behave, you stop memorising and start deriving. That shift is exactly what this chapter is designed to bring about.
In Class 9 and 10, you studied trigonometry as the ratio of sides in a right-angled triangle. That definition works beautifully for acute angles, but it breaks down the moment you ask: what is the sine of 150°? Or 270°? Or even a negative angle? Chapter 3 resolves this by redefining trigonometric functions in terms of the unit circle, extending them to any real number and not just angles between 0° and 90°. For all Chapters must, read NCERT Solutions for Class 11 Maths and subject-wise NCERT Solutions for Class 11.
NCERT's approach here is methodical. The chapter opens by connecting degrees to radians — a unit students often resist at first but come to appreciate in calculus. It then redefines all six trig functions for general angles, explores their signs in different quadrants using the ASTC rule, and builds towards a rich collection of identities that are used throughout Class 11 and 12. The solutions here are built to help you understand the derivation of each identity rather than just stating them, because CBSE increasingly rewards shown working even in objective sections.
Download PDF – All Exercises of NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions
| Exercise | Topic Covered | Number of Questions |
|---|---|---|
| Exercise 3.1 | Angle Measurement – Degrees and Radians | 7 Questions |
| Exercise 3.2 | Trig Functions and Their Values | 10 Questions |
| Exercise 3.3 | Trigonometric Identities and Compound Angles | 25 Questions |
| Exercise 3.4 | Trigonometric Equations – General Solutions | 9 Questions |
| Miscellaneous | Mixed Problems and Proof-Based Questions | 10 Questions |
Chapter 3 – Trigonometric Functions: Concepts, Explanation and Key Tables
The Radian Measure: Why Bother?
Degrees are intuitive but mathematically inconvenient. Radians make calculus clean. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Since the full circumference is 2πr, a complete revolution = 2π radians = 360°.
The conversion formula is: θ (in radians) = θ (in degrees) × π/180. Students who memorise a few key conversions (30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π) can handle most questions in Exercise 3.1 within seconds.
Sign of Trigonometric Functions in Each Quadrant (ASTC Rule)
| Quadrant | Angle Range | Positive Functions | Negative Functions |
|---|---|---|---|
| I (First) | 0° to 90° | All (sin, cos, tan, cosec, sec, cot) | None |
| II (Second) | 90° to 180° | sin, cosec | cos, tan, sec, cot |
| III (Third) | 180° to 270° | tan, cot | sin, cos, sec, cosec |
| IV (Fourth) | 270° to 360° | cos, sec | sin, tan, cosec, cot |
Memory aid: All Students Take Calculus — Quadrants I, II, III, IV.
Standard Values of Trigonometric Functions
| Angle | 0° | 30° (π/6) | 45° (π/4) | 60° (π/3) | 90° (π/2) | 180° (π) |
|---|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | –1 |
| tan | 0 | 1/√3 | 1 | √3 | Not defined | 0 |
| cosec | Not def. | 2 | √2 | 2/√3 | 1 | Not def. |
| sec | 1 | 2/√3 | √2 | 2 | Not defined | –1 |
| cot | Not def. | √3 | 1 | 1/√3 | 0 | Not def. |
Key Identities You Must Know
These identities are the engine of Exercise 3.3 and almost every trigonometry question in competitive exams:
Fundamental (Pythagorean) Identities: sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = cosec²θ
Compound Angle Formulas: sin(A + B) = sin A cos B + cos A sin B sin(A – B) = sin A cos B – cos A sin B cos(A + B) = cos A cos B – sin A sin B cos(A – B) = cos A cos B + sin A sin B tan(A + B) = (tan A + tan B) / (1 – tan A tan B) tan(A – B) = (tan A – tan B) / (1 + tan A tan B)
Double Angle Formulas: sin 2A = 2 sin A cos A cos 2A = cos²A – sin²A = 1 – 2sin²A = 2cos²A – 1 tan 2A = 2 tan A / (1 – tan²A)
Triple Angle Formulas: sin 3A = 3 sin A – 4 sin³A cos 3A = 4 cos³A – 3 cos A
General Solutions of Trigonometric Equations
| Equation | General Solution |
|---|---|
| sin θ = 0 | θ = nπ, n ∈ ℤ |
| cos θ = 0 | θ = (2n + 1)π/2, n ∈ ℤ |
| tan θ = 0 | θ = nπ, n ∈ ℤ |
| sin θ = sin α | θ = nπ + (–1)ⁿ α, n ∈ ℤ |
| cos θ = cos α | θ = 2nπ ± α, n ∈ ℤ |
| tan θ = tan α | θ = nπ + α, n ∈ ℤ |
Graphs of Trigonometric Functions – Properties
| Function | Domain | Range | Period | Odd/Even |
|---|---|---|---|---|
| sin x | ℝ | [–1, 1] | 2π | Odd |
| cos x | ℝ | [–1, 1] | 2π | Even |
| tan x | ℝ – {(2n+1)π/2} | ℝ | π | Odd |
| cosec x | ℝ – {nπ} | (–∞,–1] ∪ [1,∞) | 2π | Odd |
| sec x | ℝ – {(2n+1)π/2} | (–∞,–1] ∪ [1,∞) | 2π | Even |
| cot x | ℝ – {nπ} | ℝ | π | Odd |
Study Tips for Chapter 3
- Never just memorise sin 30° = 1/2. Understand how the unit circle produces this — you will then derive it even under exam pressure.
- Exercise 3.3 problems almost always reduce to one of three things: Pythagorean identities, compound angle formulas, or product-to-sum conversions. Identify which before calculating.
- For general NCERT solutions in Exercise 3.4, write the full general form with n ∈ ℤ — partial answers lose marks.
- Prove identities by working on only one side (usually the more complex one), never by cross-multiplying.
FAQs on NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions
Trigonometric functions are mathematical functions that relate the angles of a triangle to the ratios of its sides. The six primary trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. In Class 11 Maths, students learn how these functions are defined, evaluated, and applied to different types of problems. NCERT Solutions explain these concepts with detailed examples and solved exercises. Understanding trigonometric functions is important because they are used extensively in geometry, physics, engineering, navigation, and astronomy. Regular practice helps students become familiar with their properties and applications, making it easier to solve examination questions accurately and efficiently.
The radian is a standard unit used for measuring angles in mathematics. Unlike degrees, radians are directly related to the radius and arc length of a circle, making calculations more convenient in advanced mathematics. Chapter 3 introduces students to the conversion between degrees and radians and explains their practical significance. NCERT Solutions provide step-by-step methods for solving questions involving angle conversions and radian measures. A clear understanding of radians is essential because many higher-level mathematical concepts, including calculus and trigonometric graphs, use radian measurement. Mastering this topic helps students build a stronger foundation for future studies.
Trigonometric identities are equations involving trigonometric functions that remain true for all valid values of angles. Students can memorize these identities effectively by understanding their derivations rather than relying solely on rote learning. Regular practice, revision of formulas, and solving identity-based questions help improve retention. NCERT Solutions include numerous examples that demonstrate how identities are applied in problem-solving. Preparing a formula sheet and revising it frequently can also be helpful. Understanding the relationships among sine, cosine, tangent, secant, cosecant, and cotangent functions makes it easier to recall identities during examinations and use them correctly in mathematical proofs.
NCERT Solutions provide detailed and accurate explanations for every exercise question in the chapter. They help students understand important concepts such as angle measurement, trigonometric functions, identities, and graphs through step-by-step solutions. These solutions make complex problems easier to understand and encourage logical thinking. Students can use them for classroom learning, homework completion, revision, and exam preparation. Regular practice with NCERT Solutions improves problem-solving skills, accuracy, and confidence. Since trigonometry forms the basis of many advanced mathematical topics, mastering this chapter through NCERT Solutions is beneficial for both academic success and competitive examinations.