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.