Class 11 Maths NCERT Solutions Chapter 3 Trigonometric Functions | Boundless Maths

Chapter 3 Class 11 Maths NCERT Solutions · Unit I · Sets and Functions

Class 11 Maths NCERT Solutions Chapter 3: Trigonometric Functions

Free, step-by-step NCERT Solutions for all three exercises of this chapter — angles and their measurement, trigonometric functions defined on the unit circle, and the sum and difference formulas — solved the way CBSE awards marks, with the key definitions and formulas right on this page.

📐 Formula Cards

4Exercises (incl. Misc.)

52Total Questions

2026-27CBSE Syllabus

100%Solved

Class 11 Maths NCERT Solutions Chapter 3 — Overview

An angle can be measured in degrees or radians, related by π radians = 180°, with arc length l = rθ. Placing an angle on the unit circle defines cos x and sin x for any real x, with sin²x + cos²x = 1 and the sign of each function fixed by the quadrant (ASTC rule). This chapter covers how sine and cosine repeat every 2π, how the sum and difference formulas cos(x±y) and sin(x±y) are derived and used, and the double, triple and sum-to-product identities built from them.

How the Chapter Builds

One Idea Leads to the Next

Angles

Degree and radian measure, the relation l = rθ, and converting between the two. Exercise 3.1.

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Trigonometric Functions

Defined via the unit circle — domain, range, sign in each quadrant, and periodicity. Exercise 3.2.

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Sum & Difference Formulas

cos(x ± y), sin(x ± y), tan(x ± y) — the identities every later result is built from. Exercise 3.3.

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Multiple Angle Formulas

sin 2x, cos 2x, tan 2x and the triple-angle formulas, derived from the sum formulas. §3.3.

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Sum-to-Product Identities

Converting sums/differences of sines and cosines into products, and back. Miscellaneous Exercise.

Quick Reference

Important Formulas — Chapter 3

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Trigonometric Functions.

Angles (§3.2)

Arc length

l = r\theta

θ in radians, r the radius — an arc of length equal to the radius subtends exactly 1 radian.

Degree ⟷ radian

\pi \text{ radian} = 180^\circ

Radian measure = (π/180) × degree measure; Degree measure = (180/π) × radian measure.

Trigonometric Functions (§3.3)

Pythagorean identities

\sin^2x+\cos^2x=1 \quad 1+\tan^2x=\sec^2x \quad 1+\cot^2x=\csc^2x

All three follow from dividing the first identity by cos²x or sin²x.

Periodicity

\sin(2n\pi+x)=\sin x \qquad \cos(2n\pi+x)=\cos x

sin and cos repeat every 2π; tan and cot repeat every π.

ASTC sign rule

Quadrant I: all + II: sin,cosec + III: tan,cot + IV: cos,sec +

"All Sin Tan Cos" — remember which pair of functions is positive in each quadrant.

Sum & Difference Formulas (§3.4)

Cosine sum/difference

\cos(x{+}y)=\cos x\cos y-\sin x\sin y

\cos(x{-}y)=\cos x\cos y+\sin x\sin y

The sign inside flips the sign of the product term — easy to mix up under exam pressure.

Sine sum/difference

\sin(x{+}y)=\sin x\cos y+\cos x\sin y

\sin(x{-}y)=\sin x\cos y-\cos x\sin y

Opposite pattern to cosine — the sign inside matches the sign in front of the product term.

Tangent sum/difference

\tan(x\pm y)=\dfrac{\tan x\pm\tan y}{1\mp\tan x\tan y}

Valid whenever x, y and x±y are not odd multiples of π/2.

Multiple Angles & Sum-to-Product

Double angle

\sin2x=2\sin x\cos x

\cos2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x

Three equivalent forms of cos 2x — pick whichever matches the rest of the expression.

Triple angle

\sin3x=3\sin x-4\sin^3x

\cos3x=4\cos^3x-3\cos x

Derived by expanding sin(2x+x) and cos(2x+x) with the sum formulas.

Sum-to-product

\sin x+\sin y=2\sin\tfrac{x+y}{2}\cos\tfrac{x-y}{2}

\cos x+\cos y=2\cos\tfrac{x+y}{2}\cos\tfrac{x-y}{2}

Turns a sum or difference of two sines/cosines into a product — the key move in most Ex 3.3 proofs.

Decision Guide

Which Trigonometry Technique Applies?

A quick way to decide, once you know what the question is actually asking for.

What the question is asking	Use this	Why
Convert a degree measure to radians or back	π radian = 180° conversion	Multiply by π/180 (degree→radian) or 180/π (radian→degree) (§3.2).
Find an arc length, radius, or the angle subtended at the centre	l = rθ	θ must be in radians for this formula to apply directly (§3.2).
Given one trig ratio and a quadrant, find the other five	Pythagorean identity + ASTC	Solve for the paired ratio, then fix the sign using the quadrant (§3.3).
Evaluate a trig function of a very large or negative angle	Periodicity	Subtract/add multiples of 2π (or π for tan/cot) to reach a standard angle (§3.3).
Prove an identity involving cos(x+y) or sin(x−y)	Sum/difference formulas	Expand directly — matches the exact structure of the identity (§3.4).
Simplify an expression with sin 2x, cos 2x or tan 2x	Double angle formulas	Choose the cos 2x form (in sin, cos, or tan) that matches surrounding terms.
An identity has sin x + sin y or cos x − cos y	Sum-to-product (C+D) formulas	Converts a sum into a product, which is usually easier to simplify or cancel.
An identity has a product like sin x cos y	Product-to-sum formulas	The reverse conversion — turns a product into a sum for easy combination.

Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across all three exercises.

Mixing degree and radian measure mid-calculation — l = rθ, and every sum/difference/multiple-angle formula, requires θ in radians. Convert first, and keep the units consistent throughout a solution.
Getting the sign wrong after finding sin²x or cos²x — taking the square root gives ±, and the actual sign must be fixed using which quadrant x lies in (ASTC), not assumed positive by default.
Swapping the sign pattern between cos(x±y) and sin(x±y) — cosine flips the sign inside to flip the sign of the product term; sine does not. Writing sin(x−y) = sin x cos y + cos x sin y is a very common slip.
Assuming tan x and cot x repeat every 2π, like sine and cosine — they actually repeat every π, which changes how a large angle should be reduced.
Reducing a large angle by the wrong multiple of 2π (or π) — always check the remainder lands in [0, 2π) before reading off the quadrant and sign.
Forgetting the domain restrictions on tan, cot, sec and cosec — these are undefined wherever cos x = 0 or sin x = 0 respectively; stating the formula without this exclusion is an incomplete answer.
Trying to prove a sum/difference identity by expanding both sides independently — it's almost always faster to simplify only the more complex side (usually the LHS) down to match the other, rather than manipulating both at once.

Solve Chapter-Wise

Class 11 Maths NCERT Solutions Chapter 3 — Choose an Exercise

3.1

Exercise 3.1

Angles — degree and radian measure, converting between the two, and the arc-length formula l = rθ · 7 questions

Solve Exercise 3.1 →

3.2

Exercise 3.2

Trigonometric functions — finding the remaining ratios from one given value and quadrant, and evaluating functions of large or negative angles · 10 questions

Solve Exercise 3.2 →

3.3

Exercise 3.3

Trigonometric functions of sum and difference of two angles — proving identities, evaluating exact values, and multiple-angle formulas · 25 questions

Solve Exercise 3.3 →

Miscellaneous Exercise

Sum-to-product proofs and half-angle problems, tying the whole chapter together · 10 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Trigonometric Functions — plus every other Class 11 Maths chapter — in one printable PDF.

Get Formula Cards →

Common Questions

Frequently Asked Questions

Quick answers from Class 11 Maths NCERT Solutions Chapter 3, Trigonometric Functions.

How many exercises are there in Chapter 3, Trigonometric Functions?

There are three main exercises — 3.1 (Angles, 7 questions), 3.2 (Trigonometric Functions, 10 questions) and 3.3 (Trigonometric Functions of Sum and Difference of Two Angles, 25 questions) — plus a Miscellaneous Exercise of 10 questions applying these identities together, totalling 52 questions.

What is the difference between degree measure and radian measure?

Degree measure divides one complete revolution into 360 equal parts, each called a degree. Radian measure instead defines an angle by the arc it subtends on a unit circle — an angle of 1 radian corresponds to an arc equal in length to the radius. The two are related by π radians = 180°, which is the conversion factor used to move between them.

How do you find the values of trigonometric functions for angles greater than 360° or negative angles?

Since sine and cosine repeat every 2π radians (360°), any angle can first be reduced by adding or subtracting a suitable multiple of 2π until it falls between 0 and 2π. The resulting angle's quadrant then fixes the sign of each trigonometric function using the ASTC rule, and the reference angle gives the numerical value.

Where can I find the official NCERT textbook for this chapter?

Trigonometric Functions is Chapter 3 of the NCERT Class 11 Mathematics textbook, published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.