Chapter 3 of 14

21%

CBSE Class 11 Mathematics★ 10-12 marks · CBSE Class 11 Mathematics Chapter 3

Trigonometric Functions

Sine, cosine, tangent — the ratios that relate angles to sides and UNDERLIE everything from physics to engineering. This chapter covers degree and radian measure, trigonometric functions of any angle, graphs, and key identities. CBSE Class 11 Mathematics Chapter 3.

⏱ 28 min readHard difficulty✓ Free · NCERT-aligned

Ask AI about this

🔤Build your vocabulary from this chapterLexiGenome mines the key words, explains them in your language, and schedules them with spaced repetition.Mine words →

By the end of this chapter you'll be able to…

1Convert between degree and radian measure and recall standard angle values (0°, 30°, 45°, 60°, 90°)
2Use the unit circle definition to determine signs of sin, cos, tan in all four quadrants (ASTC rule)
3Apply all three Pythagorean identities and reciprocal identities to simplify and prove expressions
4Use sum, difference, and double-angle formulas to evaluate trigonometric expressions
5Identify the domain, range, and period of y=sin x, y=cos x, and y=tan x from their graphs

💡

Why this chapter matters

Trigonometric Functions carries the highest single-chapter weightage in Class 11 Mathematics and is indispensable for JEE — identities, compound angle formulas, and general solutions appear in nearly every competitive exam. It is also the prerequisite for calculus (derivatives of sin/cos) in Class 12.

Trigonometric Functions

"Trigonometry began with the stars. It will take you to the farthest reaches of calculus."

1. Chapter Overview

TRIGONOMETRY is the study of relationships between ANGLES and SIDES of triangles — and, more broadly, PERIODIC PHENOMENA (waves, oscillations, circular motion). This chapter covers: angle measurement (degrees AND radians), trigonometric ratios for ALL angles (not just acute), graphs of sin/cos/tan, and FUNDAMENTAL IDENTITIES.

2. Angles — Degree and Radian Measure

Degree Measure

1 complete revolution = 360° (historical — Babylonian origin)
1° = 60 minutes (60'). 1' = 60 seconds (60").

Radian Measure

1 radian = the angle subtended at the centre of a circle by an arc EQUAL IN LENGTH to the radius
π radians = 180°
Conversion: Degrees × (π/180) = Radians
Radians are the 'NATURAL' unit for calculus (limits, derivatives of trig functions only work simply with radian measure)

Key Angles

Degrees	0°	30°	45°	60°	90°	180°	270°	360°
Radians	0	π/6	π/4	π/3	π/2	π	3π/2	2π

3. Trigonometric Ratios — Beyond Acute Angles

Unit Circle Definition

Draw a circle of RADIUS 1 centred at the ORIGIN
A point P(x, y) on the circle makes an angle θ with the positive x-axis
cos θ = x (the x-coordinate of P)
sin θ = y (the y-coordinate of P)
tan θ = y/x = sin θ / cos θ (x ≠ 0)
This extends trig functions to ANY angle (not just 0° to 90°) — and shows they are PERIODIC

Signs in Quadrants

Quadrant	Angle Range	sin	cos	tan
I	0° to 90°	+	+	+
II	90° to 180°	+	—	—
III	180° to 270°	—	—	+
IV	270° to 360°	—	+	—

Mnemonic: All Students Take Calculus (I: All +ve, II: Sin +ve, III: Tan +ve, IV: Cos +ve)

4. Trigonometric Ratios of Allied Angles

Angle	sin	cos	tan
-θ	-sin θ	cos θ	-tan θ
90° ± θ	cos θ	∓ sin θ	-cot θ (for 90°+θ), cot θ (for 90°-θ)
180° ± θ	∓ sin θ	-cos θ	± tan θ
270° ± θ	-cos θ	± sin θ	∓ cot θ
360° ± θ	± sin θ	cos θ	± tan θ

5. Graphs of Trigonometric Functions

y = sin x

Domain: R (all real numbers)
Range: [-1, 1]
Period: 2π
Shape: SMOOTH WAVE starting at (0, 0), peaks at (π/2, 1), crosses at (π, 0), trough at (3π/2, -1)

y = cos x

Domain: R
Range: [-1, 1]
Period: 2π
Shape: Same as sin but shifted LEFT by π/2. Starts at (0, 1).

y = tan x

Domain: R — {π/2 + nπ, n ∈ Z} (asymptotes at odd multiples of π/2)
Range: R (all real numbers)
Period: π
Shape: Repeating curve with VERTICAL ASYMPTOTES at x = π/2 + nπ

6. Fundamental Trigonometric Identities

Pythagorean Identities

sin²θ + cos²θ = 1 (THE fundamental identity)
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ

Reciprocal Identities

cot θ = 1/tan θ = cos θ/sin θ
sec θ = 1/cos θ
cosec θ = 1/sin θ

Sum and Difference 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 = 2 cos²A — 1 = 1 — 2 sin²A
tan 2A = 2 tan A / (1 — tan²A)

7. Exam Focus

Radian measure — definition, conversion to/from degrees
Trig ratios of standard angles (0°, 30°, 45°, 60°, 90°) — MEMORISE
Signs of trig functions in all four quadrants (ASTC)
Domain and range of sin, cos, tan
Graphs and their key features (period, amplitude, asymptotes)
Fundamental identities — sin²θ + cos²θ = 1 and its variants
Sum/difference and double angle formulas

8. Key Values to Memorise

θ	0°	30°	45°	60°	90°
sin θ	0	1/2	1/√2	√3/2	1
cos θ	1	√3/2	1/√2	1/2	0
tan θ	0	1/√3	1	√3	Not defined

9. Conclusion

Trigonometry is the GEOMETRY OF CIRCLES, extended to the ALGEBRA OF WAVES:

ANGLES: Degrees for geometry, RADIANS for calculus
UNIT CIRCLE: The elegant definition — extends trig to ALL angles, reveals periodicity
IDENTITIES: The toolkit. sin²θ + cos²θ = 1 is the foundation of everything
GRAPHS: Sin, cos, tan — the repeating, oscillating, beautiful curves that describe everything from pendulums to sound waves

'There is geometry in the humming of the strings. There is music in the spacing of the spheres.' — Pythagoras. Trigonometry is where geometry and algebra meet — and sing.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Degree-Radian Conversion

Radians = Degrees × (π/180); Degrees = Radians × (180/π)

π radians = 180°; used constantly when evaluating trig functions in calculus

Pythagorean Identity (primary)

sin²θ + cos²θ = 1

THE fundamental identity — all others are derived from this

Pythagorean Identity (tan/sec)

1 + tan²θ = sec²θ

Derived by dividing sin²θ + cos²θ = 1 through by cos²θ

Pythagorean Identity (cot/cosec)

1 + cot²θ = cosec²θ

Derived by dividing sin²θ + cos²θ = 1 through by sin²θ

Compound Angle: sin(A±B)

sin(A+B) = sinA cosB + cosA sinB; sin(A−B) = sinA cosB − cosA sinB

Signs: sin of sum — same signs; sin of difference — opposite signs

Compound Angle: cos(A±B)

cos(A+B) = cosA cosB − sinA sinB; cos(A−B) = cosA cosB + sinA sinB

Signs: cos of sum — opposite signs; cos of difference — same signs

Double Angle Formulas

sin2A = 2sinA cosA; cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan2A = 2tanA/(1−tan²A)

Three forms for cos2A — choose the one that fits the given expression

Standard Values

sin30°=1/2, sin45°=1/√2, sin60°=√3/2; cos30°=√3/2, cos45°=1/√2, cos60°=1/2; tan30°=1/√3, tan45°=1, tan60°=√3

Must be memorised — form the building blocks for all other exact values

⚠️

Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Writing cos(A+B) = cosA + cosB

✓ cos(A+B) = cosA cosB − sinA sinB. Trigonometric functions do NOT distribute over addition.

WATCH OUT

✗ Confusing the sign in cos 2A — using the wrong form

✓ cos2A has three equivalent forms: cos²A−sin²A, 2cos²A−1, and 1−2sin²A. Choose based on what's given: if only cosA appears, use 2cos²A−1; if only sinA, use 1−2sin²A.

WATCH OUT

✗ Forgetting that tan 90° is undefined (not infinity)

✓ tan θ = sinθ/cosθ. At θ=90°, cosθ=0, so the expression is undefined. The domain of tan x excludes π/2+nπ for all integers n.

WATCH OUT

✗ Using degrees in calculus-based limits (lim sinx/x = 1 only in radians)

✓ The standard limit lim(x→0) sinx/x = 1 is valid ONLY when x is in radians. In degrees the limit is π/180, not 1.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 3.1

Exercise 3.1

Degree-radian conversion, arc length, and angle measurement problems

Questions

Ex 3.2

Exercise 3.2

Trigonometric functions of angles, signs in quadrants, allied angles

Questions

Ex 3.3

Exercise 3.3

Proving identities, evaluating using compound and double-angle formulas

Questions

Ex 3.4

Exercise 3.4

Principal and general solutions of trigonometric equations

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Standard Values

Evaluate: sin 30° cos 60° + cos 30° sin 60°.

▶ Show solution

Using sin(A+B) = sinA cosB + cosA sinB, this equals sin(30°+60°) = sin 90° = 1. Alternatively: (1/2)(1/2) + (√3/2)(√3/2) = 1/4 + 3/4 = 1.

Q2MEDIUM· Identities

Prove: (1 + tan²θ) / (1 + cot²θ) = tan²θ.

▶ Show solution

LHS = sec²θ / cosec²θ (using 1+tan²θ=sec²θ and 1+cot²θ=cosec²θ). = (1/cos²θ) / (1/sin²θ) = sin²θ/cos²θ = tan²θ = RHS. Hence proved.

Q3HARD· Double Angle

If tanθ = 3/4 and θ is in the first quadrant, find sin2θ, cos2θ, and tan2θ.

▶ Show solution

Step 1: From tanθ=3/4, draw right triangle with opposite=3, adjacent=4, hypotenuse=5. So sinθ=3/5, cosθ=4/5. Step 2: sin2θ = 2sinθcosθ = 2(3/5)(4/5) = 24/25. Step 3: cos2θ = cos²θ−sin²θ = 16/25−9/25 = 7/25. Step 4: tan2θ = sin2θ/cos2θ = (24/25)/(7/25) = 24/7.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•π radians = 180°; radian measure is essential for calculus (limits/derivatives of trig functions)
•Unit circle: cos θ = x-coordinate, sin θ = y-coordinate of point on circle of radius 1
•ASTC rule — quadrant I: all positive; II: sin positive; III: tan positive; IV: cos positive
•Three Pythagorean identities: sin²θ+cos²θ=1; 1+tan²θ=sec²θ; 1+cot²θ=cosec²θ
•sin(A+B) = sinA cosB + cosA sinB; cos(A+B) = cosA cosB − sinA sinB
•sin 2A = 2sinA cosA; three forms for cos 2A to handle different situations
•Period of sin and cos = 2π; period of tan and cot = π
•tan 90° and cot 0° are UNDEFINED (denominator = 0 in respective ratios)

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 10-12 marks

Question type	Marks each	Typical count	What it tests
Short Answer	2	2	Evaluating trig ratios at standard angles, degree-radian conversion
Long Answer	4-6	1-2	Proving identities, solving equations, applying compound/double-angle formulas

Prep strategy

Memorise the standard values table (sin/cos/tan for 0°,30°,45°,60°,90°) — it unlocks almost every numerical question
For identity proofs, work on ONE side only (usually the more complex side) and simplify step-by-step using known identities
Practice the ASTC quadrant rule daily until it is instinctive — it determines the sign of every trig expression

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Sound and Light Waves

All periodic waves (sound, light, radio) are modelled using sin and cos functions — the amplitude, frequency, and phase are the three parameters of y = A sin(ωt + φ).

Architecture and Structural Engineering

Engineers use trigonometric ratios to calculate forces in beams, determine angles for roof trusses, and analyse the stability of structures.

Navigation and Surveying

GPS, ship navigation, and land surveying all use the sine rule and cosine rule derived from the identities in this chapter to find unknown distances and angles.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

For identity proofs, never cross-multiply or transfer terms — work on one side only
Always check which quadrant the angle is in before assigning a sign to a trig ratio
For finding values like sin 75°, decompose as sin(45°+30°) and apply the sum formula
In the exam, write down all given data and the required formula before substituting — this reduces errors and earns method marks

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Product-to-sum formulas: 2sinA cosB = sin(A+B)+sin(A−B); used in JEE problems to simplify products into sums
Conditional identities: if A+B+C=π (angles of a triangle), then tanA+tanB+tanC=tanA·tanB·tanC — a classic olympiad result

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 11 BoardVery High

JEE MainVery High

JEE AdvancedVery High

AI helpers for this chapter

Generate more practice on the fly, or have the chapter explained at a different level.

⚡

Generate 5 more questions

Fresh MCQs + short-answer items based on this chapter, plus answer keys.

🧠

Explain at a different level

Re-explain the chapter at the level you actually want.

📝 My notes

Saved on this device — sign in to sync · how this works

0/600

Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

The derivative of sinx is cosx ONLY when x is in radians. If x is in degrees, an extra factor of π/180 appears. Radians are the 'natural' unit because they arise directly from arc length (arc length = rθ).

Use the ASTC mnemonic: All Students Take Calculus. Quadrant I: All positive. Quadrant II: Sin positive. Quadrant III: Tan positive. Quadrant IV: Cos positive.

x = nπ + (−1)ⁿα, where n is any integer. For cosx=cosα: x = 2nπ ± α. For tanx=tanα: x = nπ + α.

Practice more, ask doubts, find a tutor

CBSE Class 11 mock tests

CBSE Class 11 PYQs

Ask in Q&A

Mathematics flashcards

Related chapters

Students who read Trigonometric Functions usually read these next.

Sets

The language of all mathematics begins here — with SETS. This foundational chapter covers set notation, types of sets (empty, finite, infinite, equal, subsets), Venn diagrams, and set operations (union, intersection, difference, complement). CBSE Class 11 Mathematics Chapter 1.

Relations and Functions

How are things CONNECTED? Relations and functions formalise the idea of association — from the Cartesian product of sets to the special case where every input has EXACTLY ONE output. This chapter covers domain, range, and types of functions. CBSE Class 11 Mathematics Chapter 2.

Complex Numbers and Quadratic Equations

What is √(-1)? The answer — i — opens up an ENTIRE NEW NUMBER SYSTEM. This chapter introduces complex numbers (a + ib), their algebra, the Argand plane, and how they solve every quadratic equation. CBSE Class 11 Mathematics Chapter 4.

Linear Inequalities

Not every relationship is an EQUATION. Sometimes — often — it's an INEQUALITY. This chapter covers solving and graphing linear inequalities in one and two variables, and the crucial rule: when you multiply or divide by a NEGATIVE, the sign FLIPS. CBSE Class 11 Mathematics Chapter 5.