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Mathematics Grade XI

Grade 11 Mathematics

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Chapter 3: Trigonometric Functions

Class 11 • Mathematics • Chapter 3

Trigonometric Functions

Measuring angles and working with sine, cosine and the other ratios for every angle.

CBSE / NCERT Aligned   School Revise

Chapter Roadmap

Angles • Degree & Radian Measure • Trigonometric Functions • Signs in Quadrants • Domain & Range • Sum & Difference Formulas • Multiple-Angle Identities • Applications
1

Why Trigonometry Goes Far Beyond Triangles

In earlier classes you used sine, cosine and tangent only for right-angled triangles. In Class 11 the subject grows up. Angles are no longer trapped between 0° and 90°; they can spin round and round, in either direction, and can even be measured in a natural unit called the radian. Once angles are set free, the trigonometric ratios become true functions that repeat in smooth waves.

This is why trigonometry quietly runs the modern world. The sound coming from your earphones, the alternating current in your home, the GPS signal that locates your phone, and the orbit of a satellite are all described by sine and cosine waves. This chapter builds that machinery: how to measure angles properly, how to define the six trigonometric functions on a unit circle, how their signs change across the four quadrants, and how to combine angles with the powerful sum, difference and multiple-angle formulas.

Key relation
A straight angle of 180° equals π radians. So 1 radian ≈ 57.3°, and to convert: multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees.
2

Key Terms You Must Know
Term Meaning Example
Angle The amount of rotation of a ray about its starting point; positive if anticlockwise, negative if clockwise. A 90° turn from east to north
Degree An angle unit where one full turn is 360°. A right angle is 90°
Radian The natural angle unit; the angle that an arc equal in length to the radius subtends at the centre. π rad = 180°
Unit circle A circle of radius 1 centred at the origin, used to define trig functions for any angle. x² + y² = 1
Periodic function A function whose values repeat after a fixed interval called the period. sin x repeats every 2π
Quadrant One of the four regions of the plane; the sign of each trig function depends on it. Quadrant I: 0 to 90°
3

Core Concepts, Step by Step

1. Measuring Angles: Degrees and Radians

An angle measures rotation. Going anticlockwise gives a positive angle; clockwise gives a negative one, and the ray can turn past a full circle. Two units are used. In degree measure a full turn is 360°. In radian measure, one radian is the angle made at the centre by an arc whose length equals the radius. The bridge between them is π radians = 180°. A handy length formula follows: for a circle of radius r, an arc that subtends an angle θ (in radians) has length l = rθ.

2. Starting Point: the Right Triangle

For an acute angle θ, the three main ratios come straight from a right-angled triangle. Label the side opposite θ as a, the side next to θ (touching the right angle) as b, and the longest side facing the right angle as the hypotenuse c. The figure below shows how each ratio is just one side divided by another.

The right triangle: SOH-CAH-TOA

opposite
(side a)
         
         
         
         
□       θ
hypotenuse
(side c)
  adjacent (side b)  

The small square marks the right angle (90°); the angle θ sits at the opposite corner. Reading the sides relative to θ:
sin θ = opposite ÷ hypotenuse = a / c
cos θ = adjacent ÷ hypotenuse = b / c
tan θ = opposite ÷ adjacent = a / b
A quick way to recall this is SOH-CAH-TOA: Sin-Opp-Hyp, Cos-Adj-Hyp, Tan-Opp-Adj.

The right triangle: sides and the angle θ

Right triangle showing opposite, adjacent, hypotenuse and angle theta

3. Growing Up: the Unit Circle

To handle any angle (not just 0°-90°), we move to the unit circle: a circle of radius 1 centred at the origin. Place the angle θ at the centre, measured anticlockwise from the positive x-axis, and look at where its arm meets the circle at point P(x, y). Then cos θ = x and sin θ = y. The other four follow: tan θ = sin θ/cos θ, cot θ = cos θ/sin θ, sec θ = 1/cos θ, cosec θ = 1/sin θ. Because the radius is 1, the point obeys x² + y² = 1, which immediately gives the fundamental identity sin²θ + cos²θ = 1.

The unit circle: P(cosθ, sinθ) on a circle of radius 1

Unit circle with angle theta and point cos sin

4. Signs of the Functions in the Four Quadrants

As θ grows, P travels around the circle and the signs of x and y change, so the functions change sign too. The figure below shows which functions are positive in each quadrant. A quick way to recall it, reading Quadrants I to IV in order: All, Sine, Tangent, Cosine.

The unit circle and the four quadrants

90° (π/2) ↑  y-axis = sin

Quadrant II (90°-180°)

Positive: sin, cosec.
cos < 0, sin > 0

Quadrant I (0°-90°)

All six positive.
cos > 0, sin > 0

Quadrant III (180°-270°)

Positive: tan, cot.
cos < 0, sin < 0

Quadrant IV (270°-360°)

Positive: cos, sec.
cos > 0, sin < 0
270° (3π/2) ↓
← 180° (π) 0° / 360° (0, 2π) →  x-axis = cos

An angle θ meets the unit circle at P(cos θ, sin θ): the x-coordinate is cos θ and the y-coordinate is sin θ. The sign of each function therefore follows the quadrant in which P lands.

Standard angles on the unit circle (worth knowing by heart)

Angle Radians sin cos tan
0° 0 0 1 0
30° π/6 1/2 √3/2 1/√3
45° π/4 1/√2 1/√2 1
60° π/3 √3/2 1/2 √3
90° π/2 1 0 undefined

5. Domain and Range of the Trigonometric Functions

Since sine and cosine are coordinates on the unit circle, their outputs always lie between −1 and 1; both are defined for every real angle. So sin x and cos x have domain R and range [−1, 1]. The function tan x is undefined wherever cos x = 0 (at odd multiples of π/2) and its range is all real numbers. The reciprocals sec x and cosec x have outputs of size at least 1 and skip the angles where their denominators vanish. Sine, cosine, cosec and sec repeat every 2π, while tan and cot repeat every π.

6. Sum, Difference and Multiple-Angle Formulas

These let you handle combined angles:
• sin(A ± B) = sin A cos B ± cos 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)
Setting B = A gives the double-angle forms: sin 2A = 2 sin A cos A, cos 2A = 1 − 2 sin²A = 2 cos²A − 1, and tan 2A = (2 tan A)/(1 − tan²A). The triple-angle forms are sin 3A = 3 sin A − 4 sin³A and cos 3A = 4 cos³A − 3 cos A.
For competitive exams
Beyond the board syllabus, the general solution of a trigonometric equation is worth knowing for JEE: sin θ = sin α gives θ = nπ + (−1)ⁿ α, cos θ = cos α gives θ = 2nπ ± α, and tan θ = tan α gives θ = nπ + α, where n is any integer. Treat this as enrichment, not core Class 11 board material.
4

Key Results & Proofs

Trigonometry is built on a few results that are actually proved, not merely stated, and every identity in this chapter follows from them. Here are the central ones, derived line by line.

Result 1: The Pythagorean Identity

Statement. For every angle θ, sin²θ + cos²θ = 1.

Proof

On the unit circle, the arm of angle θ meets the circle at P(cos θ, sin θ). Because P lies on the circle, its coordinates satisfy the circle’s equation.

x² + y² = 1 equation of the unit circle (radius 1)
cos²θ + sin²θ = 1 put x = cos θ, y = sin θ

Dividing both sides by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = cosec²θ.
Result 2: Cosine of a Difference (the parent formula)

Statement. For all angles A and B, cos(A − B) = cos A cos B + sin A sin B.

Proof

On the unit circle mark P₁(cos A, sin A) and P₂(cos B, sin B). The angle between OP₁ and OP₂ is (A − B). We find the squared chord length P₁P₂² in two ways, then equate them.

Way 1, the distance between P₁ and P₂:
P₁P₂² = (cosA − cosB)² + (sinA − sinB)² distance formula
  = cos²A − 2cosAcosB + cos²B + sin²A − 2sinAsinB + sin²B expand the squares
  = (cos²A + sin²A) + (cos²B + sin²B) − 2(cosAcosB + sinAsinB) regroup
  = 2 − 2(cosAcosB + sinAsinB) each bracket = 1
Way 2, the same chord, with the angle (A − B) placed from the x-axis, so the endpoints are (1, 0) and (cos(A−B), sin(A−B)):
P₁P₂² = (cos(A−B) − 1)² + sin²(A−B) distance formula
  = cos²(A−B) + sin²(A−B) − 2cos(A−B) + 1 expand
  = 2 − 2cos(A−B) the squared terms = 1
The chord has the same length both ways, so equate:
2 − 2(cosAcosB + sinAsinB) = 2 − 2cos(A−B) Way 1 = Way 2
cos(A−B) = cosA cosB + sinA sinB cancel the 2 and rearrange

This one result is the parent of the whole family: replacing B with −B gives cos(A + B) = cosA cosB − sinA sinB, and the sine formulas follow from the link sin θ = cos(90° − θ).
Result 3: Sine of a Sum

Statement. sin(A + B) = sin A cos B + cos A sin B.

Proof

Use the complementary-angle link sin θ = cos(90° − θ) together with Result 2.

sin(A + B) = cos(90° − (A + B)) sin θ = cos(90° − θ)
  = cos((90° − A) − B) regroup the angle
  = cos(90°−A)cosB + sin(90°−A)sinB apply Result 2
  = sinA cosB + cosA sinB cos(90°−A) = sinA, sin(90°−A) = cosA

Replacing B with −B gives sin(A − B) = sinA cosB − cosA sinB.
Result 4: Double-Angle Formulas

Statement. sin 2A = 2 sin A cos A, and cos 2A = cos²A − sin²A = 1 − 2 sin²A.

Proof

Set B = A in the sum formulas just proved.

sin 2A = sin(A + A) = sinA cosA + cosA sinA put B = A in sin(A + B)
  = 2 sinA cosA combine
cos 2A = cos(A + A) = cosA cosA − sinA sinA put B = A in cos(A + B)
  = cos²A − sin²A simplify
  = 1 − 2 sin²A use cos²A = 1 − sin²A

The third form cos 2A = 2 cos²A − 1 comes the same way using sin²A = 1 − cos²A. Setting B = 2A then yields the triple-angle formulas.
5

Worked Examples
Example 1

Question: Convert 150° into radians.

▶ Show full working

To change degrees into radians we multiply by π/180, because 180° = π radians.

150° = 150 × π/180 multiply by π/180
  = 150π/180 carry out the multiplication
  = 5π/6 divide 150 and 180 by 30

Answer: 150° = 5π/6 radians.
Example 2

Question: Convert 3π/4 radians into degrees.

▶ Show full working

To change radians into degrees we multiply by 180/π so the π cancels.

3π/4 = (3π/4) × (180/π) multiply by 180/π
  = 3 × 180 / 4 the π cancels
  = 540/4 multiply the top
  = 135 divide

Answer: 3π/4 = 135°.
Example 3

Question: Find the length of an arc that subtends 60° at the centre of a circle of radius 21 cm. (Take π = 22/7.)

▶ Show full working

The formula is l = rθ, but θ must be in radians first.

θ = 60 × π/180 convert 60° to radians
  = π/3 simplify
l = rθ = 21 × π/3 apply l = rθ
  = 7π 21 ÷ 3 = 7
  = 7 × 22/7 put π = 22/7
  = 22 the 7 cancels

Answer: Arc length = 22 cm.
Example 4

Question: If sin θ = 3/5 and θ lies in Quadrant I, find cos θ and tan θ.

▶ Show full working

Use sin²θ + cos²θ = 1 to get cos θ, then build tan θ from sine and cosine.

cos²θ = 1 − sin²θ rearrange the identity
  = 1 − (3/5)² put sin θ = 3/5
  = 1 − 9/25 square it
  = 16/25 subtract
cos θ = +√(16/25) = 4/5 positive in Quadrant I
tan θ = sin θ / cos θ = (3/5)/(4/5) definition of tan
  = 3/4 the 5s cancel

Answer: cos θ = 4/5, tan θ = 3/4.
Example 5

Question: State the sign of cos θ and tan θ when θ lies in Quadrant III.

▶ Show full working

Use the A-S-T-C rule for which functions are positive in each quadrant.

In Quadrant III, the only positive functions are tangent and cot (the ‘T’ in A-S-T-C).
Cosine is not in that list, so cos θ is negative.
Tangent is in that list, so tan θ is positive.

Answer: cos θ < 0, tan θ > 0.
Example 6

Question: Find sin 75° using the sum formula.

▶ Show full working

Split 75° into 45° + 30° (two angles we know), then use sin(A + B) = sin A cos B + cos A sin B.

sin 75° = sin(45° + 30°) split into known angles
  = sin45°cos30° + cos45°sin30° apply the sum formula
  = (1/√2)(√3/2) + (1/√2)(1/2) substitute standard values
  = √3/(2√2) + 1/(2√2) multiply each term
  = (√3 + 1)/(2√2) add over the common denominator

Answer: sin 75° = (√3 + 1)/(2√2).
Example 7

Question: Find cos 15° using the difference formula.

▶ Show full working

Split 15° = 45° − 30°, then use cos(A − B) = cos A cos B + sin A sin B.

cos 15° = cos(45° − 30°) split into known angles
  = cos45°cos30° + sin45°sin30° apply the formula (note the + sign)
  = (1/√2)(√3/2) + (1/√2)(1/2) substitute values
  = (√3 + 1)/(2√2) simplify

Answer: cos 15° = (√3 + 1)/(2√2).
Example 8

Question: If cos A = 4/5 with A in Quadrant I, find sin 2A.

▶ Show full working

First find sin A from the identity, then use sin 2A = 2 sin A cos A.

sin²A = 1 − cos²A = 1 − 16/25 from the identity
  = 9/25 subtract
sin A = 3/5 positive in Quadrant I
sin 2A = 2 sin A cos A double-angle formula
  = 2 × (3/5) × (4/5) substitute
  = 24/25 multiply

Answer: sin 2A = 24/25.
Example 9

Question: Prove that (1 + tan²θ) cos²θ = 1.

▶ Show full working

Replace 1 + tan²θ with sec²θ, then write sec in terms of cos.

(1 + tan²θ)cos²θ = sec²θ · cos²θ since 1 + tan²θ = sec²θ
  = (1/cos²θ) · cos²θ since sec θ = 1/cos θ
  = 1 cos²θ cancels

Answer: Proved, the expression equals 1.
Example 10

Question: Find tan(A + B) if tan A = 1/2 and tan B = 1/3.

▶ Show full working

Use tan(A + B) = (tan A + tan B)/(1 − tan A tan B).

tan(A+B) = (tan A + tan B)/(1 − tan A tan B) the formula
  = (1/2 + 1/3)/(1 − (1/2)(1/3)) substitute
  = (5/6)/(1 − 1/6) add the top
  = (5/6)/(5/6) simplify the bottom
  = 1 a number over itself

Answer: tan(A + B) = 1.
Example 11

Question: Express 2 sin 5x cos 3x as a sum.

▶ Show full working

Use the product-to-sum identity 2 sin A cos B = sin(A + B) + sin(A − B), with A = 5x and B = 3x.

2 sin5x cos3x = sin(5x + 3x) + sin(5x − 3x) apply the identity
  = sin 8x + sin 2x add and subtract the angles

Answer: 2 sin 5x cos 3x = sin 8x + sin 2x.
Example 12

Question: Find the value of cos 2A if sin A = 1/3.

▶ Show full working

Choose the form cos 2A = 1 − 2 sin²A, since we already know sin A.

cos 2A = 1 − 2 sin²A chosen form
  = 1 − 2(1/3)² substitute sin A = 1/3
  = 1 − 2/9 square and multiply
  = 7/9 9/9 − 2/9

Answer: cos 2A = 7/9.
6

Where You Meet This in Real Life

Sound and music

Every musical note is a sine wave of a fixed frequency. Mixing notes simply adds waves, the same sum and difference idea you study here.

Electricity at home

Alternating current rises and falls as a sine wave, which is why mains voltage is written in the form V = V₀ sin(ωt).

GPS and navigation

Satellites locate your phone by triangulation, solving triangles with the very sine and cosine rules trigonometry provides.

Architecture and engineering

Ramps, roofs and bridges are designed using angle measures and trigonometric ratios to balance loads and slopes safely.

Astronomy and tides

The rise and fall of ocean tides and the apparent motion of stars are modelled with periodic sine and cosine functions.
7

Practice Sets A-D

Practice Set A – Basics

A1. Convert 90° to radians.

▶ Reveal full working

Multiply by π/180.

90° = 90 × π/180 convert
  = π/2 90/180 = 1/2

Answer: π/2 radians.

A2. Convert π/6 radians to degrees.

▶ Reveal full working

Multiply by 180/π.

π/6 = (π/6) × (180/π) convert
  = 180/6 the π cancels
  = 30 divide

Answer: 30°.

A3. State the range of sin x and of cos x.

▶ Reveal full working

sin x and cos x are coordinates of a point on a circle of radius 1.

A point on the unit circle is never more than 1 unit from the centre.
So each coordinate stays between −1 and 1.

Answer: [−1, 1] for both.

A4. In which quadrant are all six trigonometric functions positive?

▶ Reveal full working

A function is positive when its coordinate is positive.

Both x (cos) and y (sin) are positive only in the top-right region.
That region is Quadrant I.

Answer: Quadrant I.

Practice Set B – Conceptual

B1. Why is tan x undefined at x = 90°?

▶ Reveal full working

tan x is a fraction: tan x = sin x / cos x.

tan 90° = sin 90° / cos 90° definition
  = 1 / 0 cos 90° = 0
Dividing by zero is not allowed, so tan 90° has no value.

Answer: Undefined, because cos 90° = 0.

B2. State the period of sin x and of tan x.

▶ Reveal full working

The period is how far you travel before the wave repeats.

sin x repeats after a full circle, so its period is 2π.
tan x repeats after half a circle, so its period is π.

Answer: sin x: 2π; tan x: π.

B3. If cos θ = −1/2 and θ is in Quadrant II, find sin θ.

▶ Reveal full working

Use sin²θ = 1 − cos²θ.

sin²θ = 1 − (−1/2)² substitute cos θ
  = 1 − 1/4 square
  = 3/4 subtract
sin θ = +√(3/4) = √3/2 positive in Quadrant II

Answer: sin θ = √3/2.

B4. Explain why sin²θ + cos²θ = 1 holds for every angle.

▶ Reveal full working

On the unit circle cos θ is the x-coordinate and sin θ the y-coordinate.

x² + y² = 1 equation of the unit circle
cos²θ + sin²θ = 1 put x = cos θ, y = sin θ

Answer: It is the unit-circle equation x² + y² = 1.

Practice Set C – Application / Numerical

C1. Find the arc length for θ = π/4 on a circle of radius 8 cm.

▶ Reveal full working

θ is already in radians, so use l = rθ directly.

l = rθ = 8 × π/4 apply the formula
  = 2π 8 ÷ 4 = 2

Answer: 2π cm ≈ 6.28 cm.

C2. Find sin 105° using a sum formula.

▶ Reveal full working

Split 105° = 60° + 45°.

sin 105° = sin(60° + 45°) split
  = sin60°cos45° + cos60°sin45° sum formula
  = (√3/2)(1/√2) + (1/2)(1/√2) substitute
  = (√3 + 1)/(2√2) combine

Answer: sin 105° = (√3 + 1)/(2√2).

C3. If tan A = 3/4 and A is acute, find tan 2A.

▶ Reveal full working

Use tan 2A = (2 tan A)/(1 − tan²A).

tan 2A = (2 tan A)/(1 − tan²A) formula
  = (2 × 3/4)/(1 − 9/16) substitute
  = (3/2)/(7/16) simplify top and bottom
  = (3/2) × (16/7) flip and multiply
  = 24/7 simplify

Answer: tan 2A = 24/7.

C4. Express cos 7x + cos 3x as a product.

▶ Reveal full working

Use cos C + cos D = 2 cos((C+D)/2) cos((C−D)/2).

cos 7x + cos 3x = 2 cos((7x+3x)/2) cos((7x−3x)/2) sum-to-product
  = 2 cos 5x cos 2x simplify the halves

Answer: 2 cos 5x cos 2x.

Practice Set D – HOTS / Proofs

D1. Prove that cos 3A = 4 cos³A − 3 cos A.

▶ Reveal full working

Write 3A = 2A + A and expand step by step.

cos 3A = cos(2A + A) rewrite
  = cos2A cosA − sin2A sinA cosine sum formula
  = (2cos²A − 1)cosA − (2 sinA cosA)sinA double-angle forms
  = 2cos³A − cosA − 2 sin²A cosA expand
  = 2cos³A − cosA − 2cosA(1 − cos²A) sin²A = 1 − cos²A
  = 2cos³A − cosA − 2cosA + 2cos³A expand the bracket
  = 4cos³A − 3cosA collect like terms

Answer: Proved.

D2. If sin A + sin B = 1/2 and cos A + cos B = 1, find tan((A+B)/2).

▶ Reveal full working

Turn each sum into a product, then divide one equation by the other.

sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2) = 1/2 sum-to-product
cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2) = 1 sum-to-product
tan((A+B)/2) = (1/2) / 1 divide; the 2 and cos((A−B)/2) cancel
  = 1/2 simplify

Answer: tan((A+B)/2) = 1/2.

D3. Prove that (sin 3x + sin x)/(cos 3x + cos x) = tan 2x.

▶ Reveal full working

Turn the top and bottom into products, then cancel.

sin 3x + sin x = 2 sin 2x cos x sum-to-product (top)
cos 3x + cos x = 2 cos 2x cos x sum-to-product (bottom)
(sin3x+sinx)/(cos3x+cosx) = (2 sin2x cosx)/(2 cos2x cosx) divide
  = sin 2x / cos 2x cancel 2 and cos x
  = tan 2x sine over cosine

Answer: Proved.

D4. Find the maximum and minimum values of 3 sin x + 4 cos x.

▶ Reveal full working

Any a sin x + b cos x swings between −√(a²+b²) and +√(a²+b²).

amplitude = √(a² + b²) = √(3² + 4²) with a = 3, b = 4
  = √(9 + 16) = √25 add
  = 5 square root
So the expression runs from −5 to +5.

Answer: Maximum 5, minimum −5.

Chapter Summary

Everything in One Glance

Angle Measure

Positive anticlockwise; π rad = 180°; convert with π/180 and 180/π.

  

Arc Length

For radius r and angle θ in radians, l = rθ.
 

Right Triangle

sin = opp/hyp, cos = adj/hyp, tan = opp/adj (SOH-CAH-TOA).

  

Unit Circle

cos θ = x, sin θ = y; gives sin²θ + cos²θ = 1.
 

Quadrant Signs

All, Sine, Tangent, Cosine positive in Quadrants I-IV in order.

  

Combining Angles

Sum, difference, double- and triple-angle formulas rewrite expressions.
 
8

Are You Exam-Ready?

8-Point Exam Quick-Check

1 Convert 225° to radians.
 
2 What is the arc length for θ = π/2 on a circle of radius 10 cm?
 
3 State the domain and range of cos x.
 
4 In which quadrant is sine positive but cosine negative?
 
5 Write the sum formula for cos(A + B).
 
6 Express sin 2A in terms of sin A and cos A.
 
7 Why does tan x have period π rather than 2π?
 
8 Find the maximum value of 5 sin x + 12 cos x.
 

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Class 11 Mathematics Chapter 3: Trigonometric Functions, Complete Notes and Practice

These free Class 11 Maths Chapter 3 Trigonometric Functions notes follow the latest NCERT 2026-27 syllabus and give complete, exam-ready coverage for board exams and competitive foundations. Includes worked examples, step-by-step proofs and graded practice, free on SchoolRevise.com.
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