Class 11 Maths Chapter 3: Trigonometric Functions

Trigonometric Functions is Chapter 3 of NCERT Class 11 Maths. It extends the trigonometry you learnt in Class 10 from acute angles in a right triangle to angles of any size, measured in degrees or radians, and builds the identities that you will use all the way through Class 12 calculus. Watch the 12 video lessons below, then use these notes, formulas and solved examples to revise.

What you will learn

How to measure angles in degrees and radians and convert between them
The six trigonometric functions for any real number using the unit circle
Signs of the functions in the four quadrants, their domain, range and graphs
Sum and difference formulas for sin, cos and tan
Double angle, triple angle and product to sum formulas with their uses

Video Lessons

Trigonometric Functions - Part 1

Trigonometric Functions - Part 2

Trigonometric Functions - Part 3

Trigonometric Functions - Part 4

Trigonometric Functions - Part 5

Trigonometric Functions - Part 6

Trigonometric Functions - Part 7

Trigonometric Functions - Part 8

Trigonometric Functions - Part 9

Trigonometric Functions - Part 10

Trigonometric Functions - Part 11

Trigonometric Functions - Part 12

Key Concepts and Formulas

Measuring Angles

An angle is formed by rotating a ray about its initial point. Anticlockwise rotation gives a positive angle and clockwise rotation gives a negative angle. In this chapter radian measure becomes the standard unit because it links an angle directly to arc length.

360° = 2π radians, so π radians = 180°
Degree to radian: multiply by π/180
Radian to degree: multiply by 180/π
1 radian ≈ 57°16′ and 1° ≈ 0.01746 radian
1° = 60′ (minutes), 1′ = 60″ (seconds)
Arc length: l = rθ
where r = radius, θ = central angle in radians

Arc length l = rθ on a sector of radius 18 cm and angle 40°, Class 11 Trigonometric Functions — Converting 40° to radians gives an arc of 4π cm

Trigonometric Functions of a Real Number

Take a point P(a, b) on the unit circle x² + y² = 1 so that the arc from (1, 0) to P has length x. Then cos x = a and sin x = b. This definition works for every real number x, not just acute angles.

sin x = 0 when x = nπ
cos x = 0 when x = (2n + 1)π/2  (n any integer)
tan x = sin x / cos x,  cot x = cos x / sin x
sec x = 1 / cos x,  cosec x = 1 / sin x
sin²x + cos²x = 1
1 + tan²x = sec²x
1 + cot²x = cosec²x

Signs in the Four Quadrants

The sign of each function depends on the signs of the coordinates of P. A handy memory line is "All Students Take Coffee" for quadrants I, II, III and IV.

Unit circle with P(cos x, sin x) and ASTC signs in four quadrants for Class 11 Trigonometric Functions — Which functions are positive in each quadrant

Quadrant I: all six functions are positive
Quadrant II: only sin and cosec are positive
Quadrant III: only tan and cot are positive
Quadrant IV: only cos and sec are positive

Domain, Range and Periodicity

A function is periodic if its values repeat after a fixed interval. Knowing the period lets you reduce large angles to small ones quickly.

sin x, cos x: range [−1, 1], period 2π
tan x: range all reals, period π
sec x, cosec x: range (−∞, −1] ∪ [1, ∞)
sin(−x) = −sin x   (odd)
cos(−x) = cos x    (even)
tan(−x) = −tan x
sin(2nπ + x) = sin x,  cos(2nπ + x) = cos x

sin x and cos x are defined for all real numbers
tan x is not defined at odd multiples of π/2

Sum and Difference Formulas

These are the backbone of the chapter. Most identities and allied angle results can be derived from the first two.

cos(x + y) = cos x cos y − sin x sin y
cos(x − y) = cos x cos y + sin x sin y
sin(x + y) = sin x cos y + cos x sin y
sin(x − y) = sin x cos y − cos x sin y
tan(x + y) = (tan x + tan y)/(1 − tan x tan y)
tan(x − y) = (tan x − tan y)/(1 + tan x tan y)
Allied angles:
cos(π/2 − x) = sin x
sin(π − x) = sin x,  cos(π − x) = −cos x
sin(π + x) = −sin x

Double and Triple Angle Formulas

Putting y = x in the sum formulas gives the double angle results. They are used to simplify expressions and prove identities.

sin 2x = 2 sin x cos x
sin 2x = 2 tan x / (1 + tan²x)
cos 2x = cos²x − sin²x
cos 2x = 2cos²x − 1 = 1 − 2sin²x
cos 2x = (1 − tan²x)/(1 + tan²x)
tan 2x = 2 tan x / (1 − tan²x)
sin 3x = 3 sin x − 4 sin³x
cos 3x = 4 cos³x − 3 cos x
tan 3x = (3 tan x − tan³x)/(1 − 3 tan²x)

Product to Sum and Sum to Product

These formulas convert sums of sines and cosines into products and back, which is very useful in proofs.

cos x + cos y = 2 cos((x+y)/2) cos((x−y)/2)
cos x − cos y = −2 sin((x+y)/2) sin((x−y)/2)
sin x + sin y = 2 sin((x+y)/2) cos((x−y)/2)
sin x − sin y = 2 cos((x+y)/2) sin((x−y)/2)
2 sin x cos y = sin(x + y) + sin(x − y)
2 cos x cos y = cos(x + y) + cos(x − y)

Solved Examples

Example 1: Convert 75° into radians.

75° = 75 × π/180
    = 75π/180
∴ 75° = 5π/12 radians

Example 2: A circle has radius 18 cm. Find the length of the arc that subtends 40° at the centre.

θ = 40 × π/180 = 2π/9
l = rθ = 18 × 2π/9
∴ l = 4π cm ≈ 12.57 cm

Example 3: If cos x = −8/17 and x lies in the third quadrant, find the other trigonometric functions.

sin²x = 1 − 64/289 = 225/289
sin x < 0 in Q III, so sin x = −15/17
tan x = sin x / cos x = 15/8
cot x = 8/15
sec x = −17/8
∴ cosec x = −17/15

Example 4: Find the value of cos 1110°.

1110° = 3 × 360° + 30°
cos 1110° = cos 30°
∴ cos 1110° = √3/2

Example 5: Find the exact value of cos 105°.

cos 105° = cos(60° + 45°)
= cos 60° cos 45° − sin 60° sin 45°
= (1/2)(1/√2) − (√3/2)(1/√2)
∴ cos 105° = (1 − √3)/(2√2)

Example 6: If tan x = 3/4 and x is acute, find sin 2x and cos 2x.

sin 2x = 2 tan x / (1 + tan²x)
= (3/2) / (1 + 9/16)
= (3/2) × (16/25) = 24/25
cos 2x = (1 − tan²x)/(1 + tan²x)
= (7/16) / (25/16)
∴ sin 2x = 24/25, cos 2x = 7/25

Example 7: Evaluate sin 75° − sin 15°.

sin x − sin y = 2 cos((x+y)/2) sin((x−y)/2)
= 2 cos 45° sin 30°
= 2 × (1/√2) × (1/2)
∴ sin 75° − sin 15° = 1/√2

Common Mistakes to Avoid

Mixing degrees and radians in the same step. In l = rθ, the angle must be in radians.
Forgetting the quadrant while taking a square root. Always decide the sign of sin, cos or tan from the quadrant before writing the answer.
Writing sin(x + y) = sin x + sin y. Trigonometric functions are not linear, so always use the proper sum formula.
Using tan x or sec x at odd multiples of π/2, where they are not defined.

Quick Revision

π radians = 180°,  l = rθ (θ in radians)
sin²x + cos²x = 1
1 + tan²x = sec²x,  1 + cot²x = cosec²x
sin 2x = 2 sin x cos x
cos 2x = 1 − 2sin²x
tan 2x = 2 tan x / (1 − tan²x)
sin 3x = 3 sin x − 4 sin³x
cos 3x = 4 cos³x − 3 cos x

Positive functions: All (I), sin (II), tan (III), cos (IV)
Periods: sin and cos 2π, tan π