Trigonometric Functions – Class 11 Maths Chapter 3 Notes & NCERT PDF

Class 11MathematicsChapter 3NCERT book: Mathematics - Class 11

Notes and a simple summary of Chapter 3 of the NCERT Class 11 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 3: Trigonometric Functions

Introduction

In Class 10, trigonometric ratios were defined for acute angles of a right triangle. This chapter extends them to any angle, measured in degrees or radians, by defining them through the unit circle. It then develops the sum and difference formulae that let us find exact values such as sin 15° and simplify complicated expressions.

Key Concepts

3.2 Angles

An angle is the amount of rotation of a ray about its initial point; anticlockwise rotation gives a positive angle, clockwise a negative one.

Degree measure: one complete rotation = 360°; 1° = 60′ (minutes), 1′ = 60″ (seconds).

Radian measure: the angle subtended at the centre of a unit circle by an arc of length 1.

In a circle of radius r, an arc of length l subtends an angle θ = l/r radians, i.e. l = rθ.

One complete rotation = 2π radians, so π radians = 180°.

- Radian measure = (π/180) × degree measure

- Degree measure = (180/π) × radian measure

- 1 radian ≈ 57°16′; 1° ≈ 0.01746 radian

3.3 Trigonometric Functions

For an angle x, let the point where its terminal side meets the unit circle be P(a, b). Then cos x = a and sin x = b; since a² + b² = 1, cos² x + sin² x = 1.

The other functions: tan x = sin x / cos x (cos x ≠ 0), cot x = cos x / sin x (sin x ≠ 0), sec x = 1/cos x, cosec x = 1/sin x.

Identities: 1 + tan² x = sec² x and 1 + cot² x = cosec² x.

Periodicity: sin (2nπ + x) = sin x and cos (2nπ + x) = cos x; tan and cot repeat after π.

sin x = 0 when x = nπ; cos x = 0 when x is an odd multiple of π/2.

3.3.1 Sign of trigonometric functions — in quadrant I all are positive; in II only sin and cosec; in III only tan and cot; in IV only cos and sec. Also sin (–x) = –sin x and cos (–x) = cos x.

3.3.2 Domain and range — sin and cos have domain R and range [–1, 1]; tan has domain R – {(2n + 1)π/2} and range R; cosec and sec have range R – (–1, 1).

3.4 Trigonometric Functions of Sum and Difference of Two Angles

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), when the angles are not odd multiples of π/2

Allied angles: cos (π/2 – x) = sin x, sin (π/2 – x) = cos x, cos (π – x) = –cos x, sin (π – x) = sin x, cos (π + x) = –cos x, sin (π + x) = –sin x

Multiple angles:

- sin 2x = 2 sin x cos x = 2 tan x/(1 + tan² x)

- cos 2x = cos² x – sin² x = 2cos² x – 1 = 1 – 2sin² x = (1 – tan² x)/(1 + tan² x)

- tan 2x = 2 tan x/(1 – tan² x)

- sin 3x = 3 sin x – 4 sin³ x; cos 3x = 4 cos³ x – 3 cos x; tan 3x = (3 tan x – tan³ x)/(1 – 3 tan² x)

Sum-to-product: 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)

Product-to-sum: 2 cos x cos y = cos (x + y) + cos (x – y); –2 sin x sin y = cos (x + y) – cos (x – y); 2 sin x cos y = sin (x + y) + sin (x – y); 2 cos x sin y = sin (x + y) – sin (x – y)

Worked Examples

Example 1 (degree → radian). 40°20′ = 40⅓° = (121/3) × π/180 = 121π/540 radians.

Example 2 (radian → degree, π = 22/7). 6 radians = 6 × 180 × 7/22 = 343 7/11 degrees = 343°38′11″ approximately.

Example 3 (l = rθ). A central angle of 60° intercepts an arc of 37.4 cm. θ = π/3, so r = l/θ = 37.4 × 3 × 7/22 = 35.7 cm.

Example 4 (reducing large angles). sin 765° = sin (2 × 360° + 45°) = sin 45° = 1/√2. cosec (–1410°) = cosec (–1410° + 4 × 360°) = cosec 30° = 2.

Example 5 (sin 15°). sin 15° = sin (45° – 30°) = sin 45° cos 30° – cos 45° sin 30° = (√6 – √2)/4.

Example 6 (quadrant signs). If sin x = 3/5 and x lies in the second quadrant, then cos x = –√(1 – 9/25) = –4/5 (cos is negative in II) and tan x = –3/4.

Formula Sheet

l = rθ; π rad = 180°

sin² x + cos² x = 1; 1 + tan² x = sec² x; 1 + cot² x = cosec² x

sin (x ± y), cos (x ± y), tan (x ± y) as above

sin 2x, cos 2x (four forms), tan 2x, sin 3x, cos 3x, tan 3x

Sum-to-product and product-to-sum formulae

Exam Focus

Exact values: write the angle as a sum or difference of standard angles (15° = 45° – 30°, 75° = 45° + 30°, 13π/12 = π + π/12; tan 13π/12 = 2 – √3).

Proving identities: start from the more complicated side and use sum-to-product or double-angle formulae; show each step.

Given one ratio and a quadrant: find the others with the Pythagorean identity, then fix signs by the quadrant.

Key Terms

Angle (कोण), Degree (अंश), Radian (रेडियन)

Arc (चाप), Unit circle (इकाई वृत्त)

Trigonometric function (त्रिकोणमितीय फलन)

Sine (ज्या), Cosine (कोज्या), Tangent (स्पर्शज्या)

Quadrant (चतुर्थांश), Period (आवर्त)

Domain (प्रांत), Range (परिसर)

Identity (सर्वसमिका), Allied angles (संबद्ध कोण)

Common Mistakes

Forgetting to convert degrees to radians before using l = rθ.

Wrong signs from ignoring the quadrant.

Writing sin (x + y) = sin x + sin y.

Mixing up the forms of cos 2x or the signs in cos (x ± y).

Assuming tan x is defined everywhere — it is undefined at odd multiples of π/2.

💡 Key Learning Points

✓Measure angles in degrees and radians and convert between them using π rad = 180°
✓Use l = rθ to relate arc length, radius and angle
✓Define trigonometric functions of any real number using the unit circle and know their signs, domains and ranges
✓Apply sum, difference, multiple-angle, sum-to-product and product-to-sum formulae
✓Find exact values of non-standard angles and prove trigonometric identities

👨‍🏫 Teaching Tips

→Demonstrate a radian with a string equal to the radius laid along a circle's edge
→Draw the unit circle and mark the coordinates (cos x, sin x) for standard angles
→Teach the quadrant signs with a four-part diagram and plenty of quick questions
→Derive cos (x + y) once in class so students see where the other formulae come from
→Practise reducing large angles by subtracting multiples of 360° or 2π
→Build a formula sheet with the class and colour-code sum, double-angle and product formulae
→Set short daily identity-proving drills to build fluency

📋 Assessment Questions

Can the student convert between degree and radian measure accurately?
Can the student use l = rθ in word problems?
Can the student find values of trigonometric functions of any angle with correct signs?
Can the student find exact values such as sin 15° or tan 13π/12?
Can the student prove identities using sum and product formulae?

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