Home › Free notes › Class 11 Mathematics › Trigonometric Functions

Trigonometric Functions – Class 11 Mathematics notes, important questions and MCQs

NCERT Class 11 Mathematics, chapter 3. Notes, key terms, important questions with answers and MCQs.

Chapter overview

Angles are measured in degrees or radians. Trigonometric functions extend to all real numbers using the unit circle, with sum, difference and multiple-angle identities.

Measure and signs

π radians = 180°; arc length l = rθ (θ in radians).
Signs: All positive in Q1, sin in Q2, tan in Q3, cos in Q4 (ASTC).
Periods: sin and cos 2π, tan π.

Identities

sin²x + cos²x = 1; 1 + tan²x = sec²x; 1 + cot²x = cosec²x.
sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.
sin 2x = 2 sin x cos x; cos 2x = cos²x − sin²x = 1 − 2sin²x; sin 3x = 3 sin x − 4 sin³x.

Key terms

Radian: angle subtended at the centre by an arc equal to the radius
Period: smallest positive value after which a function repeats
Allied angles: angles like (90° ± θ), (180° ± θ), (270° ± θ), (360° − θ) whose ratios are expressed using θ

Formulas

l = rθ
sin 2x = 2 sin x cos x
cos 2x = 1 − 2sin²x

Important questions with answers

Find the value of tan 15°.2 marks

tan(45° − 30°) = (1 − 1/√3)/(1 + 1/√3) = (√3 − 1)/(√3 + 1) = 2 − √3.

Prove that sin 3x = 3 sin x − 4 sin³x.3 marks

sin 3x = sin(2x + x) = sin 2x cos x + cos 2x sin x = 2 sin x cos²x + (1 − 2 sin²x) sin x = 2 sin x(1 − sin²x) + sin x − 2 sin³x = 3 sin x − 4 sin³x.

Prove that cos 4x = 1 − 8 sin²x cos²x, and find the general solution of sin 2x − sin 4x + sin 6x = 0.5 marks

cos 4x = 1 − 2 sin²2x = 1 − 2(2 sin x cos x)² = 1 − 8 sin²x cos²x. For the equation: sin 6x + sin 2x = 2 sin 4x cos 2x, so 2 sin 4x cos 2x − sin 4x = 0, sin 4x(2 cos 2x − 1) = 0. sin 4x = 0 gives x = nπ/4; cos 2x = 1/2 gives 2x = 2nπ ± π/3, x = nπ ± π/6.

Quick revision

π rad = 180°.
ASTC rule for signs.
tan has period π.

Exam tips

General solution: sin x = 0 ⇒ x = nπ; cos x = 0 ⇒ x = (2n + 1)π/2.
For (90° ± θ) or (270° ± θ) the ratio changes (sin ↔ cos, tan ↔ cot); for (180° ± θ) or (360° ± θ) it stays the same; the sign comes from the quadrant.

Practice MCQs

60° in radians is
1. π/3
2. π/6
3. π/4
4. 2π/3
Show answer
Answer: (a) π/3. 60 × π/180.
sin²θ + cos²θ equals
1. 0
2. 1
3. 2
4. tan θ
Show answer
Answer: (b) 1. Identity.
cos 75° equals
1. (√6 + √2)/4
2. (√6 − √2)/4
3. √3/2
4. 1/2
Show answer
Answer: (b) (√6 − √2)/4. cos(45° + 30°).
Sign of tan 210° is
1. undefined
2. negative
3. zero
4. positive
Show answer
Answer: (d) positive. Third quadrant.
sin 120° equals
1. √3/2
2. 1/2
3. −1/2
4. −√3/2
Show answer
Answer: (a) √3/2. sin(180° − 60°).
Arc length of a circle of radius 10 cm subtending 30° at the centre is
1. 3π cm
2. 10π cm
3. π/6 cm
4. 5π/3 cm
Show answer
Answer: (d) 5π/3 cm. 10 × π/6.
Maximum value of 3 sin x + 4 cos x is
1. 5
2. 7
3. 1
4. 12
Show answer
Answer: (a) 5. √(9 + 16).
cos 2x = 1/2 has general solution
1. x = nπ ± π/3
2. x = nπ ± π/6
3. x = 2nπ ± π/3
4. x = nπ + π/6
Show answer
Answer: (b) x = nπ ± π/6. 2x = 2nπ ± π/3.

All chapters of Class 11 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.