Class 11 Maths Notes

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Trigonometric Functions Class 11 Notes

Complete, exam-ready notes on trigonometric functions: angle systems and degree–radian conversion, arc length, the six trigonometric ratios and their quadrant signs, identities, compound and double-angle formulas, trigonometric equations with general solutions, and the sine and cosine rules — written for CBSE, JEE and NEET revision.

Class:11Subject:MathematicsCovers:CBSE · JEE

7 Key Formulas4 Practice MCQs

DWritten byDeep Narayan

UpdatedSeptember 2026

Key Concept Summary

What is this chapter about in one line?

Trigonometric functions relate angles to ratios of sides, and this chapter builds the identities, compound-angle and double-angle formulas needed to solve equations and triangles.

Angles and Degree–Radian Conversion

Definition

Degree and radian

One radian is the angle subtended at the centre by an arc equal in length to the radius. A full revolution is 360° = 2π radians, so 180° = π radians. To convert, multiply degrees by π/180 or multiply radians by 180/π.

Degree–radian conversion

1^{\circ} = \frac{π}{180} rad, 1 rad = \frac{18 0 ^{\circ}}{π}

Definition

Arc length

l = r θ

The length l of an arc of a circle of radius r is l = rθ, where θ is the subtended angle measured in radians (not degrees).

The Six Ratios and Quadrant Signs

Definition

Six trigonometric ratios

For an angle θ, the six ratios are sin, cos, tan, cosec (1/sin), sec (1/cos) and cot (1/tan). Their signs depend on the quadrant of the terminal side.

Quadrant I: all six ratios are positive.
Quadrant II: only sin and cosec positive.
Quadrant III: only tan and cot positive.
Quadrant IV: only cos and sec positive.
Mnemonic: ASTC — All, Sin, Tan, Cos (anti-clockwise from Quadrant I).

Basic Trigonometric Identities

Definition

Pythagorean identities

These follow from the unit circle and are true for every angle θ. Dividing sin² + cos² = 1 by cos² or sin² gives the other two.

Pythagorean identities

sin^{2} θ + cos^{2} θ = 1, 1 + tan^{2} θ = sec^{2} θ, 1 + cot^{2} θ = csc^{2} θ

Compound, Double and Half-Angle Formulas

Definition

Compound angles

Formulas for sin, cos and tan of A ± B expand a single angle into two. Setting A = B gives the double-angle identities, from which half-angle and power-reduction formulas follow.

Sine of a sum/difference

sin (A \pm B) = sin A cos B \pm cos A sin B

Cosine of a sum/difference

cos (A \pm B) = cos A cos B \mp sin A sin B

Tangent of a sum/difference

tan (A \pm B) = \frac{tan A \pm tan B}{1 \mp tan A tan B}

Sum-to-Product and Product-to-Sum

Definition

Product-to-sum

Products of sines and cosines convert into sums of sines or cosines. These are used to integrate and to simplify expressions.

Product-to-sum

2 sin A cos B = sin (A + B) + sin (A - B)

Double angle (sine)

sin 2 A = 2 sin A cos A

Trigonometric Equations and General Solutions

Definition

General solution

Sine, cosine and tangent repeat every period, so an equation like sinθ = sinα has infinitely many solutions captured by a periodic formula.

General solution for sine

sin θ = sin α ⟹ θ = nπ + (- 1)^{n} α, n \in Z

Cosine and tangent general solutions

For cosθ = cosα, θ = 2nπ ± α. For tanθ = tanα, θ = nπ + α, for n an integer.

Solved Examples

Example: Convert 120° to radians and 2.5 radians to degrees.

Solution: 120° × π/180 = 2π/3 radians. 2.5 × 180/π ≈ 143.2°.

Example: Solve sinθ = 1/2 for the general solution.

Solution: sinθ = sin(π/6), so θ = nπ + (−1)ⁿ·(π/6) for n ∈ ℤ.

NCERT Exercise Questions

From the textbook

All NCERT questions for this chapter — NCERT Maths Part I, Chapter 3 (Trigonometric Functions). Questions follow the textbook order so you can match them to the book.

Exercises (NCERT Q1–Q51)

1) Find the radian measures corresponding to the following degree measures: (i) 25°, (ii) −47°30′, (iii) 240°, (iv) 520°.
2) Find the degree measures corresponding to the following radian measures: (i) 11/16, (ii) −4, (iii) 5π/3, (iv) 7π/6. Use π = 22/7.
3) A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?
4) Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm. Use π = 22/7.
5) In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of the minor arc of the chord.
6) If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.
7) Find the angle in radians through which a pendulum swings if its length is 75 cm and the tip describes an arc of length (i) 10 cm, (ii) 15 cm, (iii) 21 cm.
8) Find the values of the other five trigonometric functions if cos x = −1/2 and x lies in the third quadrant.
9) Find the values of the other five trigonometric functions if sin x = 3/5 and x lies in the second quadrant.
10) Find the values of the other five trigonometric functions if cot x = 3/4 and x lies in the third quadrant.
11) Find the values of the other five trigonometric functions if sec x = 13/5 and x lies in the fourth quadrant.
12) Find the values of the other five trigonometric functions if tan x = −5/12 and x lies in the second quadrant.
13) Find the value of the trigonometric function sin 765°.
14) Find the value of the trigonometric function cos (−1710°).
15) Find the value of the trigonometric function cosec (−1410°).
16) Find the value of the trigonometric function tan (19π/3).
17) Find the value of the trigonometric function cot (−15π/4).
18) Prove that sin²(π/6) + cos²(π/3) − tan²(π/4) = −1/2.
19) Prove that 2 sin²(π/6) + cosec²(7π/6) cos²(π/3) = 3/2.
20) Prove that cot²(π/6) + cosec(5π/6) + 3 tan²(π/6) = 6.
21) Prove that 2 sin²(3π/4) + 2 cos²(π/4) + 2 sec²(π/3) = 10.
22) Find the value of sin 75°.
23) Prove that cos(π/4 − x) cos(π/4 − y) − sin(π/4 − x) sin(π/4 − y) = sin(x + y).
24) Prove that tan(π/4 + x) / tan(π/4 − x) = ((1 + tan x)/(1 − tan x))².
25) Prove that cos(π + x) cos(−x) / [sin(π − x) cos(π/2 + x)] = cot² x.
26) Prove that cos(3π/2 + x) cos(2π + x) [cot(3π/2 − x) + cot(2π + x)] = 1.
27) Prove that sin(n + 1)x sin(n + 2)x + cos(n + 1)x cos(n + 2)x = cos x.
28) Prove that cos(3π/4 + x) − cos(3π/4 − x) = −√2 sin x.
29) Prove that sin² 6x − sin² 4x = sin 2x sin 10x.
30) Prove that cos² 2x − cos² 6x = sin 4x sin 8x.
31) Prove that sin 2x + 2 sin 4x + sin 6x = 4 cos² x sin 4x.
32) Prove that cot 4x (sin 5x + sin 3x) = cot x (sin 5x − sin 3x).
33) Prove that (cos 9x − cos 5x)/(sin 17x − sin 3x) = −sin 2x / cos 10x.
34) Prove that (sin 5x + sin 3x)/(cos 5x + cos 3x) = tan 4x.
35) Prove that (sin x − sin y)/(cos x + cos y) = tan((x − y)/2).
36) Prove that (sin x + sin 3x)/(cos x + cos 3x) = tan 2x.
37) Prove that (sin x − sin 3x)/(sin²x − cos²x) = 2 sin x.
38) Prove that (cos 4x + cos 3x + cos 2x)/(sin 4x + sin 3x + sin 2x) = cot 3x.
39) Prove that cot x cot 2x − cot 2x cot 3x − cot 3x cot x = 1.
40) Prove that tan 4x = 4 tan x (1 − tan²x) / (1 − 6 tan²x + tan⁴x).
41) Prove that cos 4x = 1 − 8 sin²x cos²x.
42) Prove that cos 6x = 32 cos⁶x − 48 cos⁴x + 18 cos²x − 1.
43) Find the principal and general solutions of the equation tan x = √3.
44) Find the principal and general solutions of the equation sec x = 2.
45) Find the principal and general solutions of the equation cot x = −√3.
46) Find the principal and general solutions of the equation cosec x = −2.
47) Find the general solution of cos 4x = cos 2x.
48) Find the general solution of cos 3x + cos x − cos 2x = 0.
49) Find the general solution of sin 2x + cos x = 0.
50) Find the general solution of sec² 2x = 1 − tan 2x.
51) Find the general solution of sin x + sin 3x + sin 5x = 0.

Answers at a glance

1
Q1 → (i) 5π/36 (ii) −19π/72 (iii) 4π/3 (iv) 26π/9 radians.
2
Q2 → (i) 39°22′30″ (ii) −229°5′27″ (iii) 300° (iv) 210°.
3
Q3 → 12π radians per second.
4
Q4 → 12°36′.
5
Q5 → 20π/3 cm.
6
Q6 → r₁ : r₂ = 5 : 4.
7
Q7 → (i) 2/15 (ii) 1/5 (iii) 7/25 radians.
8
Q8 → sin x = −√3/2, tan x = √3, cosec x = −2/√3, sec x = −2, cot x = 1/√3.
9
Q9 → cos x = −4/5, tan x = −3/4, cosec x = 5/3, sec x = −5/4, cot x = −4/3.
10
Q10 → tan x = 4/3, sin x = −4/5, cos x = −3/5, cosec x = −5/4, sec x = −5/3.
11
Q11 → cos x = 5/13, sin x = −12/13, cosec x = −13/12, tan x = −12/5, cot x = −5/12.
12
Q12 → cot x = −12/5, sin x = 5/13, cos x = −12/13, cosec x = 13/5, sec x = −13/12.
13
Q13 → √2/2.
14
Q14 → 0.
15
Q15 → 2.
16
Q16 → √3.
17
Q17 → 1.
18
Q18 → Proved: LHS = −1/2.
19
Q19 → Proved: LHS = 3/2.
20
Q20 → Proved: LHS = 6.
21
Q21 → Proved: LHS = 10.
22
Q22 → (√6 + √2)/4.
23
Q23 → Proved.
24
Q24 → Proved.
25
Q25 → Proved.
26
Q26 → Proved.
27
Q27 → Proved.
28
Q28 → Proved.
29
Q29 → Proved.
30
Q30 → Proved.
31
Q31 → Proved.
32
Q32 → Proved — both sides equal 2 cos 4x cos x.
33
Q33 → Proved.
34
Q34 → Proved.
35
Q35 → Proved.
36
Q36 → Proved.
37
Q37 → Proved.
38
Q38 → Proved.
39
Q39 → Proved.
40
Q40 → Proved.
41
Q41 → Proved.
42
Q42 → Proved.
43
Q43 → Principal: x = π/3, 4π/3. General: x = nπ + π/3, n ∈ Z.
44
Q44 → Principal: π/3, 5π/3. General: x = 2nπ ± π/3.
45
Q45 → Principal: 5π/6, 11π/6. General: x = nπ − π/6.
46
Q46 → Principal: 7π/6, 11π/6. General: x = nπ + (−1)ⁿ(7π/6).
47
Q47 → x = nπ/3, n ∈ Z.
48
Q48 → x = (2n+1)π/4 or x = 2nπ ± π/3, n ∈ Z.
49
Q49 → x = (2n+1)π/2 or x = nπ + (−1)ⁿ(7π/6), n ∈ Z.
50
Q50 → x = nπ/2 or x = nπ/2 − π/8, n ∈ Z.
51
Q51 → x = nπ/3 or x = nπ ± π/3, n ∈ Z.

CBSE Pattern Questions

How CBSE asks this chapter

CBSE Class 11 Mathematics board paper: Section A (20 MCQs × 1M), Section B (5 × 2M), Section C (6 × 3M) and Section D (4 × 4M). Trigonometric Functions gives 1-mark MCQs on radians and standard values, 2/3-mark questions proving identities and solving simple trig equations, and 4-mark problems on sum/difference and product-sum formulas.

MCQ Questions

The radian measure of 180° is

sin(π/6) equals

The identity sin²x + cos²x for all x equals

The period of the function cos x is

1 Mark Questions

1
Convert 45° to radians. (Ans: π/4)
2
Find the value of tan(π/4). (Ans: 1)
3
Write the general solution of sin x = 0. (Ans: x = nπ, n ∈ Z)
4
If tan x = 3/4 with x in the first quadrant, find sin x. (Ans: 3/5)

2 Mark Questions

1
Find the value of cos 15° using cos(45° − 30°). (Ans: (√6 + √2)/4)
2
A wheel makes 360 revolutions in one minute; find its angle turned per second. (Ans: 12π rad/s)
3
Prove that sin(π + x) = −sin x using the addition formula.

3 Mark Questions

1
Solve 2 sin x = 1 in the interval [0, 2π]. (Ans: x = π/6, 5π/6)
2
Prove that cos(x − y) = cos x cos y + sin x sin y using the formula for cos(x + y). State the formula first.
3
If sin x = 3/5 and x lies in the second quadrant, find the values of all other trigonometric functions. (Ans: cos x = −4/5, tan x = −3/4, cosec x = 5/3, sec x = −5/4, cot x = −4/3)

4 Mark Questions

1
Prove the product-to-sum identity 2 sin A cos B = sin(A + B) + sin(A − B) and use it to find the value of 2 sin 45° cos 15°. (Ans: sin 60° + sin 30° = √3/2 + 1/2)
2
Prove that sin⁴x + cos⁴x = 1 − 2 sin²x cos²x, then show that sin⁴(π/8) + cos⁴(π/8) = 3/4.

PDF: Assignments & Tests

Printable study material

Free downloadable PDFs for Trigonometric Functions — two topic-wise assignments covering every concept of the chapter, and two full CBSE-pattern tests (each 90 minutes, 40 marks) mirroring the board paper. No signup required.

Free PDF Downloads & Assignments

No Signup Required

Practice path: complete Assignment 1 first (topic-by-topic), then Assignment 2 (mixed difficulty with full solutions), then attempt the timed tests.

Trigonometric Functions Assignment 1 — Topic-wise Questions

Covers all six topics: angles and degree–radian conversion; the six ratios and quadrant signs; basic trigonometric identities; compound, double and half-angle formulas; sum-to-product and product-to-sum; and trigonometric equations and general solutions.

trigonometric-functions-class-11-notes-assignment-1.pdf

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Trigonometric Functions Assignment 2 — Challenge + Integrated Problems

Higher-difficulty set with multi-step problems (identity proofs, compound-angle evaluations, sum-to-product transformations, general solutions) and full worked answers. Includes an answer key.

trigonometric-functions-class-11-notes-assignment-2.pdf

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Trigonometric Functions Test 1 — CBSE Pattern (90 min, 40 marks)

Full board-pattern test: Section A (MCQ/VSA 1M × 10), Section B (2M × 4), Section C (3M × 4), Section D (5M × 2) = 40 marks. Time: 90 minutes. Includes answer key.

trigonometric-functions-class-11-notes-test-1-cbse.pdf

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Trigonometric Functions Test 2 — CBSE Pattern (90 min, 40 marks)

Second full paper with different questions, same 40-mark / 90-minute blueprint. Includes answer key.

trigonometric-functions-class-11-notes-test-2-cbse.pdf

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Quick Revision

Key formulas at a glance

Memorise these equations — direct application numericals and derivations in CBSE & JEE frequently hinge on these.

Arc length

l = r θ

Sine of sum/difference

sin (A \pm B) = sin A cos B \pm cos A sin B

Cosine of sum/difference

cos (A \pm B) = cos A cos B \mp sin A sin B

Tangent of sum/difference

tan (A \pm B) = \frac{tan A \pm tan B}{1 \mp tan A tan B}

Product-to-sum

2 sin A cos B = sin (A + B) + sin (A - B)

Sine general solution

sin θ = sin α ⟹ θ = nπ + (- 1)^{n} α

Sine rule

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

Exam Strategy

How this chapter is asked

High-yield question patterns observed across CBSE boards, JEE Main & Advanced, and NEET.

180° = π radians; multiply by π/180 to convert degrees to radians.
Arc length l = rθ needs θ in radians.
ASTC gives quadrant signs: All, Sin, Tan, Cos.
sin²θ + cos²θ = 1 is the base of the other two identities.
Double angle: sin2A = 2sinAcosA.
General solution of sinθ = sinα is θ = nπ + (−1)ⁿα.
Sine rule links sides and angles of any triangle.

FAQ

Frequently asked questions

How do you convert degrees to radians?

Multiply the angle in degrees by π/180. Since 180° = π radians, 120° becomes 2π/3 radians and a full circle is 2π radians.

What are the compound-angle formulas?

They expand sin(A±B), cos(A±B) and tan(A±B) into single-angle terms, for example sin(A+B) = sinAcosB + cosAsinB. Setting A = B gives the double-angle formulas.

What is the general solution of sinθ = sinα?

θ = nπ + (−1)ⁿα for any integer n. Since sine repeats every 2π and is symmetrical about π/2, this single formula captures all solutions.

What are the sine and cosine rules?

The sine rule a/sinA = b/sinB = c/sinC relates sides to opposite angles. The cosine rule, c² = a² + b² − 2ab·cosC, finds a side or angle in any triangle.

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