Class 11 Mathematics Notes Chapter 3 (Trigonometric functions) – Mathematics Book

Alright class, let's begin our focused session on Chapter 3: Trigonometric Functions. This chapter is fundamental not just for your Class 11 curriculum but forms the bedrock for many topics in calculus and is frequently tested in various government examinations. Pay close attention to the concepts and formulas.

Chapter 3: Trigonometric Functions - Detailed Notes for Government Exam Preparation

1. Introduction: Angles

• Angle: A measure of rotation of a given ray about its initial point. The original ray is the initial side, and the final position is the terminal side. The point of rotation is the vertex.
• Positive Angle: Measured counterclockwise from the initial side.
• Negative Angle: Measured clockwise from the initial side.
• Units of Measurement:
  • Degree: If a rotation from the initial side to the terminal side is (1/360)th of a revolution, the angle is said to have a measure of 1 degree (1°).
    • 1° = 60 minutes (60')
    • 1' = 60 seconds (60")
  • Radian: The angle subtended at the centre of a circle by an arc equal in length to the radius of the circle. It's a constant value.
    • If 'l' is the length of the arc and 'r' is the radius, the angle θ (in radians) is given by θ = l/r.
    • A complete circle subtends 2π radians at the centre.
• Relation between Degree and Radian:
  • 2π radians = 360°
  • π radians = 180° (This is the most important conversion factor)
  • To convert degrees to radians: Multiply by π/180.
    • Example: 60° = 60 × (π/180) = π/3 radians.
  • To convert radians to degrees: Multiply by 180/π.
    • Example: π/6 radians = (π/6) × (180/π) = 30°.
  • Approximate value: 1 radian ≈ 57° 16'

2. Trigonometric Functions using the Unit Circle

• Unit Circle: A circle with radius 1 unit (r=1) centred at the origin (0,0) of the Cartesian coordinate system.
• Let P(a, b) be any point on the unit circle such that the angle made by OP with the positive x-axis is x radians.
• Then, the coordinates of P are defined as: a = cos x and b = sin x.
• Therefore, for any point P(cos x, sin x) on the unit circle:
  • cos x: x-coordinate of the point P.
  • sin x: y-coordinate of the point P.
• Other Trigonometric Functions:
  • tan x = sin x / cos x (provided cos x ≠ 0)
  • cosec x = 1 / sin x (provided sin x ≠ 0)
  • sec x = 1 / cos x (provided cos x ≠ 0)
  • cot x = cos x / sin x = 1 / tan x (provided sin x ≠ 0)
• Fundamental Identity: Since P(a, b) lies on the unit circle x² + y² = 1, we have cos²x + sin²x = 1. This holds true for all values of x.

3. Signs of Trigonometric Functions in Quadrants (ASTC Rule)

• Quadrant I (0 < x < π/2): All T-functions are positive. (A)
• Quadrant II (π/2 < x < π): sin x and cosec x are positive. (S)
• Quadrant III (π < x < 3π/2): tan x and cot x are positive. (T)
• Quadrant IV (3π/2 < x < 2π): cos x and sec x are positive. (C)
• Mnemonic: All Silver Tea Cups or Add Sugar To Coffee.

4. Domain and Range of Trigonometric Functions

5. Periodicity of Trigonometric Functions

• A function f(x) is periodic if f(x + T) = f(x) for some positive constant T (the period).
• sin(x + 2π) = sin x, cos(x + 2π) = cos x, cosec(x + 2π) = cosec x, sec(x + 2π) = sec x. Period is 2π.
• tan(x + π) = tan x, cot(x + π) = cot x. Period is π.

6. Basic Trigonometric Identities

• sin²x + cos²x = 1
• 1 + tan²x = sec²x
• 1 + cot²x = cosec²x

7. Trigonometric Functions of Allied Angles (Negative, Complementary, Supplementary etc.)

• sin(-x) = -sin x
• cos(-x) = cos x
• tan(-x) = -tan x
• sin(π/2 - x) = cos x
• cos(π/2 - x) = sin x
• tan(π/2 - x) = cot x
• sin(π/2 + x) = cos x
• cos(π/2 + x) = -sin x
• tan(π/2 + x) = -cot x
• sin(π - x) = sin x
• cos(π - x) = -cos x
• tan(π - x) = -tan x
• sin(π + x) = -sin x
• cos(π + x) = -cos x
• tan(π + x) = tan x
• sin(2π - x) = -sin x
• cos(2π - x) = cos x
• tan(2π - x) = -tan x

(Remember the ASTC rule and the function changes (sin↔cos, tan↔cot, sec↔cosec) for π/2 ± x and 3π/2 ± x)

8. Sum and Difference Formulas (Compound Angles)

• cos(A + B) = cos A cos B - sin A sin B
• cos(A - B) = cos A cos B + sin A sin B
• sin(A + B) = sin A cos B + cos A sin B
• sin(A - B) = sin A cos B - cos A sin B
• tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
• tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
• cot(A + B) = (cot A cot B - 1) / (cot B + cot A)
• cot(A - B) = (cot A cot B + 1) / (cot B - cot A)

9. Multiple Angle Formulas

• Double Angles:
  • sin(2A) = 2 sin A cos A = (2 tan A) / (1 + tan²A)
  • cos(2A) = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A = (1 - tan²A) / (1 + tan²A)
  • tan(2A) = (2 tan A) / (1 - tan²A)
• Triple Angles:
  • sin(3A) = 3 sin A - 4 sin³A
  • cos(3A) = 4 cos³A - 3 cos A
  • tan(3A) = (3 tan A - tan³A) / (1 - 3 tan²A)
• Derived Half-Angle relations (from cos 2A):
  • cos²A = (1 + cos 2A) / 2
  • sin²A = (1 - cos 2A) / 2
  • tan²A = (1 - cos 2A) / (1 + cos 2A)

10. Product-to-Sum Formulas

• 2 sin A cos B = sin(A + B) + sin(A - B)
• 2 cos A sin B = sin(A + B) - sin(A - B)
• 2 cos A cos B = cos(A + B) + cos(A - B)
• 2 sin A sin B = cos(A - B) - cos(A + B) (Note the order in the last one)

11. Sum-to-Product Formulas (C-D Formulas)

• sin C + sin D = 2 sin((C+D)/2) cos((C-D)/2)
• sin C - sin D = 2 cos((C+D)/2) sin((C-D)/2)
• cos C + cos D = 2 cos((C+D)/2) cos((C-D)/2)
• cos C - cos D = -2 sin((C+D)/2) sin((C-D)/2) = 2 sin((C+D)/2) sin((D-C)/2)

12. Trigonometric Equations

• Principal Solutions: Solutions lying in the interval [0, 2π).
• General Solutions: Expression involving integer 'n' (n ∈ Z) which gives all possible solutions.
  • If sin x = sin y => x = nπ + (-1)ⁿ y, where n ∈ Z
  • If cos x = cos y => x = 2nπ ± y, where n ∈ Z
  • If tan x = tan y => x = nπ + y, where n ∈ Z
• Special Cases:
  • sin x = 0 => x = nπ
  • cos x = 0 => x = (2n+1)π/2
  • tan x = 0 => x = nπ
  • sin x = 1 => x = (4n+1)π/2 or 2nπ + π/2
  • cos x = 1 => x = 2nπ
  • sin x = -1 => x = (4n+3)π/2 or 2nπ + 3π/2
  • cos x = -1 => x = (2n+1)π

Key Tips for Exams:

1. Master the Formulas: Trigonometry is formula-intensive. Write them down, revise them daily.
2. Understand the Unit Circle: It helps visualize values, signs, and relationships.
3. Practice Conversions: Be quick with Degree ↔ Radian conversions.
4. ASTC Rule: Never forget the signs in different quadrants.
5. Identities are Tools: Learn to recognize when and how to apply identities to simplify expressions.
6. General Solutions: Understand the derivation and application of general solution formulas.
7. Domain & Range: Know the restrictions on inputs and outputs of T-functions.
8. Practice Problems: Solve a variety of problems, including simplification, proving identities, and solving equations.

Multiple Choice Questions (MCQs)

1. The value of tan(19π/3) is:
   (A) √3
   (B) -√3
   (C) 1/√3
   (D) -1/√3

2. If cos x = -1/2 and x lies in the third quadrant, then the value of sin x is:
   (A) 1/2
   (B) √3/2
   (C) -1/2
   (D) -√3/2

3. The value of sin 75° is:
   (A) (√3 + 1) / 2√2
   (B) (√3 - 1) / 2√2
   (C) (√2 + 1) / 2√3
   (D) (√2 - 1) / 2√3

4. The expression (sin 2A) / (1 + cos 2A) simplifies to:
   (A) tan A
   (B) cot A
   (C) sin A
   (D) cos A

5. The general solution of the equation tan x = √3 is:
   (A) x = nπ + π/6, n ∈ Z
   (B) x = 2nπ ± π/3, n ∈ Z
   (C) x = nπ + π/3, n ∈ Z
   (D) x = nπ + (-1)ⁿ π/3, n ∈ Z

6. The value of cos 1° cos 2° cos 3° ... cos 179° is:
   (A) 1
   (B) 0
   (C) -1
   (D) 1/√2

7. What is 7π/6 radians in degrees?
   (A) 150°
   (B) 210°
   (C) 240°
   (D) 120°

8. The minimum value of 3 cos x + 4 sin x + 5 is:
   (A) 0
   (B) 5
   (C) 10
   (D) -5

9. If tan A = 1/2 and tan B = 1/3, then tan(A + B) is:
   (A) 1
   (B) 5/6
   (C) 1/6
   (D) 7/6

10. Which of the following is not possible?
    (A) sin θ = 5/4
    (B) cos θ = -1/2
    (C) tan θ = 20
    (D) sec θ = 10

Answer Key & Explanations:

1. (A) √3

   • tan(19π/3) = tan(6π + π/3) = tan(π/3) = √3. (Using periodicity tan(x+nπ)=tan x)

2. (D) -√3/2

   • sin²x = 1 - cos²x = 1 - (-1/2)² = 1 - 1/4 = 3/4. So, sin x = ±√3/2. Since x is in the third quadrant, sin x is negative. Hence, sin x = -√3/2.

3. (A) (√3 + 1) / 2√2

   • sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°
   • = (1/√2)(√3/2) + (1/√2)(1/2) = (√3 + 1) / 2√2.

4. (A) tan A

   • (sin 2A) / (1 + cos 2A) = (2 sin A cos A) / (1 + (2 cos²A - 1)) = (2 sin A cos A) / (2 cos²A) = sin A / cos A = tan A.

5. (C) x = nπ + π/3, n ∈ Z

   • tan x = √3 = tan(π/3). The general solution for tan x = tan y is x = nπ + y. So, x = nπ + π/3.

6. (B) 0

   • The sequence includes cos 90°. Since cos 90° = 0, the entire product becomes zero.

7. (B) 210°

   • (7π/6) × (180/π) = 7 × 30 = 210°.

8. (A) 0

   • The range of a cos x + b sin x is [-√(a²+b²), √(a²+b²)].
   • So, the range of 3 cos x + 4 sin x is [-√(3²+4²), √(3²+4²)] = [-5, 5].
   • The minimum value of 3 cos x + 4 sin x is -5.
   • The minimum value of 3 cos x + 4 sin x + 5 is -5 + 5 = 0.

9. (A) 1

   • tan(A + B) = (tan A + tan B) / (1 - tan A tan B) = (1/2 + 1/3) / (1 - (1/2)(1/3))
   • = ((3+2)/6) / (1 - 1/6) = (5/6) / (5/6) = 1.

10. (A) sin θ = 5/4

    • The range of sin θ is [-1, 1]. 5/4 = 1.25, which is outside this range. All other values are possible (cos θ is in [-1, 1], tan θ is in R, sec θ is in R - (-1, 1)).

Make sure you understand these concepts thoroughly. Keep practicing! Let me know if any part needs further clarification.