Trigonometric Functions – Complete Guide For Class 11 Math Chapter 3

Our learning resources for Mathematics Class 11 ‘Trigonometric Functions’ chapter 3 are designed to ensure that you grasp this concept with clarity and perfection. Whether you’re studying for an upcoming exam or strengthening your concepts, our engaging animated videos, practice questions, and notes offer you the best of integrated learning with interesting explanations and examples.

Our comprehensive approach ensures that you have access to everything you need to have an in-depth understanding of the chapter Trigonometric Functions. From detailed notes to interactive exercises, our materials are tailored to meet your learning needs and help you excel in your studies. Get ready to dive into an enriching educational experience that will make mastering this chapter a breeze.

Objectives Of Learning The Chapter

In this chapter, students explore the fundamental concepts of trigonometric functions, extending their understanding from right-angled triangles to real numbers. The unit covers angles in radians and degrees, the signs of trigonometric functions in different quadrants, and their periodicity. Students learn about the graphs of sine, cosine, and tangent functions, along with their properties and transformations. Additionally, the chapter includes the study of identities, inverse trigonometric functions, and the solution of trigonometric equations, providing a solid foundation for advanced mathematical concepts.

This chapter covers the fundamental concepts of trigonometric functions, including angles, their measures in degrees and radians, the signs of trigonometric functions in different quadrants, and the transformation of sums and products. It also introduces trigonometric equations and their solutions.

Introduction to Trigonometric Functions

The term Trigonometry is derived from two Greek words:

• Trigonon = Triangle
• Metron = Measure

Thus, Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles.

Angles

• An angle is a figure formed by joining the endpoints of two half-lines called rays.
• Positive angles: If the direction of rotation is anticlockwise, the angle is positive.
• Negative angles: Angles measured clockwise are negative.
• Initial side: The side you measure from.
• Terminal side: The side you measure to.

Notation of Angles

Angles are commonly denoted using the Greek alphabet:

• α (alpha), β (beta), γ (gamma), θ (theta), δ (delta), φ (phi).

The measure of an Angle

The measure of an angle is the amount of rotation needed to move from the initial side to the terminal side. Angles can be measured in:

• Degrees (°)
• Radians (c)

Degree Measure

• 1° = (1/360)th​ of a revolution
• 1 minute (1′) = (1/60)th​ of a degree
• 1 second (1”) = (1/60)​ of a minute
• 1° = 60′
• 1′ = 60”

Radian Measure

One radian is the angle subtended by an arc whose length is equal to the radius of the circle.

• If θ (in radians) is the angle subtended at the center of a circle of radius r by an arc of length l, then:

θ = l/r​

Relation between Degree and Radian

2π radians = 360°

1 radian =180/π radian ≈ 57°16′

1° = π/180 radian ≈ 0.01746 

Degree and Radian Measures of Common Angles

Example: Conversion of Angles

Convert 75° into radian measure.
Solution:
We know that 180° = π radian.

75° = 75×(π/180) radian 5π/12​ radians.

Sign of Trigonometric Functions in Different Quadrants

Let A (a,b) be a point on the unit circle with the center at the origin, such that ∠AOP = x.

Since, for every point A( a,b ) on the unit circle,

-1 ≤ a ≤ 1, -1 ≤ b ≤ 1 

-1 ≤ sin x ≤ 1, -1 ≤ cos x ≤ 1

• In the first quadrant (0<x<π/2): All trigonometric ratios are positive.
• In the second quadrant (π/2<x<π): Only sin⁡ x and cosec x⁡ are positive.
• In the third quadrant (π<x<3π/2​): Only tan ⁡x and cot⁡ x are positive.
• In the fourth quadrant (3π/2<x<2π): Only cos⁡ x and sec ⁡x are positive.

Table: Sign of Trigonometric Functions in Different Quadrants

Domain and Range of Trigonometric Functions

Periods of Trigonometric Functions

For any function f(x), if f(x+t) = f(x), then t is the period of the function.

Behavior of Trigonometric Functions in Different Quadrants

The behavior of sin x:

The behavior of cos x:

Graph of tan x

The behavior of tan x:

Graph of cot x

Behavior of cot x:

Graph of sec x

The behavior of sec x:

Graph of cosec x:

The behavior of Cosec x:

Trigonometric Functions of Sum and Difference of Two Angles

1. cos⁡(x+y) = cos⁡ x cos ⁡y − sin ⁡x sin ⁡y
2. cos⁡(x−y) = cos ⁡x cos ⁡y + sin ⁡x sin ⁡y
3. sin⁡(x+y) = sin ⁡x cos ⁡y + cos ⁡x sin⁡ y
4. sin⁡(x−y) = sin ⁡x cos ⁡y − cos ⁡x sin ⁡y
5. tan(x+y) = (tan x + tan y/1 − tan x tan y)​
6. tan⁡(x−y) = (tan⁡x − tan⁡ y/1 + tan ⁡x tan ⁡y)
7. cot (x+y) = (cot x cot y -1/ cot x + cot y )
8. cot (x-y) = (cot x cot y +1/ cot y – cot x )

Trigonometric Ratios of Multiple Angles

1. sin⁡ 2x = 2 sin ⁡x cos ⁡x = (2 tan ⁡x/1+tan⁡²x)​
2. cos⁡ 2x = cos⁡²x − sin²x = 2 cos²x − 1 = 1 − 2 sin²x = (1 – tan ² x/ 1 + tan² x) 
3. tan⁡ 2x = (2 tan⁡ x/1 − tan²x)
4. sin⁡ 3x = 3 sin ⁡x − 4 sin³ x
5. cos⁡3x = 4cos³ x − 3 cos ⁡x
6. tan ⁡3x = (3 tan ⁡x − tan³ x/1 – 3 tan² x)​

Formulae to Transform the Sum of Difference into Product

1. 2 sin x cos y = sin(x+y ) + sin(x-y )
2. 2 cos x sin y = sin(x+y ) – sin(x-y )
3. 2 cos x cos y = cos(x+y ) + cos(x-y )
4. 2 sin x sin y = cos(x-y ) – cos(x+y )

Trigonometric Equations

Equations involving trigonometric functions are called trigonometric equations.

• Principal solution: 0 ≤ x < 2π
• General solution: Expression giving all solutions, typically expressed in terms of an integer n.

This concludes the Trigonometric Functions for Class 11 Mathematics. Stay tuned for more chapters on advanced mathematical concepts!

Let’s Conclude

In summary, CBSE Chapter 3 of Class 11 Math, Trigonometric Functions, lays a strong foundation for understanding the relationships between angles and sides in triangles. This chapter covers essential concepts such as the measurement of angles in degrees and radians, the signs of trigonometric functions in different quadrants, and the periodicity of trigonometric functions. With detailed notes, interactive exercises, and practical examples, mastering these trigonometric functions becomes an engaging and comprehensive learning experience.

Whether you’re reviewing for exams or diving deeper into mathematical principles, our resources for Trigonometric Functions are designed to help you achieve a clear and thorough understanding. The chapter’s focus on the fundamental properties and applications of trigonometric functions will equip you with the skills necessary for tackling more complex topics in mathematics. Keep practicing, and you’ll find that Trigonometric Functions are a key step in your journey toward mathematical proficiency.