Exercise 11.2 | New 11th Class Math 2025 | Trigonometric Functions and Their Graphs | Sir Khawar

Exercise 11.2 | New 11th Class Math 2025 | Trigonometric Functions and Their Graphs | Sir Khawar

Mathematics in 11th class becomes more practical and logical because here we learn concepts that are directly connected with real life, science, and engineering. One of the most important chapters in the new 11th Class Math 2025 Syllabus is Trigonometric Functions and Their Graphs. This chapter not only continues the journey of trigonometry that we studied in class 9th and 10th, but also takes it to a higher level where we analyze trigonometric functions in detail, explore their properties, and learn how to represent them graphically.

In this article, we are going to cover Exercise 11.2 in detail, prepared and explained by Sir Khawar in a very simple and easy method. This explanation will help students understand the core concepts of trigonometric functions, their periodicity, and graphical behavior.

Importance of Trigonometric Functions

Trigonometry is the backbone of mathematics when it comes to angles and periodic motion. Functions like sin θ, cos θ, tan θ, cot θ, sec θ, and cosec θ are called trigonometric functions. They are used in:

Physics (waves, oscillations, electricity, sound, and light).

Engineering (mechanical design, electrical circuits, civil constructions).

Computer graphics and animations.

Astronomy and navigation.

Daily life problems related to heights, distances, and cycles.

When we represent these trigonometric functions in the form of graphs, we actually visualize their nature. A graph gives us an idea of how a function behaves, where it increases, decreases, reaches maximum, or minimum values. This is why Exercise 11.2 is an important practical part of this chapter.

What Do We Study in Exercise 11.2?

In this exercise, the main focus is on:

1. Graphs of Basic Trigonometric Functions

Graph of y = sin x

Graph of y = cos x

Graph of y = tan x

Graph of y = cot x

Graph of y = sec x

Graph of y = cosec x

2. Domain and Range of Trigonometric Functions

Domain means the set of all possible values of x for which a function is defined.

Range means the set of all possible output values (y-values) that a function can take.

3. Periodicity of Functions

Sin and Cos functions are periodic with period 2π.

Tan and Cot functions are periodic with period π.

Sec and Cosec functions are also periodic with period 2π.

4. Transformation of Graphs

Shifting the graph up or down.

Stretching or compressing.

Reflection about x-axis or y-axis.

Graphs of Trigonometric Functions

Let us explain the graphs one by one in simple words.

1. Graph of y = sin x

The sine function starts at (0, 0).

At π/2, sin x = 1 (maximum value).

At π, sin x = 0 again.

At 3π/2, sin x = –1 (minimum value).

At 2π, sin x = 0 and the cycle repeats.

So, the graph is a smooth wave that repeats every 2π.

Domain: (–∞, ∞)

Range: [–1, 1]

Period: 2π

2. Graph of y = cos x

The cosine function starts at (0, 1).

At π/2, cos x = 0.

At π, cos x = –1.

At 3π/2, cos x = 0.

At 2π, cos x = 1 and the cycle repeats.

Graph of cos x is also a wave, similar to sine but shifted.

Domain: (–∞, ∞)

Range: [–1, 1]

Period: 2π

3. Graph of y = tan x

The tangent function is sin x / cos x.

It is undefined when cos x = 0 → at x = π/2, 3π/2, 5π/2, etc.

The graph increases from –∞ to +∞ between vertical asymptotes.

The function repeats after every π.

Domain: x ≠ (2n+1)π/2

Range: (–∞, ∞)

Period: π

4. Graph of y = cot x

The cotangent function is cos x / sin x.

It is undefined when sin x = 0 → at x = 0, π, 2π, etc.

The graph decreases from +∞ to –∞ between vertical asymptotes.

The function repeats after every π.

Domain: x ≠ nπ

Range: (–∞, ∞)

Period: π

5. Graph of y = sec x

The secant function is 1 / cos x.

It is undefined when cos x = 0 → at x = π/2, 3π/2, etc.

Graph looks like a set of U-shaped and inverted-U curves.

The graph never touches between –1 and 1.

Domain: x ≠ (2n+1)π/2

Range: (–∞, –1] ∪ [1, ∞)

Period: 2π

6. Graph of y = cosec x

The cosecant function is 1 / sin x.

It is undefined when sin x = 0 → at x = 0, π, 2π, etc.

Graph looks like upward and downward U-shaped curves.

The graph never touches between –1 and 1.

Domain: x ≠ nπ

Range: (–∞, –1] ∪ [1, ∞)

Period: 2π

Why Graphs are Important?

Graphs of trigonometric functions tell us many things:

Where the function increases or decreases.

Maximum and minimum values of the function.

Symmetry of the function (even or odd).

Periodic nature and cycles of trigonometric motion.

Applications in solving real-life problems like sound waves, electrical waves, population growth, seasonal cycles, etc.

Step-by-Step Method to Draw Graphs

1. Write the equation of the function (like y = sin x).

2. Make a table of values for x and y.

3. Choose important points like 0, π/2, π, 3π/2, 2π.

4. Plot these points on graph paper.

5. Join them smoothly (never use straight lines).

6. Extend the graph on both sides because these functions repeat infinitely.

In this exercise, Sir Khawar explains each trigonometric graph step by step in very simple language. He uses easy examples and clear drawings so that every student, even weak in mathematics, can understand. The exercise also includes:

Plotting sin, cos, tan, cot, sec, and cosec functions.

Finding their domain, range, and period.

Explaining transformations of graphs.

Solving related short questions and conceptual problems.

This makes the learning process smooth and effective.

PDF SOLUTION:

📘 First Year Math Notes (Chapter-wise) – New Syllabus 2025

Applications in Real Life

To make learning meaningful, we also connect trigonometric graphs with real-life uses:

Sin and Cos graphs represent oscillations like sound waves, light waves, and pendulum motion.

Tan and Cot graphs are useful in navigation, slope calculation, and angles of elevation or depression.

Sec and Cosec graphs are used in advanced science like optics and astronomy.

Conclusion

Exercise 11.2 of New 11th Class Math 2025 (Trigonometric Functions and Their Graphs) is one of the most interesting and useful exercises. Here, students learn how to represent trigonometric functions graphically, understand their domain, range, and periodicity, and explore their real-world applications.

With the help of Sir Khawar’s easy and simple method, students can easily master these concepts. Once you understand these graphs, you will find trigonometry much easier in higher studies, especially in calculus, physics, and engineering.

If you are preparing for exams, make sure to practice each graph carefully, learn the basic shapes, and understand how transformations affect the graphs. This will not only help in board exams but also in future competitive exams.