Exercise 11.1 – First Year Math 2025 | Trigonometric Functions and Their Graphs| New 11th Class Math 2025
Trigonometry is one of the most important parts of mathematics because it connects numbers, angles, and geometry in a very natural way. In the new first year mathematics 2025 syllabus, Chapter 11 is about Trigonometric Functions and Their Graphs. This chapter builds the foundation for advanced studies in mathematics, physics, engineering, astronomy, and computer science.
These topics will help you understand how trigonometric functions behave, how they repeat, and how they can be represented in graphs. Let’s go step by step.
1. Trigonometric Functions – A Quick Recall
The six basic trigonometric functions are:
sin θ
cos θ
tan θ
cot θ
sec θ
csc θ
These functions are defined using the unit circle or right-angled triangles. For example:
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = sin θ / cos θ
Trigonometric functions are not limited to just 0°–90°; they are defined for all real numbers using the unit circle. This is why domain, range, and periodicity are important.
2. Domain and Range of Trigonometric Functions
The domain of a function means all possible input values (x or θ) for which the function is defined.
The range of a function means all possible output values (y values) the function can take.
(i) sin θ
Domain: All real numbers (−∞ < θ < ∞)
Range: −1 ≤ sin θ ≤ 1
(ii) cos θ
Domain: All real numbers (−∞ < θ < ∞)
Range: −1 ≤ cos θ ≤ 1
(iii) tan θ
Domain: All real numbers except θ = (2n+1)π/2, where n ∈ Z
Range: (−∞, ∞)
(iv) cot θ
Domain: All real numbers except θ = nπ, where n ∈ Z
Range: (−∞, ∞)
(v) sec θ
Domain: All real numbers except θ = (2n+1)π/2, where n ∈ Z
Range: (−∞, −1] ∪ [1, ∞)
(vi) csc θ
Domain: All real numbers except θ = nπ, where n ∈ Z
Range: (−∞, −1] ∪ [1, ∞)
3. Even and Odd Functions
Functions can be classified as even or odd.
A function f(x) is even if f(−x) = f(x).
A function f(x) is odd if f(−x) = −f(x).
Examples in Trigonometry:
cos(−θ) = cos θ → cos θ is an even function.
sec(−θ) = sec θ → sec θ is also even.
sin(−θ) = −sin θ → sin θ is an odd function.
tan(−θ) = −tan θ → tan θ is odd.
cot(−θ) = −cot θ → cot θ is odd.
csc(−θ) = −csc θ → csc θ is odd.
👉 Remember:
cos θ and sec θ are even.
sin θ, tan θ, cot θ, and csc θ are odd.
This classification makes graphing much easier because you can use symmetry:
Even functions are symmetric about the y-axis.
Odd functions are symmetric about the origin.
4. Period of Trigonometric Functions
The period of a function is the smallest interval after which the function repeats itself.
sin θ and cos θ repeat every 2π.
tan θ and cot θ repeat every π.
sec θ and csc θ repeat every 2π.
Examples:
sin(θ + 2π) = sin θ
cos(θ + 2π) = cos θ
tan(θ + π) = tan θ
This property shows that trigonometric functions are periodic functions, which makes them very useful in studying waves, circular motion, alternating current, and vibrations.
5. Graphical Understanding
When we draw the graphs of trigonometric functions, the domain, range, even/odd nature, and periodicity guide us:
sin θ: Wave starting at (0,0), max = 1, min = −1, period = 2π.
cos θ: Wave starting at (0,1), max = 1, min = −1, period = 2π.
tan θ: Undefined at (π/2), (3π/2), etc., goes from −∞ to ∞, period = π.
cot θ: Undefined at (0, π, 2π, …), period = π.
sec θ and csc θ:
Have vertical asymptotes and their ranges are outside −1 to 1.
6. Importance in Real Life
Understanding these properties is not just for exams – they are used in real life:
Engineering: Analysis of alternating current (AC), vibration, and sound waves.
Astronomy: Motion of planets and satellites.
Economics: Cyclical growth and decline can be modeled with periodic functions.
Physics: Study of oscillations, waves, and circular motion.
PDF SOLUTION:
1. Identify the domain and range of each trigonometric function.
2. Classify functions as even or odd.
3. Determine the period of trigonometric functions.
4. Use properties to simplify problems and sketch graphs.
Conclusion
Exercise 11.1 of the First Year Math 2025 syllabus is all about building a strong base in trigonometric functions. By mastering domain and range, even and odd functions, and periods, you will be ready to handle graphs and advanced applications of trigonometry. These concepts are not only part of your exams but also the foundation of real-world applications in science and technology. Practice every question carefully, and always try to visualize the function graphically for deeper understanding.
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