Introduction
Prepare Class 11 Math Unit 9: Trigonometric Functions with complete solved exercises based on the FBISE NBF New Book for Session 2026–27. This unit focuses on the properties and graphical behaviour of trigonometric functions, including their range, maximum and minimum values, period, odd and even functions, equations, and graphs.
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HSA Notes provides concise, exam-focused solutions for Exercise 9.1, Exercise 9.2, and Review Exercise 9. The notes are designed for FBISE and Punjab Board students who want to understand the important concepts and practise key questions before their HSSC Part-I Mathematics examinations.
Unit 9 Trigonometric Functions Solved Notes
Trigonometric functions such as sin x, cos x, and tan x are important mathematical functions used to describe periodic phenomena and relationships between angles. In this unit, students study how these functions behave over their domains and how their graphs can be used to identify important properties.
The major concepts include domain and range, maximum and minimum values, period, asymptotes, odd and even functions, trigonometric equations, and graphical representation. Understanding these properties makes it easier to solve both conceptual and numerical questions.
The HSA Notes PDF presents the solutions in a short and exam-ready format so students can revise the important results without unnecessary complexity.
Exercise 9.1 Range and Period
Exercise 9.1 focuses mainly on determining the range, maximum value, minimum value, and period of different trigonometric functions.
For example, some solved questions have the following ranges:
Q1(i): Range = [0, 4]
Maximum = 4
Minimum = 0
Q1(ii): Range = [1/6, 7/6]
Maximum = 7/6
Minimum = 1/6
Q1(iii): Range = [−9/5, 11/5]
Maximum = 11/5
Minimum = −9/5
Q1(iv): Range = [32/5, 38/5]
Maximum = 38/5
Minimum = 32/5
The exercise also includes functions for which the maximum and minimum values need to be determined from their mathematical form.
For Q2(i), the maximum value is 1 and the minimum value is 1/7.
For Q2(ii), there is no finite maximum or minimum because the function has asymptotic behaviour.
Period of Trigonometric Functions
Another important part of Exercise 9.1 is determining the period of functions. Students should remember that the basic sine and cosine functions have period:
2π
For example:
Q3(i): Range = [−7, 7], Period = 2π
Q3(ii): Range = [−1, 1], Period = 2π
Q3(iv): Range = R, Period = 2
Q3(vi): Range = [cos 1, 1], Period = 2π
Other period questions in the exercise include answers such as π and 4π. These problems help students recognize how changes inside a trigonometric function affect its period.
Exercise 9.2 Odd and Even Functions and Graphs
Exercise 9.2 introduces the classification of functions as odd or even and provides practice with trigonometric graphs.
A function is even if:
f(−x) = f(x)
A function is odd if:
f(−x) = −f(x)
Students should carefully substitute −x into the given function and compare the resulting expression with f(x).
The solved answers for the odd/even questions include:
Q1(i): Odd
Q1(ii): Even
Q1(iii): Even
Q1(iv): Even
Q1(v): Odd
Q1(vi): Even
Q1(vii): Odd
Q1(viii): Odd
These questions are useful for quickly identifying symmetry and understanding the relationship between algebraic functions and their graphs.
Solving Trigonometric Equations
Exercise 9.2 also contains important trigonometric equations.
For example:
sin x = cos x
gives:
x = π/4 + kπ, k ∈ Z
Another important equation is:
cos x = x
which has the numerical solution:
x ≈ 0.739
These questions demonstrate how trigonometric equations may sometimes require numerical methods or graphical interpretation rather than direct application of standard angle formulas.
Applications and Graphs
Unit 9 also connects trigonometric functions with practical situations involving periodic motion. The solved notes include equations such as:
V(t) = 180 sin(112πt)
and
h(t) = −8.5 cos(πt/25) + 10.5
Another application is represented by:
h(t) = 4 cos(πt) + 16
Students can use these equations to understand how amplitude, period, vertical shifts, and trigonometric graphs describe changing quantities.
Graphs are particularly important in this unit. Students should understand the basic shapes of sine, cosine, and tangent graphs, including their intercepts, maximum and minimum values, periods, and asymptotes where applicable.
Review Exercise 9
The Review Exercise 9 provides additional practice covering the main concepts of the unit. It is useful for revising range, period, odd/even functions, trigonometric equations, and graphical questions.
Before the FBISE examination, students should revise the standard properties of sine, cosine, and tangent functions and practise identifying changes caused by coefficients and transformations.
HSA Notes Features
HSA Notes provides:
- Complete solved Exercise 9.1 and Exercise 9.2
- Complete Review Exercise 9
- Short and exam-ready solutions
- Range and maximum/minimum questions
- Period of trigonometric functions
- Odd and even functions
- Sine, cosine, and tangent graphs
- Trigonometric equations
- Application-based questions
- Important calculations and explanations
- Free PDF access
These notes are suitable for FBISE and Punjab Board students, including students studying at schools and colleges such as Beaconhouse, Roots, LGS, The City School, Karachi Grammar School, Aitchison College, Lawrence College, ICG, APS, and FPS.
View the Complete Unit 9 PDF
Students can view the complete Class 11 Trigonometric Functions Unit 9 solved PDF below. The PDF includes the detailed calculations, graphs, proofs, tables, and complete solutions for the exercises and review questions.
For effective preparation, students should first understand the properties of each trigonometric function and then practise questions involving range, period, symmetry, equations, and graphs. Regular revision of these concepts can improve accuracy and confidence in FBISE HSSC Part-I Mathematics.
For information about live online classes, contact hsanotes48@gmail.com.
Conclusion
Class 11 Math Unit 9 Trigonometric Functions covers important concepts such as range, maximum and minimum values, period, odd and even functions, trigonometric equations, and graphical representation. Exercises 9.1, 9.2, and the Review Exercise provide focused practice for understanding these concepts and applying them to exam questions. The complete HSA Notes PDF follows the FBISE NBF New Book 2026–27 and provides concise, exam-ready solutions to help students strengthen their concepts, improve accuracy, and prepare effectively for HSSC Part-I Mathematics examinations.