1st Year Math Chapter 11 Exercise 11.1 Notes: Domains, Ranges, and Periods of Trigonometric Functions (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Chapter 11, Trigonometric Functions and their Graphs, follows Trigonometric Identities in the 14-unit Mathematics 11 (PECTAA) textbook. Exercise 11.1 establishes the domain, range, and period of each of the six trigonometric functions before their graphs are studied.
What Does This Exercise Cover?
This exercise teaches exactly which angles each trigonometric function is defined for, what values it can output, and how often its pattern repeats — laying the groundwork needed to sketch accurate graphs in Exercises 11.2 and 11.3.
Key Concepts
1. Domain and Range of Sine and Cosine
Both sine and cosine are defined for every real number, so their domain is all real numbers (R). Both functions only ever output values between −1 and 1, so their range is [−1, 1].
2. Domain and Range of Tangent and Cotangent
Tangent is undefined wherever cos θ = 0 (at θ = π/2 + nπ), so its domain excludes those values; its range is all real numbers. Cotangent is undefined wherever sin θ = 0 (at θ = nπ), with a similarly unrestricted range.
3. Domain and Range of Secant and Cosecant
Secant is undefined wherever cos θ = 0, with range (−∞, −1] ∪ [1, ∞). Cosecant is undefined wherever sin θ = 0, with the same range (−∞, −1] ∪ [1, ∞).
4. Period of a Trigonometric Function
The period of a function is the smallest positive value after which its pattern repeats exactly. Sine, cosine, secant, and cosecant all have period 2π, while tangent and cotangent have the shorter period π.
Domain, Range, and Period Summary
| Function | Domain | Range | Period |
| sin θ | All real numbers | [−1, 1] | 2π |
| cos θ | All real numbers | [−1, 1] | 2π |
| tan θ | R except θ = π/2 + nπ | All real numbers | π |
| cot θ | R except θ = nπ | All real numbers | π |
| sec θ | R except θ = π/2 + nπ | (−∞,−1] ∪ [1,∞) | 2π |
| csc θ | R except θ = nπ | (−∞,−1] ∪ [1,∞) | 2π |
Solved Examples
Example 1: State the domain of tan θ.
tan θ = sin θ / cos θ, which is undefined wherever cos θ = 0.
This happens at θ = π/2 + nπ for any integer n.
Answer: All real numbers except θ = π/2 + nπ
Example 2: Find the period of y = sin 2θ.
The standard period of sin θ is 2π.
For sin(kθ), the period becomes 2π/k. Here k = 2, so the period is 2π/2.
Answer: π
Example 3: State the range of sec θ.
sec θ = 1/cos θ, and cos θ takes values in [−1, 1], excluding 0.
Taking the reciprocal of values in [−1, 1] (excluding 0) gives values ≤ −1 or ≥ 1.
Answer: (−∞, −1] ∪ [1, ∞)
Sample MCQs
1. The domain of sin θ is:
a) [−1, 1] b) All real numbers c) All real numbers except 0 d) Positive real numbers only
Answer: b) All real numbers
2. The period of the tangent function is:
a) π b) 2π c) π/2 d) 4π
Answer: a) π
3. The range of sec θ is:
a) [−1, 1] b) All real numbers c) (−∞,−1] ∪ [1,∞) d) [0, 1]
Answer: c) (−∞,−1] ∪ [1,∞)
4. tan θ is undefined at:
a) θ = 0, π, 2π, … b) θ = π/2, 3π/2, … c) All values of θ d) No value of θ
Answer: b) θ = π/2, 3π/2, …
5. The period of sin θ and cos θ is:
a) π b) 2π c) π/2 d) 4π
Answer: b) 2π
Important Short Questions
- State the domain and range of the sine function.
- State the domain and range of the tangent function.
- Define the period of a trigonometric function.
- State the period of the cosine function.
- Explain why sec θ is undefined at θ = π/2.
Important Long Questions
- Find the domain and range of the cotangent function, explaining your reasoning.
- Find the period of y = cos 3θ and y = tan(θ/2).
- Explain why the range of cosecant is (−∞,−1] ∪ [1,∞) rather than all real numbers.
- Summarize the domain, range, and period of all six trigonometric functions, with a brief justification for each.
How to Approach This Exercise Effectively
- Link every domain restriction back to a zero denominator: tan and sec fail where cos θ = 0; cot and csc fail where sin θ = 0.
- Memorize sin/cos/sec/csc share period 2π, while tan/cot share the shorter period π — this pairing makes the table easier to recall.
- For period of sin(kθ) or cos(kθ), remember the shortcut: divide the standard period (2π) by k.
- Picture the unit circle definition (cos θ = x, sin θ = y) to quickly re-derive any domain or range fact you’re unsure about.
- Build the full six-function table from memory as practice, rather than memorizing rows in isolation — the patterns reinforce each other.
FAQs
Q: Why do tan and cot have a different period from sin and cos?
A: Because tan θ = sin θ/cos θ repeats its exact pattern after only half the rotation that sine or cosine need, since tan(θ + π) = tan θ.
Q: What does ‘period’ mean for a trigonometric function?
A: It’s the smallest positive length of input after which the function’s output pattern repeats identically.
Q: Why are sec and csc undefined at certain points?
A: Because they are reciprocals of cosine and sine respectively, and reciprocals are undefined wherever the original function equals zero.
Q: How is the period of y = sin(kθ) related to k?
A: The period is the standard period (2π) divided by k, so larger values of k compress the wave into a shorter repeating interval.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 11: Trigonometric Functions and their Graphs, Exercise 11.1.