1st Year Math Chapter 10 Exercise 10.4 Notes: Domains of Trigonometric Functions and Fundamental Identities (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.
Exercise 10.4 closes the chapter by examining the domain and range of each trigonometric function, and applying the fundamental identities from Exercise 10.2 to simplify and prove trigonometric expressions.
What Does This Exercise Cover?
This exercise teaches exactly where each trigonometric function is defined (and where it isn’t), what values it can take, and how to use the fundamental identities as tools for simplifying more complex expressions.
Key Concepts
1. Domain of the Trigonometric Functions
Sine and cosine are defined for every real number. Tangent and secant are undefined wherever cos θ = 0 (at 90°, 270°, and so on). Cotangent and cosecant are undefined wherever sin θ = 0 (at 0°, 180°, and so on).
2. Range of the Trigonometric Functions
Sine and cosine both have range [−1, 1]. Tangent and cotangent can take any real value. Secant and cosecant have range (−∞, −1] ∪ [1, ∞).
3. Applying Fundamental Identities to Simplify Expressions
The three fundamental identities from Exercise 10.2 — sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ — are used as substitution tools to simplify trigonometric expressions or prove that two expressions are equal.
Where Each Function Is Undefined
| Function | Undefined When |
| tan θ, sec θ | cos θ = 0 (θ = 90°, 270°, …) |
| cot θ, csc θ | sin θ = 0 (θ = 0°, 180°, …) |
Solved Examples
Example 1: State the domain restriction for tan θ.
tan θ = sin θ / cos θ, which is undefined wherever the denominator cos θ = 0.
This occurs at θ = 90° + 180°n for any integer n.
Answer: tan θ is undefined at θ = 90°, 270°, 450°, … (and their negative counterparts)
Example 2: Simplify sin θ × sec θ.
sec θ = 1/cos θ.
So sin θ × sec θ = sin θ / cos θ.
Answer: tan θ
Example 3: Prove that (1 − cos²θ) / sin θ = sin θ.
From the fundamental identity, 1 − cos²θ = sin²θ.
Substitute: sin²θ / sin θ.
Answer: sin θ, confirming the identity
Sample MCQs
1. The function tan θ is undefined when:
a) sin θ = 0 b) cos θ = 0 c) tan θ = 0 d) θ = 0°
Answer: b) cos θ = 0
2. The domain of sin θ and cos θ is:
a) All real numbers b) Only positive numbers c) Only angles between 0° and 90° d) Undefined for negative angles
Answer: a) All real numbers
3. The range of sin θ is:
a) All real numbers b) [−1, 1] c) [0, 1] d) (−∞, −1] ∪ [1, ∞)
Answer: b) [−1, 1]
4. sec θ is undefined wherever:
a) sin θ = 0 b) cos θ = 0 c) tan θ = 1 d) θ = 90° only
Answer: b) cos θ = 0
5. Simplifying sin θ × sec θ gives:
a) 1 b) cos θ c) tan θ d) cot θ
Answer: c) tan θ
Important Short Questions
- State the domain restriction for tan θ and sec θ.
- State the domain restriction for cot θ and csc θ.
- State the range of the sine and cosine functions.
- Simplify cos θ × csc θ using the fundamental identities.
- Explain why csc θ is undefined at θ = 0°.
Important Long Questions
- Simplify the expression (1 − sin²θ) / cos²θ using the fundamental identities.
- Prove that sec θ − cos θ = sin θ × tan θ.
- Explain the domain and range of all six trigonometric functions, and how they connect to the fundamental identities.
- Simplify (sin θ + cos θ)² + (sin θ − cos θ)², using the identity sin²θ + cos²θ = 1.
How to Approach This Exercise Effectively
- Link each domain restriction back to a zero denominator — tan and sec fail where cos θ = 0, cot and csc fail where sin θ = 0.
- When simplifying an expression, rewrite every function in terms of sin θ and cos θ first — this often reveals a straightforward cancellation.
- For ‘prove’ questions, work from the more complicated side of the equation toward the simpler side, substituting identities as needed.
- Keep the range facts (−1 to 1 for sin/cos, everything else for tan/cot, outside −1 to 1 for sec/csc) handy for quickly spotting impossible answers.
- Practice a mix of simplification and proof questions, since both styles are common in this exercise’s short and long questions.
FAQs
Q: Why are tan and sec undefined at certain angles?
A: Because both are defined using cos θ in the denominator (directly for sec θ, and indirectly for tan θ = sin θ/cos θ), and division by zero is undefined.
Q: How is the range of secant and cosecant different from sine and cosine?
A: Since secant and cosecant are reciprocals of cosine and sine, and those base functions range only between −1 and 1, their reciprocals can never fall strictly between −1 and 1, giving the range (−∞,−1] ∪ [1,∞).
Q: Why is it useful to know the domain of a trigonometric function?
A: It tells you immediately which angles must be excluded from a problem or graph, preventing invalid calculations involving undefined values.
Q: Can fundamental identities be used to simplify any trigonometric expression?
A: They can simplify many expressions, especially those involving squares of sine, cosine, tangent, or their reciprocals, though not every expression reduces neatly — practice helps recognize when an identity applies.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 10: Trigonometric Identities, Exercise 10.4.