1st Year Math Chapter 10 Exercise 10.1 Notes: Units of Measurement of Angles (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 10, Trigonometric Identities, follows Division of Polynomials in the 14-unit Mathematics 11 (PECTAA) textbook. Exercise 10.1 introduces the two systems used to measure angles — degrees and radians — and how to convert between them.
What Does This Exercise Cover?
This exercise builds the foundation for the whole chapter: measuring angles in the sexagesimal (degree) and circular (radian) systems, converting between the two, and relating radian measure to arc length.
Key Concepts
1. The Sexagesimal System (Degrees)
In the sexagesimal system, a complete rotation is divided into 360 degrees (360°). Each degree is further divided into 60 minutes (60′), and each minute into 60 seconds (60″).
2. The Circular System (Radians)
One radian is the angle subtended at the center of a circle by an arc whose length equals the circle’s radius. A complete rotation measures 2π radians.
3. The Relationship Between Degrees and Radians
Since a complete rotation is both 360° and 2π radians, it follows that π radians = 180°. This single relationship is used to convert between the two systems in both directions.
4. Arc Length Formula
For a circle of radius r, the length s of an arc subtending a central angle θ (measured in radians) is given by s = rθ.
Common Degree-Radian Conversions
| Degrees | Radians |
| 0° | 0 |
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 180° | π |
| 270° | 3π/2 |
| 360° | 2π |
Solved Examples
Example 1: Convert 45° to radians.
Multiply by π/180: 45° × (π/180).
Answer: π/4 radians
Example 2: Convert 5π/6 radians to degrees.
Multiply by 180/π: (5π/6) × (180/π).
Answer: 150°
Example 3: Find the length of an arc that subtends a central angle of 60° in a circle of radius 10 cm.
Convert 60° to radians: 60° × (π/180) = π/3.
Apply s = rθ: s = 10 × (π/3).
Answer: s = 10π/3 cm ≈ 10.47 cm
Sample MCQs
1. In the sexagesimal system, one degree equals:
a) 60 minutes b) 100 minutes c) 60 seconds d) 1 radian
Answer: a) 60 minutes
2. π radians equals:
a) 90° b) 180° c) 270° d) 360°
Answer: b) 180°
3. To convert degrees to radians, multiply by:
a) 180/π b) π/180 c) 90/π d) π
Answer: b) π/180
4. The formula for arc length (θ in radians) is:
a) s = r/θ b) s = rθ c) s = r + θ d) s = θ/r
Answer: b) s = rθ
5. 90° expressed in radians is:
a) π/6 b) π/4 c) π/3 d) π/2
Answer: d) π/2
Important Short Questions
- Define the sexagesimal system of angle measurement.
- Define the circular (radian) system of angle measurement.
- State the relationship between degrees and radians.
- Convert 120° to radians.
- Convert 3π/4 radians to degrees.
Important Long Questions
- Convert 75°30′ into decimal degree form, then into radians.
- Find the length of an arc that subtends a central angle of 45° in a circle of radius 14 cm.
- Convert 210° to radians and 7π/4 radians to degrees.
- Explain the relationship between the length of an arc, the radius of a circle, and the central angle measured in radians.
How to Approach This Exercise Effectively
- Memorize π radians = 180° as your single conversion anchor — every other conversion follows from it.
- Always double-check whether you’re converting degrees to radians (multiply by π/180) or radians to degrees (multiply by 180/π).
- When converting minutes and seconds to decimal degrees, divide minutes by 60 and seconds by 3600 before adding to the whole degree part.
- For arc length problems, convert the angle to radians first — the formula s = rθ only works with θ in radians, not degrees.
- Practice the standard angle conversions (30°, 45°, 60°, 90°, etc.) until they’re automatic, since they appear constantly throughout the rest of the chapter.
FAQs
Q: Why do we use both degrees and radians?
A: Degrees are intuitive and widely used in everyday and geometric contexts, while radians connect angle measure directly to arc length and are essential for calculus and advanced trigonometry.
Q: What is the significance of π radians = 180°?
A: It’s the fundamental bridge between the two systems, derived from the fact that a full circle (360° or 2π radians) is the same rotation measured two different ways.
Q: How is the arc length formula s = rθ derived?
A: It follows directly from the definition of a radian: since one radian corresponds to an arc equal in length to the radius, θ radians corresponds to an arc θ times as long, or rθ.
Q: Can radian measure be negative?
A: Yes — like degrees, radian measure can be negative, representing rotation in the opposite (clockwise) direction.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 10: Trigonometric Identities, Exercise 10.1.