2. Introduction

Trigonometry, a fundamental branch of mathematics in the Class 10 WBBSE syllabus, begins with grasping how angles are measured. This chapter, “Trigonometry: Concept of Measurement of Angle”, introduces learners to measurement systems essential for solving trigonometric problems. Through this foundation, students can later apply sine, cosine, and tangent in right-angled triangles. This overview ensures clarity on radian and degree systems, conversion techniques, and their practical use.

3. Key Concepts and Structure

3.1. What Is an Angle?

An angle forms when two rays share a common endpoint (vertex). In this chapter, students focus on two measurement systems:

• Degrees (°): Standard unit—365° in a full circle. 
• Radians (rad): Based on the arc length—2π radians in a full circle. 

3.2. Degree Measure

The degree system divides a circle into 360 equal parts. Key points:

• 1 full revolution = 360°, half circle = 180°, quarter circle = 90°. 
• 1 degree (1°) is 1/360 of a full circle. 
• Each degree has 60 minutes (′); each minute has 60 seconds (″). 
  • Example: 45° 30′ 15″ is a measure combining degrees, minutes, and seconds. 
  • Conversion example: To convert 30° 15′ into degrees:
    30 + (15 ÷ 60) = 30.25° 

3.3. Radian Measure

The radian is the standard unit in mathematics and science:

• Defined as the ratio of arc length to radius. 
• In a full circle, the circumference (2πr) yields 2π radians. 
• Hence 180° = π radians, so 1° = π/180 radians, and 1 rad ≈ 57.2958°. 
  • Example: Convert 60° to radians:
    (π/180) × 60 = π/3 radians 
  • Convert π/4 radians to degrees:
    (180/π) × (π/4) = 45° 

3.4. Degree-Radian Conversion Techniques

Mastery of conversions is crucial for subsequent trigonometry:

• Degrees to Radians: Multiply by π/180 
  • Example: 150° = (150 × π/180) = 5π/6 radians 
• Radians to Degrees: Multiply by 180/π 
  • Example: 2 radians = 2 × (180/π) ≈ 114.59° 

3.5. Positive and Negative Angles

Angles can be measured in both directions:

• Positive angles: Measured counterclockwise from the initial ray. 
• Negative angles: Measured clockwise. 
  • Example: –30° indicates a 30° rotation clockwise. 
  • Similarly, –π/6 radians is the negative counterpart of π/6. 

3.6. Standard Position of Angles

An angle in standard position has its vertex at the origin, initial side along the positive x-axis:

• The terminal side indicates the angle’s measure. 
• Helps visualize angles and understand circular motion principles. 

3.7. Co-Terminal Angles

Angles that share the same terminal side, differing by full rotations:

• 360° or 2π radians added/subtracted yields co-terminal angles. 
  • Example: 30° is co-terminal with 30° + 360° = 390°, or 30° – 360° = –330°. 
  • In radians: π/6 is co-terminal with π/6 + 2π = 13π/6. 

4. Practical Examples for Clarity

Task	Example
Convert degrees to decimals	45° 30′ = 45 + (30 ÷ 60) = 45.5°
Convert degrees to radians	120° = (120 × π/180) = 2π/3 rad
Convert radians to degrees	4π/5 rad = (4π/5 × 180/π) = 144°
Identify co-terminal angles	θ = 45° → 45° + 360° = 405°, 45° – 360° = –315°
Negative angle direction	–π/3 rad corresponds to a 60° clockwise rotation

5. Why It Matters 

Understanding angle measurement is essential to:

• Accurately define trigonometric ratios later in the curriculum. 
• Interpret math and physics problems involving waveforms, rotations, and circular motion. 
• Transition smoothly to unit circle, graphs of sine and cosine, and solving trigonometric equations. 

6. Conclusion

This overview of Chapter 20: Trigonometry – Concept of Measurement of Angles aligns with the Class 10 WBBSE syllabus. It clearly explains degree and radian systems, conversion steps, angle representation, and co-terminal and negative angles. Vivid examples help learners internalize these foundations—essential for advanced trigonometry. Including targeted keywords enhances visibility and ensures the content is both accessible and optimized for educational discovery.