Lecture Notes: Radians in Mathematics

Introduction to Radians

• Radians: An alternative way to measure angles, apart from degrees.
• Definition: The radian measure (\theta) of an angle is the ratio of the length of the arc it spans on the circle to the length of the radius.
  • Formula: (\theta = \frac{s}{r})
    • (s) represents arc length
    • (r) represents radius

Understanding the Geometry

• Standard Position: Initial side on the positive x-axis, terminal side in quadrant one.
• Arc Length ( s): Distance from the intersection point of the initial side to the terminal side.
• Radius ( r): Distance from the center of the circle to any point on the circle.

Arc Length Theorem

• Formula: (s = r \cdot \theta)
  • (\theta) in radians
• Example: Calculate arc length for a circle with:
  • Radius = 4 meters
  • Central angle = 0.5 radians
  • Calculation: (s = 4 \cdot 0.5 = 2) meters

Converting Between Degrees and Radians

• Circumference: (C = 2 \pi r)
  • Unit circle: (C = 2\pi)
  • Full revolution: (2\pi) radians = 360 degrees
• Conversion:
  • (\pi) radians = 180 degrees
  • Converting from radians to degrees: Multiply by (\frac{180}{\pi})
  • Converting from degrees to radians: Multiply by (\frac{\pi}{180})

Conversion Examples

• Degrees to Radians:

  • 80 degrees to radians: (80 \times \frac{\pi}{180} = \frac{4\pi}{9}) radians
  • -135 degrees to radians: (-135 \times \frac{\pi}{180} = -\frac{3\pi}{4}) radians

• Radians to Degrees:

  • Negative value: Multiply by (\frac{180}{\pi})
  • Example: (-\frac{540}{5} = -108) degrees
  • Another example: (\frac{1440}{3} = 480) degrees

Conclusion

• Understanding and converting between degrees and radians is essential for measuring angles in mathematics.