Lecture Notes on Angle Measurement in Degrees and Radians

Degrees

• Degree: Unit of measurement for angles.
• Common Angles:
  • 30°: Example angle measurement
  • 90°: Right angle
  • 180°: Straight line
  • 360°: Complete rotation
• Reason for 360° in complete rotation: Somewhat arbitrary, possibly related to the base 60 number system.

Radians

• Radians: Alternative unit of angle measurement.
• Common Misunderstanding: Many people know how to convert between degrees and radians but do not understand the concept of radians.
• Concept Explanation:
  • Draw a Circle with Center Point O and Radius R.
  • Take the length R and place it on the circumference of the circle, forming an arc with length R.
  • The angle formed when drawing radii to the endpoints of this arc is one radian.
  • There are radii everywhere around this angle (r, r, and r).
• Logical Measurement: Radians measure the angle in relation to the radius of the circle.

Relationship Between Radians and Degrees

• Circumference of Circle: 2πr (where r is the radius).
• Number of Arcs (length R) to Cover Circumference: 2π arcs.
• Full Rotation in Radians: 2π radians equal one full rotation (360°).

Conversion Between Degrees and Radians

• Key Equations for Conversion:

  • 2π radians = 360°
  • π radians = 180° (half rotation)
  • π/2 radians = 90° (right angle)

• One Radian in Degrees:

  • Derivation: 1 radian = 360° / (2π)
  • Approximate Value: ~57.2 degrees.

• Remembering Conversions: Knowing one key equation (2π radians = 360°) allows derivation of others.