Angles and Radian Measure

Section 4.1: Angles and Radian Measure

  • Definition of an Angle
      - An angle in a rectangular coordinate system is said to be in standard position if:
        - Its vertex is located at the origin of the coordinate system.
        - Its initial side is aligned along the positive x-axis.
      - Positive Angles: Generated by rotating the terminal side counterclockwise from the initial side.
      - Negative Angles: Generated by rotating the terminal side clockwise from the initial side.

  • Measurement of Angles
      - Angles can be quantified in two primary ways: degrees and radians.
        - Degrees: Commonly used for practical applications involving angles.
        - Radians: Preferred for theoretical applications, particularly in geometry and calculus.

  • Radian Definition
      - A radian is defined as the ratio of an angle’s arc length (s) to its radius (r):
    θ=sr\theta = \frac{s}{r}
      - Important Note: Radians do not have units, making them particularly useful in mathematical expressions involving circles and triangles.

Conversion Between Degrees and Radians

  • To effectively work with angles in different contexts, we need to convert between degrees and radians:

      - Degree to Radian Conversion:
        - To convert from degrees to radians, multiply by:
    π radians180∘\frac{\pi \text{ radians}}{180^{\circ}}

  - Radian to Degree Conversion:
    - To convert from radians to degrees, multiply by:
180∘π radians\frac{180^{\circ}}{\pi \text{ radians}}

  • Examples:
      - Convert 65° to radians.
      - Convert -135° to radians.
      - Convert (\frac{\pi}{9}) to degrees.
      - Convert (\frac{11\pi}{6}) to degrees.

Coterminal Angles

  • Definition:
      - Two angles that have different measures but occupy the same position in the coordinate system are termed coterminal angles.
  • Calculation of Coterminal Angles:
      - Coterminal angles can be found by adding or subtracting a full rotation of a circle:
        - For angles measured in degrees, a full rotation is 360°.
        - For angles measured in radians, a full rotation is 2π.
  • Examples:
      - Find a positive angle less than 360° that is coterminal with 780°.
      - Find a positive angle less than 360° that is coterminal with -225°.
      - Find a positive angle less than 2π that is coterminal with (\frac{9\pi}{4}).
      - Find a positive angle less than 2π that is coterminal with -(\frac{\pi}{2}).