Lecture: Working with Radians

Key Concepts

• Radians and Degrees:
  • 180 degrees = ( \pi ) radians
  • 90 degrees = ( \pi/2 ) radians
• Angles on the coordinate plane:
  • Positive x-axis: 0 degrees/radians
  • Positive y-axis: 90 degrees/( \pi/2 ) radians
  • Negative x-axis: 180 degrees/( \pi ) radians
  • Negative y-axis: 270 degrees/( 3\pi/2 ) radians

Example Problems

Example 1: Plotting ( \frac{7\pi}{8} )

• Objective: Plot angle ( \frac{7\pi}{8} ) on the coordinate plane.
• Method:
  • Convert ( \frac{7\pi}{8} ) to a fraction of ( \pi ): ( \frac{7}{8} \pi )
  • Compare ( \frac{7}{8} ) to other fractions:
    • ( \frac{7}{8} > \frac{1}{2} )
    • ( \frac{1}{2} < \frac{7}{8} < 1 )
  • Conclusion: The terminal side is in Quadrant II.
  • Sketch: Initial side on positive x-axis, terminal side in Quadrant II.

Example 2: Plotting ( -\frac{3\pi}{5} )

• Objective: Plot negative angle ( -\frac{3\pi}{5} ) with clockwise rotation.
• Method:
  • Mark angles in clockwise orientation:
    • Positive x-axis: 0
    • Negative y-axis: ( -\pi/2 )
    • Negative x-axis: ( -\pi )
    • Positive y-axis: ( -3\pi/2 )
  • Compare ( -\frac{3}{5} ) to fractions:
    • ( -1/2 > -3/5 > -1 )
  • Conclusion: Terminal side is in Quadrant III.

Example 3: Plotting ( \frac{5\pi}{4} )

• Objective: Plot angle ( \frac{5\pi}{4} ) with counterclockwise rotation.
• Method:
  • Mark angles in counterclockwise orientation:
    • Positive y-axis: ( \pi/2 )
    • Negative x-axis: ( \pi )
    • Negative y-axis: ( 3\pi/2 )
  • Compare ( \frac{5}{4} ) to fractions:
    • ( 1 < \frac{5}{4} < \frac{3}{2} )
  • Conclusion: Terminal side is in Quadrant III.

Tips for Success

• Remember to adjust for positive or negative rotation when plotting angles.
• Use fractional comparisons to identify the correct quadrant for terminal sides.

Practice

• Try plotting angles on your own to reinforce understanding of radians and their positions on the coordinate plane.

Good luck with your studies!