Lecture Notes: Memorizing the Unit Circle

Introduction to the Unit Circle

• The Unit Circle is a circle with:
  • Radius = 1
  • Center at the origin of the coordinate plane
• Key angles:
  • Multiples of π/6 and π/4
• Need to memorize sine and cosine values for these angles

Sine and Cosine Basics

• From SOHCAHTOA:
  • Sine: Opposite leg over hypotenuse
  • Cosine: Adjacent leg over hypotenuse
• In the unit circle:
  • Hypotenuse = 1
  • Coordinates (x, y):
    • x-coordinate = cosine θ
    • y-coordinate = sine θ

Key Points on the Unit Circle

• Coordinates for key angles:
  • 0 radians (0 degrees):
    • Cosine = 1
    • Sine = 0
  • π/2 radians (90 degrees):
    • Cosine = 0
    • Sine = 1
  • π radians (180 degrees):
    • Cosine = -1
    • Sine = 0
  • 3π/2 radians (270 degrees):
    • Cosine = 0
    • Sine = -1
  • 2π radians (360 degrees):
    • Cycle repeats

Angles and Their Coordinates

• π/6 radians (30 degrees):
  • Coordinates: (√3/2, 1/2)
• π/4 radians (45 degrees):
  • Coordinates: (√2/2, √2/2)
• π/3 radians (60 degrees):
  • Coordinates: (1/2, √3/2)

Simplifying the Memorization Process

• Create a consistent numerical pattern for sine and cosine values:
  • From 0 to 1 in forms of √n/2 for sine:
    • Sine values: √0/2, √1/2, √2/2, √3/2, √4/2
  • Cosine values follow the same pattern in reverse along x-axis
• Quadrants:
  • Quadrant II:
    • Sine positive, cosine negative
  • Quadrant III:
    • Both sine and cosine negative
  • Quadrant IV:
    • Cosine positive, sine negative

Application in Calculating Trigonometric Functions

• Example: Tangent of 14π/3
  • Reduce angle: Subtract cycles of 2π
  • Tangent = Sine/Cosine
• Example: Cosecant of -17π/4
  • Move in negative direction, reduce angle
  • Cosecant = 1/Sine

Conclusion

• Practice is key to mastering the unit circle and quickly evaluating trigonometric functions for common angles.
• Remember the patterns and coordinate representations to avoid calculations.