Understanding the Unit Circle

Introduction

• The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane.
• It's crucial for understanding trigonometric functions as it relates to angles in radians.

Basic Concepts

• Sine (sin): Ratio of the length of the opposite leg to the hypotenuse in a right triangle (SOHCAHTOA).
• Cosine (cos): Ratio of the length of the adjacent leg to the hypotenuse.
• On the unit circle, the X-coordinate represents cos(θ) and the Y-coordinate represents sin(θ).

Key Angles and Coordinates

• 0 radians (0°): Coordinates (1, 0)
  • cos(0) = 1, sin(0) = 0
• π/2 radians (90°): Coordinates (0, 1)
  • cos(π/2) = 0, sin(π/2) = 1
• π radians (180°): Coordinates (-1, 0)
  • cos(π) = -1, sin(π) = 0
• 3π/2 radians (270°): Coordinates (0, -1)
  • cos(3π/2) = 0, sin(3π/2) = -1
• 2π radians (360°): Back to (1, 0)

Memorizing the Unit Circle

• First Quadrant:
  • π/6 (30°): (√3/2, 1/2)
  • π/4 (45°): (√2/2, √2/2)
  • π/3 (60°): (1/2, √3/2)
• Establish a pattern:
  • Use square roots over 2 for consistency: √0/2, √1/2, √2/2, √3/2, √4/2

Extending to Other Quadrants

• Second Quadrant:
  • sin values remain positive (Y > 0)
  • cos values become negative (X < 0)
• Third Quadrant:
  • Both sin and cos become negative (X, Y < 0)
• Fourth Quadrant:
  • cos turns positive again (X > 0), sin remains negative (Y < 0)

Visualizing and Calculating

• Understand angles in reference to the unit circle for quick calculation.
• Example:
  • For 4π/3, find reference angle π/3 in the third quadrant.
  • sin(4π/3) = -√3/2, cos(4π/3) = -1/2

Practice Problems

• Tangent Example: tan(14π/3)
  • Reduce to 2π/3, then tan(2π/3) = sin(2π/3) / cos(2π/3) = -√3
• Cosecant Example: csc(-17π/4)
  • Reduce to -π/4, then csc(-π/4) = 1/sin(-π/4) = -√2

Conclusion

• Familiarity with the unit circle allows for quick evaluation of trigonometric functions.
• Practice is key to mastery; visualize and memorize positions and coordinates on the unit circle.