Memorizing the Unit Circle

Introduction

• Focus on memorizing the unit circle.
• Understand angles in radians.

Angles in Radians

• Full circle: 2π
• Half circle: π
• Quarter circle: π/2
• Break the unit circle into eight equal parts:
  • 1π/4, 2π/4 (π/2), 3π/4, 4π/4 (π)
  • 5π/4, 6π/4 (3π/2), 7π/4, 8π/4 (2π)

Additional Angles

• 1π/6
• 2π/6 reduces to π/3
• 3π/6 (π/2)
• 4π/6 reduces to 2π/3
• 5π/6 (no reduction)
• 6π/6 (π)
• 7π/6 (no reduction)
• 8π/6 reduces to 4π/3
• 9π/6 (3π/2)
• 10π/6 reduces to 5π/3
• 11π/6 (no reduction)

Values on the Axes

• X-axis:
  • Right: x=1
  • Left: x=-1
• Y-axis:
  • Top: y=1
  • Bottom: y=-1

Quadrants

• Quadrant I: Upper right, x and y positive
• Quadrant II: Upper left, x negative, y positive
• Quadrant III: Bottom left, x and y negative
• Quadrant IV: Lower right, x positive, y negative

Values in Quadrants

• Quadrant I: Increase x from 0 to 1, y from 0 to 1 using square roots
• Reflection across the y-axis modifies signs in other quadrants:
  • Quadrant II: x negative, y positive
  • Quadrant III: x and y negative
  • Quadrant IV: x positive, y negative

Angles in Degrees

• Convert radians to degrees: π = 180°
• Examples:
  • π/6 = 30°
  • π/4 = 45°
  • π/3 = 60°
  • π/2 = 90°
  • Use multiplication for others (e.g., 5π/4)

Evaluating Trigonometric Functions

• Sine Function: Use y-value
  • Example: sin(π/3) = √3/2
• Cosine Function: Use x-value
  • Example: cos(7π/6) = -√3/2
• Tangent Function: y/x or sin/cos
  • Example: tan(π/3) = √3/1 = √3

Conclusion

• The unit circle helps evaluate trigonometric functions.
• Practice using the unit circle for fast calculations.

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