Memorizing the Unit Circle

Introduction to the Unit Circle

• Angles are measured in radians.
• Full Circle: 2π
• Half Circle: π
• Quarter Circle: π/2
• Circle divided into 8 parts:
  • π/4, π/2, 3π/4, π
  • 5π/4, 3π/2, 7π/4, 2π

Breaking Down the Unit Circle Further

• Angles in Sixth Parts:
  • π/6, π/3, π/2
  • 2π/3, 5π/6, π
  • 7π/6, 4π/3, 3π/2
  • 5π/3, 11π/6, 2π

Values on Axes

• X-axis:
  • Right: x = 1
  • Left: x = -1
  • Center: x = 0
• Y-axis:
  • Top: y = 1
  • Bottom: y = -1

Quadrants Overview

• Quadrant 1 (Upper Right): x and y positive
• Quadrant 2 (Upper Left): x negative, y positive
• Quadrant 3 (Bottom Left): x and y negative
• Quadrant 4 (Bottom Right): x positive, y negative

Calculating Values in Quadrants

• Quadrant 1 Values:
  • Pi/6: (( \frac{\sqrt{1}}{2} ), ( \frac{\sqrt{3}}{2} ))
  • Pi/4: (( \frac{\sqrt{2}}{2} ), ( \frac{\sqrt{2}}{2} ))
  • Pi/3: (( \frac{\sqrt{3}}{2} ), ( \frac{\sqrt{1}}{2} ))
• Reflections in Other Quadrants:
  • Quadrant 2: Reflect x, ( x < 0 )
  • Quadrant 3: x and y both negative
  • Quadrant 4: Reflect y, ( y < 0 )

Converting Radians to Degrees

• Pi corresponds to 180 degrees
• Conversion Examples:
  • ( \frac{\pi}{6} = 30^\circ )
  • ( \frac{\pi}{4} = 45^\circ )
  • ( \frac{\pi}{3} = 60^\circ )
  • ( \frac{3\pi}{2} = 270^\circ )

Evaluating Trigonometric Functions

• Sine: Use y-value
  • Example: ( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} )
• Cosine: Use x-value
  • Example: ( \cos(\frac{7\pi}{6}) = -\frac{\sqrt{3}}{2} )
• Tangent: y over x (sine/cosine)
  • Example: ( \tan(\frac{\pi}{3}) = \sqrt{3} )

Conclusion

• Understanding the unit circle is crucial for evaluating trigonometric functions.
• Practice converting angles and using the unit circle to build intuition for trigonometric evaluations.

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