Unit Circle and Trigonometric Functions

Introduction to the Unit Circle

• Definition: A circle with a radius of 1.
• Key Concept: In a unit circle, the hypotenuse of a right triangle formed at any angle is 1.

Trigonometric Functions on the Unit Circle

• Sine (sin θ): Equal to the y-coordinate of a point on the unit circle.
• Cosine (cos θ): Equal to the x-coordinate of a point on the unit circle.
• Key Relationship: When radius (r) = 1, cos θ = x and sin θ = y.

Common Angles in Quadrant 1

• 30 degrees:
  • x = ( \frac{\sqrt{3}}{2} )
  • y = ( \frac{1}{2} )
• 45 degrees:
  • x = ( \frac{\sqrt{2}}{2} )
  • y = ( \frac{\sqrt{2}}{2} )
• 60 degrees:
  • x = ( \frac{1}{2} )
  • y = ( \frac{\sqrt{3}}{2} )
• 0 and 90 degrees:
  • 0 degrees: (1, 0)
  • 90 degrees: (0, 1)

Evaluating Trigonometric Functions

• Examples:
  • sin(45) = ( \frac{\sqrt{2}}{2} )
  • cos(270) = 0
  • sin(60) = ( \frac{\sqrt{3}}{2} )
  • cos(180) = -1
  • sin(30) = ( \frac{1}{2} )

Angles Beyond Quadrant 1

• Using Reference Angles: Angles such as 135, 225, and 315 degrees can be evaluated by understanding reference angles.
  • 135 degrees: Reference angle is 45 degrees (x negative, y positive)
  • 225 degrees: Reference angle is 45 degrees (x and y negative)
  • 315 degrees: Reference angle is 45 degrees (x positive, y negative)

Radians and the Unit Circle

• Conversion: π radians = 180 degrees.
• Common Radian Values:
  • π/3 corresponds to 60 degrees.
  • 2π/3, 4π/3, and 5π/3 share the reference angle of π/3.

Evaluating Functions in Radians

• Examples:
  • sin(π/3) = ( \frac{\sqrt{3}}{2} )
  • cos(2π/3) = -( \frac{1}{2} )
  • sin(4π/3) = -( \frac{\sqrt{3}}{2} )
  • cos(5π/3) = ( \frac{1}{2} )

Additional Examples

• Evaluate:
  • sin(π/6): ( \frac{1}{2} )
  • cos(5π/6): -( \frac{\sqrt{3}}{2} )
  • cos(7π/6): -( \frac{\sqrt{3}}{2} )
  • sin(11π/6): -( \frac{1}{2} )

Conclusion

• Key Takeaway: Knowing the angles and values in quadrant 1 allows the calculation of values in other quadrants by considering symmetry and sign changes.