Unit Circle and Trigonometric Functions

Quadrants and Signs

• Quadrants:
  • Quadrant I: All functions positive (All Students Take Calculus)
  • Quadrant II: Sine positive
  • Quadrant III: Tangent positive
  • Quadrant IV: Cosine positive

Key Angles in Degrees and Radians

• Degrees and their Radian equivalents:
  • 0° = 0
  • 30° = π/6
  • 45° = π/4
  • 60° = π/3
  • 90° = π/2
  • 120° = 2π/3
  • 135° = 3π/4
  • 150° = 5π/6
  • 180° = π
  • 210° = 7π/6
  • 240° = 4π/3
  • 270° = 3π/2
  • 300° = 5π/3
  • 315° = 7π/4
  • 330° = 11π/6
  • 360° = 2π

Coordinates on the Unit Circle

• First Quadrant:

  • 0°: (1, 0)
  • 30°: (√3/2, 1/2)
  • 45°: (√2/2, √2/2)
  • 60°: (1/2, √3/2)
  • 90°: (0, 1)

• Second Quadrant: (X negative)

  • 120°: (-1/2, √3/2)
  • 135°: (-√2/2, √2/2)
  • 150°: (-√3/2, 1/2)
  • 180°: (-1, 0)

• Third Quadrant: (X and Y negative)

  • 210°: (-√3/2, -1/2)
  • 225°: (-√2/2, -√2/2)
  • 240°: (-1/2, -√3/2)
  • 270°: (0, -1)

• Fourth Quadrant: (X positive, Y negative)

  • 300°: (1/2, -√3/2)
  • 315°: (√2/2, -√2/2)
  • 330°: (√3/2, -1/2)
  • 360°: (1, 0)

Evaluating Sine, Cosine, and Tangent

• Sine function relates to Y-coordinate.
• Cosine function relates to X-coordinate.
• Tangent function is calculated as y/x.

Example Evaluations:

• Sine(60°): √3/2
• Cosine(60°): 1/2
• Tangent(60°): √3

Negative Angles and Coterminal Angles

• To find coterminal angles, add or subtract 360°.

Special Triangles

• 30-60-90 Triangle:

  • Opposite 30°: 1
  • Opposite 60°: √3
  • Hypotenuse: 2

• 45-45-90 Triangle:

  • Both legs: 1
  • Hypotenuse: √2

Reference Angles

• Reference angles help in evaluating sine and cosine for angles outside Quadrant I.
• Formulae:
  • Quadrant II: 180° - angle
  • Quadrant III: angle - 180°
  • Quadrant IV: 360° - angle

Inverse Trigonometric Functions

• Restricted Domains:
  • Inverse Sine: (-90°, 90°)
  • Inverse Cosine: (0°, 180°)
  • Inverse Tangent: (-90°, 90°)

Example Evaluations for Inverse Functions

• Inverse sine(1/2): 30°
• Inverse cosine(1/2): 60°
• Inverse tangent(1): 45°

Summary of Key Relationships

• Sine, cosine, and tangent can be evaluated using special triangles or the unit circle.
• Always consider the signs based on the quadrant of the angle.
• Use reference angles for evaluations outside of the first quadrant.