The Unit Circle

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Introduction

The unit circle is a circle with radius 1 centered at the origin (0,0) in the coordinate plane. It provides a powerful way to understand trigonometric functions and their properties for all angles.

Definition and Properties

The unit circle has the following properties:

  • Center at (0,0)
  • Radius = 1
  • Equation: x² + y² = 1
  • Any point on the circle has coordinates (cos θ, sin θ)
  • θ is measured from the positive x-axis

Example 1: Finding Coordinates

Find the coordinates of the point on the unit circle corresponding to θ = 30°

Solution:

For θ = 30°:

cos 30° = √3/2 ≈ 0.866

sin 30° = 1/2 = 0.5

Therefore, the coordinates are (√3/2, 1/2)

Answer: (√3/2, 1/2)

Example 2: Finding Angles

Find the angle θ if the point on the unit circle has coordinates (1/2, √3/2)

Solution:

cos θ = 1/2

sin θ = √3/2

This corresponds to θ = 60°

Answer: θ = 60°

Example 3: Negative Angles

Find the coordinates for θ = -45°

Solution:

cos(-45°) = cos(45°) = 1/√2

sin(-45°) = -sin(45°) = -1/√2

Therefore, the coordinates are (1/√2, -1/√2)

Answer: (1/√2, -1/√2)

Key Properties

  • Quadrant I (0° to 90°): Both sin θ and cos θ are positive
  • Quadrant II (90° to 180°): sin θ is positive, cos θ is negative
  • Quadrant III (180° to 270°): Both sin θ and cos θ are negative
  • Quadrant IV (270° to 360°): sin θ is negative, cos θ is positive
  • Reference Angles: The acute angle between the terminal side and x-axis

Example 4: Using Reference Angles

Find cos 150° using the unit circle

Solution:

150° is in Quadrant II

The reference angle is 180° - 150° = 30°

cos 150° = -cos 30° = -√3/2

Answer: cos 150° = -√3/2

Example 5: Periodicity

Find sin 390°

Solution:

390° = 360° + 30°

Since sin is periodic with period 360°:

sin 390° = sin 30° = 1/2

Answer: sin 390° = 1/2

Key Points to Remember

  • The unit circle has radius 1 and center at (0,0)
  • Any point on the circle has coordinates (cos θ, sin θ)
  • Use reference angles to find values in other quadrants
  • Remember the signs in each quadrant
  • Trigonometric functions are periodic

Problem-Solving Strategy

  1. Draw the unit circle and locate the angle
  2. Determine which quadrant the angle is in
  3. Find the reference angle if needed
  4. Use the appropriate sign for the quadrant
  5. Check your answer is reasonable