Lesson 92: Angles and the Unit Circle
Trigonometry begins with understanding angles and the unit circle. The unit circle is a circle of radius 1 centered at the origin of a coordinate plane. Every point on the circle corresponds to an angle measured from the positive x-axis.
Angles
Angles can be measured in degrees or radians.
- Degrees: A full circle = 360°
- Radians: A full circle = 2Ï€ radians
- Conversion: \( 180° = π \text{ radians} \) → \( \text{radians} = \frac{π}{180} × \text{degrees} \)
Example: Convert 60° to radians: \( 60 × \frac{π}{180} = \frac{π}{3} \) radians.
The Unit Circle
The unit circle has radius 1. A point on the circle can be written as (cos θ, sin θ), where θ is the angle measured from the positive x-axis.
- Quadrant I: x > 0, y > 0
- Quadrant II: x < 0, y > 0
- Quadrant III: x < 0, y < 0
- Quadrant IV: x > 0, y < 0
Special Angles:
- 0° (0 rad): (1, 0)
- 30° (π/6): (√3/2, 1/2)
- 45° (π/4): (√2/2, √2/2)
- 60° (π/3): (1/2, √3/2)
- 90° (π/2): (0, 1)
Examples
Example 1 — Converting Angles: Convert 150° to radians: \( 150 × \frac{π}{180} = \frac{5π}{6} \).
Example 2 — Finding Coordinates: Find the coordinates on the unit circle for θ = 210°: 210° is in Quadrant III → cos θ = −√3/2, sin θ = −1/2 → point = (−√3/2, −1/2)
Example 3 — Quadrants: Determine the quadrant for θ = 330°: Quadrant IV.
Practice Problems
- Convert 120° to radians.
- Find the coordinates on the unit circle for θ = 45°.
- Convert 7Ï€/6 radians to degrees.
- Identify the quadrant for θ = 5π/3.
- Find the sine and cosine for θ = 135°.