Overview
This lecture covers measuring angles in degrees and radians, defining radians, converting between degrees and radians, and the locations of common radian angles on the coordinate plane.
Degrees and Radians
- Angles can be measured in degrees (° symbol) or radians (no symbol).
- If an angle is given as a number without a °, it is in radians.
- 1 radian is the central angle of a circle when the arc length equals the radius.
Definition and Calculation of Radians
- The central angle in radians: θ = arc length (s) / radius (r).
- Example: If arc length = 12 cm and radius = 6 cm, θ = 12 ÷ 6 = 2 radians.
- Example: If arc length = 16 in and radius = 4 in, θ = 16 ÷ 4 = 4 radians.
Converting Degrees to Radians
- 360° = 2π radians, so 180° = π radians.
- To convert degrees to radians: multiply degrees by π/180.
- Example: 60° × π/180 = π/3 radians.
- Example: 150° × π/180 = 5π/6 radians.
- Example: -30° × π/180 = -π/6 radians.
- Example: -225° × π/180 = -5π/4 radians.
Converting Radians to Degrees
- To convert radians to degrees: multiply radians by 180/π.
- Example: (2π/3) × 180/π = 120°.
- Example: (11π/6) × 180/π = 330°.
- Example: -3π/2 × 180/π = -270°.
- Example: -7π/4 × 180/π = -315°.
Drawing Angles in Radians
- To draw π/4 radians: convert to 45°, halfway between 0° and 90°.
- To draw 7π/6 radians: convert to 210°, place between 180° and 270°.
- To draw -π/6 radians: convert to -30°, place in Quadrant IV.
Common Radian Angles and Locations
- π/4 = 45° (Quadrant I), 3π/4 = 135° (Quadrant II), 5π/4 = 225° (Quadrant III), 7π/4 = 315° (Quadrant IV).
- π/3 = 60°, 2π/3 = 120°, 4π/3 = 240°, 5π/3 = 300°.
- π/6 = 30°, 5π/6 = 150°, 7π/6 = 210°, 11π/6 = 330°.
- On axes: 0° (0), π/2 = 90°, π = 180°, 3π/2 = 270°.
Key Terms & Definitions
- Radian — the angle where arc length equals the radius.
- Arc Length (s) — distance along the circle between two points.
- Central Angle — angle with vertex at circle center.
- Degree — 1/360 of a full circle.
Action Items / Next Steps
- Memorize the conversions between degrees and radians.
- Practice converting angles between degrees and radians.
- Learn common radian angles and their locations on the circle.
- Draw angles in standard position using both radian and degree measures.