Overview
This lecture introduces radians as a unit for measuring angles, explains the definition and relationship between radians and degrees, and demonstrates how to convert between the two.
Degrees and Circles
- There are 360 degrees in a full circle.
- A right angle measures 90 degrees, and half of a right angle is 45 degrees.
Definition of Radian
- A radian is an angle that subtends an arc equal in length to the radius of the circle.
- One radian is the angle created when the arc length equals the radius.
Radians in a Circle
- The circumference of a circle is (2\pi r).
- There are (2\pi) radians in a full circle.
Relationship Between Radians and Degrees
- (2\pi) radians = 360 degrees in a full circle.
- 1 radian = (180/\pi) degrees.
- 1 degree = (\pi/180) radians.
- Remember: (\pi) radians = 180 degrees.
Converting Between Radians and Degrees (Examples)
- 45 degrees = (45 \times \pi/180 = \pi/4) radians.
- (\pi/2) radians = ((\pi/2) \times (180/\pi) = 90) degrees.
- 30 degrees = (30 \times \pi/180 = \pi/6) radians.
Key Terms & Definitions
- Degree — A unit of angle measurement, with 360 degrees in a full circle.
- Radian — The angle that subtends an arc equal in length to the radius of the circle.
- Arc — A portion of the circumference of a circle.
- Subtend — To draw or be opposite to; an angle subtends an arc if the arc is between the two sides of the angle.
Action Items / Next Steps
- Practice converting between degrees and radians using the conversion formulas.
- Memorize that (\pi) radians = 180 degrees for quick conversions.