Radians and Degrees Conversion

Overview

This lecture explains the concept of radians as a way to measure angles, how radians relate to degrees, and how to convert between the two.

Degrees and Circles

• A right angle measures 90 degrees, and a full circle is 360 degrees.
• Degrees are the most familiar unit for measuring angles.

Definition of a Radian

• A radian is the angle that subtends an arc equal in length to the circle's radius.
• If a circle has radius r, and the arc length is also r, the angle is 1 radian.

Total Radians in a Circle

• The circumference of a circle is 2πr.
• Since one radian subtends an arc of length r, a full circle is 2π radians.

Relationship Between Radians and Degrees

• 2π radians = 360 degrees.
• 1 radian = 180/π degrees.
• 1 degree = π/180 radians.
• π radians = 180 degrees (useful reference).

Conversion Examples

• 45 degrees = 45 × (π/180) = π/4 radians.
• π/2 radians = (π/2) × (180/π) = 90 degrees.
• 30 degrees = 30 × (π/180) = π/6 radians.

Key Terms & Definitions

• Degree — a unit for measuring angles, with 360 degrees in a full circle.
• Radian — the angle subtending an arc equal to the circle's radius.
• Circumference — the distance around a circle, calculated as 2πr.

Action Items / Next Steps

• Practice converting angles between degrees and radians using the given formulas.
• Memorize the key conversions: 1 radian = 180/π degrees and 1 degree = π/180 radians.