Understanding Radians and Degrees
Degrees and radians are two different units used to measure angles. A degree is a unit of measurement where a full circle is divided into 360 parts. A radian, on the other hand, is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. A full circle contains 2π (approximately 6.28) radians.
Section 2: The Fundamental Conversion
The fundamental relationship that connects the two units is based on a full circle. Since a full circle is 360 degrees and also 2π radians, we can state that 360° = 2π radians. This relationship can be simplified to its most common form: 180° = π radians. This simple equality is the key to converting between the two units.
Section 3: A Practical Example
To convert from degrees to radians, you multiply the angle in degrees by (π / 180°). For example, to convert 90°, you would calculate: 90° × (π / 180°) = 90π / 180 = π/2 radians. To convert from radians to degrees, you multiply by (180° / π). For example, π/4 radians × (180° / π) = 180° / 4 = 45°.
Section 4: Why Are Radians Important?
While degrees are commonly used in everyday contexts, radians are preferred in higher-level mathematics and physics. This is because radians are a more 'natural' unit for angles that simplifies many important formulas in calculus and physics, especially those involving trigonometric functions, waves, and circular motion.