Understanding Angle Units
What Are Radians?
A radian is the angle subtended by an arc equal in length to the radius. One full rotation = 2π radians ≈ 6.283. Radians are the natural unit for mathematics because they simplify calculus formulas.
What Are Degrees?
Degrees divide a full rotation into 360 equal parts. This comes from ancient Babylonian astronomy. One degree = π/180 radians. Degrees are more intuitive for everyday use.
What Are Gradians?
Gradians (also called gon) divide a full rotation into 400 parts. This makes right angles exactly 100 grad. Used primarily in surveying and some European engineering contexts.
Conversion Formulas
Degrees to radians: rad = deg × π/180. Radians to degrees: deg = rad × 180/π. Gradians to degrees: deg = grad × 180/400. The key relationship: 2π rad = 360° = 400 gon.
Why Use Radians in Calculus?
In calculus, the derivative of sin(x) is cos(x) only when x is in radians. With degrees, an extra factor of π/180 appears. Radians make all calculus formulas cleaner and more natural.