Radian

A radian is an angle measured by the ratio $\frac{\text{arc length}}{\text{radius}}$ — pure number, no units. One radian is the angle subtended at the center of a circle by an arc whose length equals the radius.

Conversions:

• Full circle: $2\pi$ rad $= 360°$
• Half circle: $\pi$ rad $= 180°$
• Right angle: $\pi/2$ rad $= 90°$
• $1$ rad $\approx 57.296°$
• Conversion: $\theta_{\text{rad}} = \theta_{\text{deg}} \times \pi/180$.

Why mathematicians prefer radians over degrees:

• $\frac{d}{dx}\sin x = \cos x$ holds only when $x$ is in radians (otherwise you'd need a $\frac{\pi}{180}$ factor).
• Arc length is simply $s = r\theta$.
• Taylor series have clean coefficients.

Degrees are an arbitrary historical convention (Babylonian base-60). Radians arise naturally from the geometry of the circle, which is why every physics formula, calculus textbook, and computer-graphics shader uses them.