## Homework Statement

Why didn't the pick 45 deg. or 90 deg. or any other random degree to equal one rad.?

## The Attempt at a Solution

Gib Z
Homework Helper
Google "radians" and just read how they are defined. Whats important is that $\pi$ radians is equal to 180 degrees, not that 1 radian is approximately 57 degrees.

tiny-tim
Homework Helper
Hi Miike012!

It isn't exactly 57°, it's exactly (180/π)°.

It's to avoid having an awkward constant in various equations …

with angle θ measured in radians, we have the simple equations:

arc-length = rθ (in particular, circumference = r(2π))

speed = r dθ/dt

acceleration = r d2θ/dt2

Also, with ω measured in radians per second:

d/dt (sinωt) = ωcosωt.​

This is similar to the reason mathematicians prefer natural logarithms (base e) to logarithms base 10.

(Similarly, we define the steradian to measure "solid angle", giving:
surface area = r2 times solid angle)​