Why 57 Degrees Was Chosen for One Radian

[Miike012]

Homework Statement

What made them pick one rad equal to 57 deg.
Why didn't the pick 45 deg. or 90 deg. or any other random degree to equal one rad.?

Homework Equations

The Attempt at a Solution

 

[Gib Z]

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]

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)​