Radians

Date: 03/22/2001 at 19:12:41
From: Maddie
Subject: Trigonometry - Radians

Dr. Math -

I have searched everywhere for the history behind radians. Why is it 
that they replaced degrees? Is there a history behind this? I thought 
it was because of the ability to have zero degrees in a circle, but I 
don't think this is the reason.

Date: 03/22/2001 at 22:56:45
From: Doctor Peterson
Subject: Re: Trigonometry - Radians

Hi, Maddie.

I wouldn't exactly say that radians replaced degrees - degrees are 
still very much in use. Each has its place: degrees work well for 
naming common angles with whole numbers (30, 45, 60, 90, rather than 
pi/6, etc.), while radians work well in calculus. You can find several 
explanations of the latter by searching our archives for the words 
radians degrees . Here's one:

  Why Radians?
  http://mathforum.org/dr.math/problems/munroe12.3.97.html   

The basic idea is that radians measure the arc length of an angle, 
which for small angles is close to the sine. That means that if you 
measure angles in radians, the slope of the graph of the sine is 1 at 
the origin. When you get to calculus, you find that the derivative of 
the sine is the cosine - it wouldn't be nearly as simple using 
degrees. And you can write infinite series and other formulas that 
give the value of the sine, which are much simpler using radians. 
Radians simply turn out to be the "obvious" unit to use for angles, 
once you've had plenty of experience with them.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/