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June 30th, 2019, 09:44 AM  #1 
Senior Member Joined: Oct 2015 From: Greece Posts: 138 Thanks: 8  Why angles have units since they are defined by a ratio of distances? Screenshot_20190630_203138.jpg Since $\displaystyle \theta = \frac{s}{r}$ the units of s and r shouldn't cancel out? So what are radians? Also, how did mathematicians said that π rads=180 degrees? why not π rads=1000 degrees? thanks 
June 30th, 2019, 12:19 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,835 Thanks: 733 
Radian definition is based on the unit circle. The length of an arc is a fraction of $2\pi$, and the angle defining that arc is defined to be the same number as the length of the arc.

June 30th, 2019, 01:13 PM  #3 
Senior Member Joined: Jun 2019 From: USA Posts: 314 Thanks: 165 
I consider radians to be a dimensionless unit (they do, as you point out, represent a fraction or ratio). When you get to physics, you will find in a lot of calculations that radians will seemingly appear or disappear from equations. Centripetal acceleration: $\displaystyle m\omega^2 r=(10 kg)(3 rad/s)^2(0.1 m)=9 kg\bullet m/s^2=9 N$ If you used degrees or revolutions, you would have to use $\displaystyle 2\pi$ or $\displaystyle \pi/180$ correction factors every time. Oh, and the 360° to a circle thing has been around for millennia. Not only is it an easy number to make fractions from, but it is also very close to the number of days in a year (before a bird played a game with the moon for some extra light, or something like that). Last edited by DarnItJimImAnEngineer; June 30th, 2019 at 01:27 PM. 
June 30th, 2019, 01:22 PM  #4 
Senior Member Joined: Oct 2015 From: Greece Posts: 138 Thanks: 8 
So what about my second question? How were degrees invented? Why 180 degrees are equal to 1 π radians?

June 30th, 2019, 01:28 PM  #5 
Senior Member Joined: Jun 2019 From: USA Posts: 314 Thanks: 165 
Added that to my answer above. 
June 30th, 2019, 06:09 PM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,691 Thanks: 2670 Math Focus: Mainly analysis and algebra  
June 30th, 2019, 06:23 PM  #7  
Math Team Joined: Jul 2011 From: Texas Posts: 3,044 Thanks: 1627  Quote:
http://mathforum.org/library/drmath/view/59075.html  

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angles, defined, distances, ratio, units 
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