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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_2019-06-30_20-31-38.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. Thanks from babaliaris
 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). Thanks from topsquark and babaliaris 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. Thanks from babaliaris
June 30th, 2019, 06:09 PM   #6
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Quote:
 Originally Posted by babaliaris Also, how did mathematicians said that π rads=180 degrees?
Radians turn out to be natural units in the same way as the natural logarithm is the natural base for logarithms. They are the unit that makes much mathematics easier.

June 30th, 2019, 06:23 PM   #7
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Quote:
 Originally Posted by babaliaris So what about my second question? How were degrees invented? Why 180 degrees are equal to 1 π radians?

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