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June 30th, 2019, 09:44 AM   #1
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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?

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June 30th, 2019, 12:19 PM   #2
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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.
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June 30th, 2019, 01:13 PM   #3
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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).
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Last edited by DarnItJimImAnEngineer; June 30th, 2019 at 01:27 PM.
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June 30th, 2019, 01:22 PM   #4
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So what about my second question? How were degrees invented? Why 180 degrees are equal to 1 π radians?
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June 30th, 2019, 01:28 PM   #5
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Added that to my answer above.
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June 30th, 2019, 06:09 PM   #6
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Originally Posted by babaliaris View Post
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.
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June 30th, 2019, 06:23 PM   #7
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Quote:
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So what about my second question? How were degrees invented? Why 180 degrees are equal to 1 π radians?

http://mathforum.org/library/drmath/view/59075.html
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