Why Radians oppose to degrees?
Matthew Harrington 
Degrees seem to be so much easier to work with and more useful than something with a $\pi $ in it. If I say $33\:^\circ $ everyone will be able to immediately approximate the angle, because its easy to visualize 30 degrees from a right angle(split right angle into three equal parts), but if I tell someone $\frac{\pi }{6}$ radians or .5236 radians...I am pretty sure only math majors will tell you how much the angle will approximately be.
Note: When I say approximately be, I mean draw two lines connected by that angle without using a protractor.
Speaking of protractor. If someone were to measure an angle with a protractor, they would use degrees; I haven't seen a protractor with radians because it doesn't seem intuitive.
So my question is what are the advantages of using degrees? It seems highly counterproductive? I am sure there are advantages to it, so I would love to hear some.
PS: I am taking College Calc 1 and its the first time I have been introduced to radians. All of high school I simply used degrees.
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$\begingroup$Radians are a dimensionless measure; being initially defined as the arc length of a circle circumscribed by the angle divided by the radius of that circle.
This comes into play when dealing with derivatives of trigonometric functions.
$$\dfrac{\mathrm d \sin(x)}{\mathrm d x} = \cos (x)$$
versus:
$$\dfrac{\mathrm d \sin (x^\circ)}{\mathrm d x} = \dfrac{\pi \cos(x^\circ)}{180^\circ}$$
It's a lot more convenient to use radian measures in calculus.
$\endgroup$ 5 $\begingroup$The radian is the standard unit of angular measure, and is often used in many areas of mathematics. Recall that $C = 2\pi r$. If we let $r =1$, then we get $C = 2\pi$. We are essentially expressing the angle in terms of the length of a corresponding arc of a unit circle, instead of arbitrarily dividing it into $360$ degrees.
The reason that you believe degrees to be a more intuitive way of expressing the measure of an angle, is simply because this has been the way you have been exposed to them up until this point in your education. In calculus and other branches of mathematics aside from geometry, angles are universally measured in radians. This is because radians have a mathematical "naturalness" that leads to a more elegant formulation of a number of important results. Trigonometric functions for instance, are simple and elegant when expressed in radians.
For more information: Advantages of Measuring in Radians, Degrees vs. Radians
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