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.
Best Answer
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.