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Q: Why radians?
When you first start doing trigonometry the choice between radians, degrees, turns, or hexacontades is a matter of personal preference. But when you get to calculus using radians is the most natural choice; anything else is just a headache waiting to happen. When you use radians you’re describing the angle by using the length of the arc it traces out on the edge of the unit circle. The circumference of a circle or radius R is 2πR, so (since R=1 on the unit circle) the full circle is 2π radians around. If it weren’t for the fact that (when using radians) we wouldn’t have .
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