Reducing Angle Measures

I can subtract 360's, or I can grab my calculator and do the division:

The "3" tells me that 360° fits into 1275° three times. The 0.541666... is the part that's left over. This will be my reduced-angle measure. But how many degrees is this? First, I'll find out how many degrees are taken care of in the first three cycles:

How much is left after these three cycles are deducted?

Then my answer is:

I need to figure out how many cycles of 2π fit in . Grabbing my calculator, I do a quick division:

So I see that there are seven cycles contained in this angle measure, plus another 0.8333...π radians more. How much is seven cycles?

I can subtract to find out how much is left over:

Then the corresponding reduced angle is:

A negative angle is one that went around "backwards": instead of rotating the "right" way, they went around the "wrong" way. But I can find the corresponding angle by going back around the "right" way or, which is the same thing for such a small angle, subtracting (the positive size of) the negative angle from "once around":

The corresponding angle measure is found by:

Then the corresponding angle is:

This one works just like the previous one, but in radians. So I'll work in terms of 2π radians, instead of in terms of 360°.

Then the corresponding angle is:

This works somewhat similarly to the previous examples. First I'll find how often 360° fits inside 3742°:

How much is just the ten cycles?

How much then is left over?

Um... that doesn't look right. What did I do wrong?

Oh, yes! This angle was negative, so I actually need one extra "once around" to carry me into the positive angle values (just as in the previous two examples, when I added the negative angles to the value of one cycle in the right direction). So I need to use 11 cycle instead of only 10:

Now I'll add my negative angle to these eleven cycles:

Then my corresponding angle is:

This is where a calculator really comes in handy, because this number is just ridiculously large. I'll start by doing the division:

So 360 fits into 15736 forty-three times, with a little left over. This gives me:

The left-over portion is:

Then the corresponding reduced angle is: