For reasons you'll learn more about in trigonometry and calculus, it is generally helpful to have your angles be between 0° and 360°. But not all angles will fall within this interval. However, since "once around" takes you right back where you'd started, you can delete revolutions until you get down to an angle between 0° and 360°.
For instance, an angle of 370° is 10° more than "once around". If you subtract that extra "once around", you'll end up facing in the exact same direction as before, but your angle measure will be the more-manageable 370° – 360° = 10°.
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I can subtract 360's, or I can grab my calculator and do the division:
1275 ÷ 360 = 3.541666...
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:
3 × 360° = 1080°
How much is left after these three cycles are deducted?
1275° – 1080° = 195°
Then my answer is:
195°
If you're working "by hand", you can do the long division of 1275 by 360, getting a 3 across the top and a remainder of 195 at the bottom. This gives you the exact same result as the calculator method described above.
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?
7 × 2 = 14
I can subtract to find out how much is left over:
Then the corresponding reduced angle is:
radians.
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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:
360 – 17 = 343
Then the corresponding angle is:
343°
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:
radians.
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This works somewhat similarly to the previous examples. First I'll find how often 360° fits inside 3742°:
3742 ÷ 360 = 10.39444...
How much is just the ten cycles?
10 × 360 = 3600
How much then is left over?
3600 + (–3742) = –142
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:
11 × 360 = 3960
Now I'll add my negative angle to these eleven cycles:
–3742 + 3960 = 218
Then my corresponding angle is:
218°
This is where a calculator really comes in handy, because this number is just ridiculously large. I'll start by doing the division:
15736 ÷ 360 = 43.711...
So 360 fits into 15736 forty-three times, with a little left over. This gives me:
43 × 360 = 15480
The left-over portion is:
15736 – 15480 = 256
Then the corresponding reduced angle is:
256°
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