Radians
Why do we have to learn radians, when we already have perfectly good degrees? Because degrees,
technically speaking, are not actually numbers, and we can only do math with numbers. This is somewhat
similar to the difference between decimals and percentages. Yes, "83%" has a clear meaning, but to do
mathematical computations, you first must convert to the equivalent decimal form, 0.83. Something
similar is going on here (which will make more sense as you progress further into calculus, etc).
The 360° for one revolution ("once around") is messy enough. Why is the value for one revolution in
radians the irrational value 2π? Because this value makes the math work out right. You know that the
circumference C of a circle with radius r is given by C = 2πr. If r = 1, then C = 2π. For reasons you'll
learn later, mathematicians like to work with the "unit" circle, being the circle with r = 1. For the math to
make sense, the "numerical" value corresponding to 360° needed to be defined as (that is, needed to be
invented having the property of) "2π is the numerical value of 'once around'."
Converting Between Radians and Degrees
Each of radians and degrees has its place. If you're describing directions to me, I'd really rather you said,
"Turn sixty degrees to the right when you pass the orange mailbox", rather than, "Turn one-third πradians"
at that point. but if I need to find the area of a sector of a circle, I'd rather you gave me the numerical
radian measure that I can plug directly into the formula, rather than the degree measure that I'd have to
convert first.
But you won't always be given angle measures in the form you'd prefer, so you'll need to be able to
convert between radians and degrees. To do this, you'll use the fact that 360° is "once around", and so
is 2π. However, you'll use this fact in the form of the somewhat simplified correspondence of 180° to π.

Convert 270° to radians.
Since 180° equates to π, then:
The equivalent angle is

Convert
radians to degrees.
The equivalent angle is 30°
Note that the way I used the correspondence varied with what I was given. If I needed to end up with
radians, I put π on top; if I needed to end up with degrees, I put 180° on top. That's all there is to that .
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°.

Find the angle between 0° and 360° that corresponds to 1275°.
I can subtract 360's, or I can grab my calculator and do the division: 1275 ÷ 360 = 3.541666...
The only part I care about is the "3", which tells me that 360° fits into 1275° three times:
1275° – 3×360° = 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. Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved

Find the angle between 0 radians and 2π radians that corresponds to
I need to figure out how many cycles of
2π fit in
:
So I see that there are seven times of "one around", with
The corresponding angle is

left over.
radians.
Find the angle between 0° and 360° that corresponds to –17°.
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 negative angle from
"once around":
The corresponding angle is 360 – 17 = 343°

Find the angle between 0 radians and 2π radians that corresponds to
This one works just like the previous one, but in radians. So I'll work in terms of
instead of in terms of 360°.
The corresponding angle is

2π radians,
radians.
Find an angle between 0° and 360° that corresponds to –3742°.
This works somewhat similarly to the previous examples. First I'll find how often 360° fits inside
3742°:
3742 ÷ 360 = 10.39444...
But this angle was negative, so I actually need one extra "once around" to carry me into the
positive angle values, so I'll use 11 instead of 10:
–3742 + 11 × 360 = –3742 + 3960 = 218
The corresponding angle is 218°

Find the angle between 0° and 360° that corresponds to 15736°.
This is where a calculator really comes in handy, because this number is just ridiculously large.
So I'll do 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
15736 – 15480 = 256
The corresponding angle is 256°