Let's see if we can give ourselves a little bit of practice converting between radians and degrees and degrees and radians. And just as a review, let's just remind ourselves of a relationship. And I always do this before I have to convert between the two. If I do one revolution of a circle, how many radians is that going to be?
Well, we know one revolution of a circle is 2 pi radians. And How many degrees is that if I do one revolution around a circle? Well, we know that that's 360. I can either write it with a little degree symbol right like that, or I could write it just like that. And this is really enough information for us to think about how to convert between radians and degrees.
If we want to simplify this a little bit, we can divide both sides by 2. two, and you could have pi radians are equal to 180 degrees. Or another way to think about it, going halfway around a circle in radians is pi radians. Or you've the arc. that subtends that angle is pi radiuses, and that's also 180 degrees.
And if you want to really think about, well, how many degrees are there per radian, you can divide both sides of this by pi. So if you divide both sides of this by pi, you get one radian. It would have to go from plural to singular.
One radian is equal to 180 over pi degrees. So all I did is I divided. divided both sides by pi.
And if you wanted to figure out how many radians are there per degree, you could divide both sides by 180. So you'd get pi over 180 radians is equal to 1 degree. So now I think we are ready to start converting. So let's convert 30 degrees to radians. So let's think about it.
So I'm going to write it out. And actually, this might remind you of kind of unit analysis that you might do when you first did unit conversion. But it also works out here. So if I were to write 30 degrees, and this is how my brain likes to work with it. I like to write out the word degrees.
And then I say, well, I want to convert to radians. So I really want to figure out how many radians are there per degree. So let me write this down. I want to figure out how many radians do we have per degree.
How many radians? do we have per degree? And I haven't filled out how many that is, but we see just the units will cancel out.
If we have degrees times radians per degree, the degrees will cancel out and I'll be just left with radians. If I multiply the number of degrees I have times the number of radians per degree, we're going to get radians. And hopefully that makes intuitive sense as well.
And here we just have to think about, well, if I have, think of it this way, if I have pi radians, pi radians, How many degrees is that? Well, that's 180 degrees. Come straight out of this right over here.
Pi radians for every 180 degrees, or pi over 180 radians per degree. But then this is going to get us to, we're going to get 30 times pi over 180, which will simplify to 30 over 180 is 1 over 6. So this is equal to pi over over 6, actually let me write the units out. This is 30 radians, which is equal to pi over 6 radians. Now let's go the other way.
Let's think about if we have pi over 3 radians. And I want to convert that to degrees. So what am I going to get if I convert that to degrees? Well, here we're going to want to figure out how many degrees are there? Huh?
How many degrees are there per radian? And one way to think about it is, well, think about the pi and the 180. For every 180 degrees, you have pi radians. 180 degrees over pi radians, these are essentially the equivalent thing.
Essentially, you're just multiplying this quantity by 1, but you're changing the units. The radians cancel out, and then the pi's cancel out, and you're left with 180 over 3 degrees. So 180 over 3 is 60, and we can We could either write out the word degrees, or you can write degrees just like that. Now let's think about 45 degrees.
So what about 45 degrees? And I'll write it like that just so you can figure it out with that notation as well. How many radians will this be equal to?
Well, once again, we're going to want to think about how many radians do we have per degree. So we're going to multiply. This times, well, we know we have pi radians for every 180 degrees. Or we could even write it this way.
Pi radians for every 180 degrees. And here, this might be a little less intuitive, the degrees cancel out. And that's why I'd like to usually write out the word. we're left with 45 pi over 180 radians. Actually, let me write this with the words written out.
For me, that's more intuitive when I'm thinking about it in terms of using the notation. So 45 degrees times. We have pi radians for every 180 degrees. So we are left with, when you multiply, 45 times pi over 180. The degrees have canceled out, and you're just left with the radians. Which is equal to what?
45 is half of 90, which is half of 180. So this is 1. 1 fourth, so this is equal to pi over 4 radians. Let's do one more over here. So let's say that we had negative pi over 2 radians.
What's that going to be in degrees? Well, once again, we have to figure out how many degrees are each of these radians. We know that there are 180 degrees.
degrees for every pi radians. So we're going to get the radians cancel out, the pi's cancel out. And so you have negative 180 over 2. This is negative 90 degrees. Or we could write it as negative 90 degrees.
Anyway, hopefully you found that helpful. And I'll do a couple more example problems here. Because the more example for this, the better. And hopefully it'll become a little bit easier.
bit intuitive.