 Hello, and welcome back. As we continue our study of trigonometry, we're going to be dealing with angles more and more, and we're going to have, we have two ways to measure angles. One is degree measure, where we basically take one complete revolution and divide it into 360 equal parts, each of which is a degree. And so, one revolution is 360 degrees. The other one is radian measure. And for that, we basically work with arc length on a unit circle, and when we go one complete revolution, the arc length is then 2 pi radians. And so, one revolution is 2 pi radians. This is kind of a typical picture of what you'll see, kind of a circle divided up into angles in both degrees and radians, and this can look overwhelming. We always should be able to construct this type of picture and so forth, but it's not one that we want to work with all the time or have to be able to reproduce whenever we want to convert from degrees to radians or vice versa. So, what I like to focus on is the idea of one complete revolution if we start here at zero and go counterclockwise, although it's not written here, when we get all the way back around, this is 360 degrees, or in radian measure, 2 pi radians. And that's kind of defining conversion on it, and actually what I really like to focus on is going a half a revolution like that in on this right here. So, 180 degrees is basically the same thing as pi radians. The other important ones to always keep in mind are the ones that kind of define the quadrants if we consider this to be the origin, and the usual x-axis is horizontal, and the y-axis vertical, 90 degrees and pi over 2 radians is a quarter of a circle, a quarter of a revolution, and as we go half a revolution, we get to this one, and then again, if we go 3 quarters of a revolution, we get to this one. Sometimes those are called quadrantial angles, kind of a mouthful, 90 degrees, 180 degrees, 270 and 360 degrees, or pi over 2 radians, 3 pi over 2 radians, or 2 pi radians. So, that really, the formula is that we can use to make our conversions between radians and degrees, and this is probably the one we focus on most often. This is another way of writing that equation, but actually the way I like to work with this equation here is to kind of think of this and take that 180 degrees, and if you want to divide both sides of this, I'm going to shortcut this a little bit, but if you take pi radians and divide it by 180 degrees, then for this, I'm going to write out dEg for degrees, and the other side of the equation, 180 degrees over 180 degrees, you can see what we get is pi radians over 180 degrees is equal to 1, and that's a real nice way to deal with it, because we know if we multiply by 1, we don't change the value. The other way to look at this then, to do other conversions is to take this equation, and again, think of it as a reciprocal, and you get 180 degrees over pi radians is also equal to 1. So when I do angle conversions, these are the two that I tend to work with, and it really helps me keep straight. Do I multiply by pi? Do I divide by 180? Which one do I do? And you don't have to try to memorize any special rule. What you want to remember are these two things. So let's try a couple of examples. A couple of converting radians to degrees, and so what we're going to do, we'll take with a 5 pi over 8 radians, and what we do is we multiply by the number 1, but we multiply it by what I call one of these conversion fractions, and so what I'm going to do is make sure that the radians, this radian here is in the numerator, so what I'm going to do is put pi radians here in the denominator in 180 degrees in the numerator, and that's just multiplying by 1. So I haven't changed the value, and in fact I can look at this and say, in the essence, the radians cancel, and in this case too, the pi's cancel, and so what we end up with is 5 8's times 180 degrees, and now I might go back to the more usual notation for degrees, and now what we have to do is carry out that multiplication, and we can use our calculator to do that. Really take 5 times 180, and then divide that by 8, and what you end up with is a 112.5 degrees. So 5 pi over 8 radians is the same angle as 112.5 degrees, and that's basically the procedure to convert radians to degrees. So whatever one you have, that's kind of how we work with, so if I take this minus 2.5, radians, I do exactly the same thing. Take this fraction equal to 1 pi radians, and I put that in the denominator so that the radians cancel 180 degrees. So again, we can think of radians as cancel. Notice in this case we don't have another pi to cancel, so what we end up with is minus 2.5 times 180 degrees divided by pi, and at this point we really do have to use our calculator. We could leave this in this form or maybe multiply the negative 2.5 times 180, which I believe would be minus 450, and have it minus 450 over pi. That would be in degrees, but usually it's nicer to have some form of decimal approximation get a better sense of the angle. So carry out that on your calculator, minus 450 divided by pi, and you'll get minus 143.24 degrees to the nearest hundredth of a degree. So there's two examples of converting radian measure to degrees, and here's kind of a summary showing how we did this, and again notice the use of this conversion fraction in both cases, and how it's set up so that the radians cancel, and then we just have to do the computations and so forth, and again in the first example these pi's canceled, but that's a small part of that. We basically have to do that computation, and we get 112.5 degrees, and down here we get minus 143.24 degrees. The process for converting degrees to radians is basically the same, except instead of using this particular conversion fraction we use its reciprocal. So here's an example of that. So if I take the 72 degrees, and I'm going to write out DEG, it helps me see the units better, and multiply that by the conversion fraction, and what I want to do is put degrees down here and radians up here, that's so that the degrees will cancel, and that pretty much sets things up for me, because I know that pi radians is 180 degrees. And so what we end up with in this case is 72 pi over 180, and that will be in radians. And there's nothing wrong with that answer. Most often we do like to reduce that fraction to lowest terms, and we'll kind of show maybe how to do that in just a minute, but what it does come out equal to is 2 pi over 5 radians, and we now have converted an angle of 72 degrees to be equivalent to an angle of 2 pi over 5 radians. This fraction, 72 over 180, there are a couple ways you can handle that. Notice I'm leaving off the pi here, I'm just going to leave the pi in the answer. What I'm focusing on is that fraction, 72 over 180. And we can go through reducing that fraction. Another way, if you have difficulty reducing fractions, just do. On your calculator, 72 divided by 180, and then use the convert to fraction function on your calculator, and it'll come up to be, and use that, and you'll see it is two-fifths. One thing we can do here, again, is we can see that each of 72 and 180 is divisible by 2. So we get 36 over 90. And there are many ways to proceed from here. Which is to kind of look at that and say, oh yeah, I can divide both of those by 6. And we get 6 15ths. And finally, we can see that both of those are divisible by 3. And so we get 2 times 3 and 5 times 3. And you can see the threes will cancel and you'll get two-fifths. So there are several different ways to do that. And 20 degrees is handled in exactly the same way. We just end up with a slightly different fraction to deal with. But again, my first thought I do is, oh, degree in the denominator so that they cancel. Radians then go in the numerator. And pi radians is the same as 180 degrees. And so now what we get is 320 pi over 180 radians. Perhaps this one you can easily see that both nominator and denominator are divisible by 10. And we can quickly change that to 32 over 18 pi radians. And reduce that fraction a little further. Both 32 and 18 are divisible by 2. And if we do that, we'll get 16 pi over 9 radians. And that's as far as we can go with reducing that fraction. So there you have two examples. Again, very typical examples. The same procedure should work anytime you want to convert degrees to radians. A summary of our work with these conversions. And again, notice the use of this particular conversion fraction. And again, the reason we put the degrees in the denominator so they would cancel, the degrees there and the answer would be left as radians. And that's what happened in both cases. So there you have it. And good luck with this and the rest of the trigonometry course. See you later.