 Welcome back to our lecture series, Math 1060 Trigonometry for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. In this first part of lecture seven, I want to introduce the idea of radiant measure and show that trigonometry, which literally translates as the study of triangles, trigonometry involves circles just as much as they did the right triangles, if not even more. And in this lecture, we really want to start understanding what do circles in trigonometry have to do with each other. So imagine we do have a circle. So I'm going to draw one on the screen. It'll probably look hideous, not perfectly circular. But you get the idea here. Consider the center of the circle. We say that an angle associated to a circle is central if the angle has as its vertex the center of the circle. So this would be an example of what we call a central angle of this circle. And for convenience, we like to think of this angle in standard position so that the initial side right here is the positive x-axis. And then the terminal side will terminate wherever it so chooses. So in a circle, a central angle that cuts off an arc, so this is what we mean by an arc. So this little piece on the circle itself that's kind of cut off by the angle, right? This is what we mean by an arc. So we say that the arc is equal in length to the radius of the circle. If the arc is equal to the radius of the circle, so if the distance of the radius, if that's the same distance right here, and I know this gets kind of funky because it's curved, but imagine we have a shoestring that we lay across the circles, we measure the radius, and then we also measure that same shoestring along the curve, and it's the same length, that's what we call one radian. Because after all, when you have the arc of the circle, it's also associated to the angle theta. And so one radian, so to speak, one radian, this happens, what this means here is that the arc, the arc has a length equal to the radius of the circle arc. Now, if you consider the entire arc, that is, if you take the circumference of the circle, then it's been known for millennia that the circumference of a circle is directly proportional to its diameter, you know, if you go all the way across here. Now the diameter of course is just twice the radius, and it's known that the proportion between the circumference and the diameter, if you take the circumference of a circle and divided by its diameter, this is always equal to this number pi, which pi is an irrational number approximately equal to 3.14159. It keeps on going, right? Cause this is a non-repeating decimal expansion. Therefore, if you solve for the circumference, we get a formula for the circumference of a circle, circumference is equal to two pi r, recognizing the diameter of the circle is just two times its radius right there. And so this is a very important formula for computing the circumference of a circle, but we're gonna use this to connect this idea of radiance with the degree measure that we're already used to, okay? It's not just the circumference and the diameter that are proportional to each other. Because of the symmetry of the circle, it turns out that the angle measure right here is gonna be proportional to the arc length of the circle, where that proportion is always gonna equal r. So in other words, if we take any arc, s, let's say that this thing has length s, if we take any arc, s, and we divide it by its radian measure, this is always gonna equal r, the radius of the circle, they're always gonna be proportional to each other. So if we take the entire circle, right, the entire circle, it has an arc length of two pi r at circumference. If we take the angle that gives us one complete rotation, call theta for a moment, then by the proportionality of the circle, we see that two pi r divided by theta will always equal r. We'll take this equation times both sides, excuse me, divide both sides by r times both sides by theta. So we're gonna times by theta over r. We see the r's cancel, the theta's cancel, and then we get this equation right here, theta equals two pi r over r. But since there's an arc on top and bottom, they cancel out and we end up with theta equals two pi. So what this tells us right here is this gives us a baseline, one complete rotation, one complete revolution of a circle in the radian measure is equal to two pi, okay? But then that gives us a way of connecting it to degrees because as we learned previously, the degree measure of a one complete revolution is 360 degrees. So two pi radians coincides with 360 degrees. If we take this equation and divide both sides by 180, excuse me, by 360, we divide both sides by 360, then the left-hand side would just become one degree, and then the right-hand side, two goes into 360, 180 times, that's where the 180 came from. And we see that one degree is equal to pi over 180 radians. Likewise, if we take this equation right here and divide both sides by two pi, then the left-hand side would become one radian and then two goes into 360, 180 times. So one radian is equal to 180 degrees over pi. So this gives us a way of converting back and forth between radians and degrees. This formula right here shows us how we convert from degrees into radians because you change one radian, a one degree to be a pi over 180 radians. And then likewise, this one over here tells us how to change from radians to degrees because you can replace one radian with 180 over pi degrees. Let's see a quick example of such a thing. So if we take, for example, 45 degrees, 45 degrees means 45 times one degree. And by the conversion chart we saw in the previous slide, one degree is the same thing as pi over 180. So 45 goes into 184 times, this would simplify to be pi over four. So 45 degrees is the same thing as pi over four radians. Similarly, if we take 450 degrees, 450 degrees right here, this is just gonna be 450 times one degree. One degree is pi over 180 radians. And so then we have this fraction, 450 over 180. You can divide both sides, both are divisible by 10. You can see the zero, so kind of take that out. You get 45 over 18. They're still divisible by nine. So you can simplify the fraction to be five pi over two radians. So this shows us how to convert from degrees to radians. If we wanna convert from radians to degrees, we do the same thing basically, double times by the reciprocal conversion factor. So if you take pi over six here, pi over six radians, one thing I should mention is that if the angle measure is not specified, if the units are not specified, by default that will mean radians. So you always need to put the little circle if you wanna represent degrees, but if you write nothing, it should be assumed that it's a radian measure. So pi over six radians means pi over six times one radian, but one radian is 180 degrees over pi. The pi's will always cancel out, and then six goes into 180, that'll go in there 30 times. So pi over six coincides with 30 degrees. Likewise, if we take four pi thirds, all right? Four pi thirds, if you multiply by 180 degree over pi, then the pi should cancel out. You get four times 180 over three. Three goes into 180 60 times. So you get four times 60, which is 240 degrees. A nice little thing to remember is when it comes to conversion, you're either gonna times by pi over 180, or you're gonna get 180 over pi. That's always the factor you use here. And to remember which one you want, the radian measures we have, typically have a pi in it, and we want the pi's to cancel out when you go to degrees. So you want, when you're going from radians to degrees, the pi should probably be in the denominator, so it cancels out with the pi that's in the numerator so that your degree measure doesn't have a pi in it whatsoever. Now, if we take some of those special angles that we've learned about previously in this lecture series, we can find the corresponding radian measure. So like zero degrees coincides with zero radians. 30 degrees coincides with pi over six that we saw right here. 45 degrees coincides with pi over four that we saw right here as well. 60 degrees is the same thing as pi third radians. 90 degrees is pi over two. 180 is pi radians. 270 degrees is three pi over two. And then 360 that started this whole conversation coincides with two pi radians.