WEBVTT 00:00:07.108 --> 00:00:11.352 In our study of trigonometry, one of the first things we'll need to be comfortable with is drawing angles 00:00:11.354 --> 00:00:16.366 in standard position. We're going to start by learning what standard position is, and we'll go ahead and 00:00:16.368 --> 00:00:20.366 start with degree measure which is something we should be familiar with from geometry. 00:00:20.368 --> 00:00:29.868 So we're going to draw a coordinate system. An angle that's drawn in standard position will have its vertex 00:00:29.870 --> 00:00:38.639 at the origin, and it's initial side will be on the positive x-axis. Then we're going to rotate the positive x-axis 00:00:38.641 --> 00:00:50.648 whatever the size of the angle is, and this will be the terminal side of the angle. 00:00:50.650 --> 00:00:57.407 Notice that I've drawn this angle in a counter-clockwise direction. This angle, which we'll call theta, 00:00:57.409 --> 00:01:06.916 has a positive measure. If I start with my initial side on the x-axis and I rotate in a clockwise direction, 00:01:06.918 --> 00:01:12.426 then my angle will have a negative measure. So let's draw a couple of angles in standard position. 00:01:12.428 --> 00:01:30.184 We're going to start with 120 degrees. So this is 0 degrees. If I rotate the positive x-axis one complete revolution, 00:01:30.186 --> 00:01:38.949 how far around would that be? 360 degrees, right. Ok, so if I do half of a revolution it would be 180 degrees, 00:01:38.951 --> 00:01:45.454 and I think I'm going to label this for you on our original picture. So all the way around is 360 degrees. 00:01:45.456 --> 00:01:56.202 Halfway around would be 180, so a quarter of the way around 90 degrees. Ok, so this must be 270 degrees. 00:01:56.204 --> 00:02:05.454 That mental picture is going to help us to decide where to draw 120 degrees. So we start on the positive x-axis. 00:02:05.456 --> 00:02:16.454 If we go this far, we've gone 90 degrees. We need to go about 30 degrees more, so this is my angle theta, 00:02:16.456 --> 00:02:23.235 and notice how the terminal side lands in quadrant number two. Now let's try another angle. 00:02:23.237 --> 00:02:35.240 Let's go with 400 degrees. Ok, so what do you notice first? It's more than 360 degrees, right? 00:02:35.242 --> 00:02:43.240 So that means that I start my initial side here on the positive x-axis. I have to go 360 degrees takes me to here. 00:02:43.242 --> 00:02:56.026 How much is left to get to 400? Right, 40 more degrees. So this is 400 degrees drawn in standard position. 00:02:56.028 --> 00:03:08.285 Now let's try one that's a negative angle. Let's do theta equals negative 200 degrees. Negative angles go in 00:03:08.287 --> 00:03:17.796 we want to rotate the positive x-axis in a clockwise direction. If I go halfway around that's negative 180 degrees, 00:03:17.798 --> 00:03:22.799 so it needs to go just about 20 degrees more and would be about right here. 00:03:24.795 --> 00:03:33.562 Let's look at the angle theta equals negative 270, in degrees. So we start with the initial side along the positive 00:03:33.564 --> 00:03:45.325 x-axis, and we're going to rotate three-quarters of a revolution. And this is our angle negative 270 degrees. 00:03:45.327 --> 00:03:52.581 Notice that there are a lot of angles whose terminal side would be in exactly the same place. In particular, 00:03:52.583 --> 00:04:04.330 there's an easy one to spot. 90 degrees in the positive direction has its terminal side in exactly the same place. 00:04:04.332 --> 00:04:12.849 Are there other angles? Yes. For instance, suppose that we went in the negative direction completely around, 00:04:12.851 --> 00:04:23.850 that would be negative 360, and then went negative 270 more. That's going to be negative 270 minus the 360. 00:04:23.852 --> 00:04:34.602 That would be 630 degrees, negative. That angle, all the way around plus the negative 270 also has the same 00:04:34.604 --> 00:04:42.880 terminal side. Ok, and you could see, I could go around two complete negative revolutions and then stop at 00:04:42.882 --> 00:04:51.390 negative 270, and I'm going to get yet another angle with exactly the same terminal side. 00:04:51.392 --> 00:04:57.891 Don't like negative angles? We can go the other direction. Suppose that I start with, I have the 90, I could make 00:04:57.893 --> 00:05:11.642 a complete revolution all the way around. 360 degrees plus another 90. I'm going to stop in the same place. 00:05:11.644 --> 00:05:19.140 Or I can make 5 revolutions around and then 90 degrees more, and again I'm going to get a terminal side in 00:05:19.142 --> 00:05:30.429 exactly the same place. These are all examples of coterminal angles. They have exactly the same terminal side, 00:05:30.431 --> 00:05:35.430 the same starting point, but they have different measures. 00:05:35.432 --> 00:05:40.451 You need to practice finding coterminal angles to help us with our study of trig. 00:05:41.437 --> 00:05:48.438 Often it will be helpful to be able to find specific coterminal angles for a given angle. Let's look at some examples. 00:05:48.440 --> 00:05:54.937 We're asked to find two positive angles and two negative angles that are coterminal with a given angle. Let's start 00:05:54.939 --> 00:06:04.935 with 210 degrees. This angle is already positive. I know for sure that if I add 360 degrees it's adding one revolution 00:06:04.937 --> 00:06:18.726 on my angle. I will have the same terminal side, and it's still going to be positive. So this is 570 degrees, and this is 00:06:18.728 --> 00:06:28.234 a positive angle that is coterminal with 210 degrees. If I add another 360 to this, then I'll get one more revolution on 00:06:28.236 --> 00:06:43.986 my angle and a bigger angle that has the same terminal side. So this is 720 plus 210. It's going to be 930 degrees. 00:06:43.988 --> 00:06:51.487 And yes, you could have just added 360 to this one. You get to the same place. Well I have enough positive angles, 00:06:51.489 --> 00:07:03.269 so let's try to get some negative angles. What happens if I subtract 360 degrees from this angle? 00:07:03.271 --> 00:07:13.021 Ok, so I subtract this, it's negative 150 degrees. There's one negative angle that is coterminal with my given angle. 00:07:13.023 --> 00:07:21.520 How do I find a second one? Right, subtract another 360. You can subtract 360 from the negative 150, or you can 00:07:21.522 --> 00:07:38.808 210 degrees minus 2 times 360 will give you the same result. 00:07:38.810 --> 00:07:48.306 Ok. Let's look at negative 100 degrees. What happens if I add 360 to this? You remember coterminal angles just 00:07:48.308 --> 00:08:02.830 differ by multiples of 360 degrees. So let's take negative 100 degrees and add 360 degrees to it, 260 degrees. 00:08:02.832 --> 00:08:05.334 That's one of the positive angles that I need. 00:08:05.336 --> 00:08:24.605 Now let's get another positive angle by adding 360 degrees one more time, and you can see that if I do 00:08:24.607 --> 00:08:32.104 negative 100 degrees plus 3 times 360 I'm just going to get yet another bigger, positive angle and that's not 00:08:32.106 --> 00:08:36.853 what I'm asked to do. I need to look for some negative angles. So now, let's go back to our negative 100 00:08:36.855 --> 00:09:00.141 and subtract. We'll subtract 360 degrees one time, this is negative 460, and do that. Subtract 360 again, 00:09:00.143 --> 00:09:05.141 and I've satisfied the requirements of the problem. I've found two positive angles that are coterminal 00:09:05.143 --> 00:09:11.389 with negative 100 degrees and two negative angles. I'd like for you to put a little star by this one though because 00:09:11.391 --> 00:09:19.393 as it turns out this 260 degree angle is very important to us. It is the angle of least non-negative measure that is 00:09:19.395 --> 00:09:27.417 coterminal with the given angle. It's so important we're going to give this a name, theta c. 00:09:27.419 --> 00:09:35.416 Alright, we have one more example, 700 degrees. What do you notice about this one? It's bigger than 360. Ok. 00:09:35.418 --> 00:09:49.434 So certainly if I add multiples of 360, I'll get bigger, positive angles, and that's fine. But if I subtract 360, I'll get a 00:09:49.436 --> 00:10:02.449 positive angle there too. So let's go ahead and start with that. It's 340 degrees. So there's one positive angle 00:10:02.451 --> 00:10:17.966 coterminal with my given angle. Let's subtract 360 again, so we're subtracting two 360's. That's negative 20 degrees. 00:10:17.968 --> 00:10:33.734 Ok, so to get another angle you can see that I'm going to need to subtract another 360 from that, 00:10:33.736 --> 00:10:39.484 and I'm going the wrong directions now. Alright, the instructions were two positive angles and two negative angles 00:10:39.486 --> 00:10:45.738 that are coterminal with my given angle. So how am I going to find that other positive angle? So we can go back 00:10:45.740 --> 00:10:57.200 here to this 700 and add 360. 00:10:59.000 --> 00:11:06.755 Often we'd like to be able the find the angle of least non-negative measure that's coterminal with our given angle, 00:11:06.757 --> 00:11:14.518 and remember we've called this theta c. Let's try that for negative 765 degrees. The first thing we notice is we're 00:11:14.520 --> 00:11:21.769 looking for a non-negative measure, and we have a negative angle. Coterminal angles, remember, differ by 00:11:21.771 --> 00:11:30.534 multiples of 360 degrees, so we can simply start by adding 360 degrees to this and see what happens. 00:11:30.536 --> 00:11:43.549 What size angle do we get? Negative 405 degrees. That's more than one revolution, right, in the negative direction, 00:11:43.551 --> 00:11:55.799 so that's not enough. Let's take negative 765 degrees plus 2 times 360, so that would be negative 765 degrees 00:11:55.801 --> 00:12:10.551 plus 720 degrees. That's negative 45 degrees. Unfortunately, it's still negative, so we just go again. 00:12:10.553 --> 00:12:21.089 Ok, so negative 765 degrees plus 3 times 360 degrees. That's going to be 360 degrees more than this angle, 00:12:21.091 --> 00:12:31.838 and that would be 315 degrees. So this is theta c, the angle of least non-negative measure coterminal with our 00:12:31.840 --> 00:12:42.837 given angle. Alright, let's try this again for 1180 degrees. This is a positive angle. A really big positive angle, right? 00:12:42.839 --> 00:12:50.838 So we need to subtract multiples of 360. Now I could start with subtracting 360 degrees, but I can tell that's not 00:12:50.840 --> 00:13:04.087 nearly big enough. So let's try 1180 degrees minus 2 times 360, that's 720. Well that's only 460 degrees, and that's 00:13:04.089 --> 00:13:15.586 still more than one revolution, so that wasn't enough. Let's try 1180 degrees minus 3 times 360. That's going to be 00:13:15.588 --> 00:13:20.628 360 less than that. It's 100 degrees. And that will be our theta c, 00:13:20.630 --> 00:13:27.000 the angle of least non-negative measure coterminal with our given angle. 00:13:28.644 --> 00:13:34.394 We're all pretty comfortable with degree measure for angles, but in our study of trigonometry we'll find another 00:13:34.394 --> 00:13:41.922 measure for angles that's even more useful, and that's radian measure. So let's start by defining what a radian is. 00:13:41.922 --> 00:13:50.177 So we start with a circle of radius r, and we've drawn a central angle theta that intercepts an arc of length r. 00:13:50.177 --> 00:13:58.442 Alright so that means that the arc is the same length as the radius of the circle. In that case, theta is an angle 00:13:58.444 --> 00:14:05.197 of measure 1 radian. It would be helpful to know how many radians are in one complete revolution 00:14:05.199 --> 00:14:12.448 just like we know there are 360 degrees in a complete revolution. So how do we figure this out? 00:14:12.450 --> 00:14:19.709 One complete revolution is the same length as the circumference of the circle. Do you remember the formula 00:14:19.711 --> 00:14:30.956 for the circumference of a circle? Yes, 2 pi times the radius. Ok, so how many r's are there as you go around 00:14:30.958 --> 00:14:49.240 the circle? There are 2 pi r's as you go around the circle, so one revolution is 2 pi radians. Now, just like in degrees, 00:14:49.242 --> 00:14:55.498 you notice that if you want to know a quarter of a revolution, a half of a revolution, three quarters of a revolution, 00:14:55.498 --> 00:15:06.008 so let's write those down. One quarter of a revolution, we'll come back to that one. Let's do half of a revolution first. 00:15:06.010 --> 00:15:15.266 Half of a revolution is half of 2 pi. That one's easy, pi radians. So a quarter of a revolution is going to be half of that 00:15:15.266 --> 00:15:30.020 or pi over 2 radians, and three quarters of a revolution is 3 pi over 2 radians. I want you to get used to those on 00:15:30.022 --> 00:15:38.015 your coordinate system because our next step is to draw angles in standard position given a radian measure. 00:15:39.529 --> 00:15:46.278 Let's look at drawing angles in standard position when the angle is given in radian measure. 00:15:46.280 --> 00:15:54.803 So it will be helpful to remember that a quarter of a revolution is pi over 2 radians, half of a revolution is pi radians, 00:15:54.805 --> 00:16:01.802 and 3 pi over 2 radians. In standard position we start with the initial side on the positive x-axis, and 00:16:01.804 --> 00:16:10.057 counterclockwise is for positive angles, clockwise rotation for negative angles. That hasn't changed. 00:16:10.059 --> 00:16:26.830 Ok, so pi over 6. It's positive, that's a good start. Pi over 6, it's a lot smaller than pi over 2, right. So, in fact, 00:16:26.832 --> 00:16:42.100 it's a third of pi over 2, so it should be about a third of the way into the first quadrant. Now let's look at 2 pi over 3. 00:16:42.102 --> 00:16:49.860 Well, it's not quite all the way halfway around because halfway around would be 3 pi over 3. We haven't quite 00:16:49.862 --> 00:17:02.610 made it that far. In fact, I've made it two thirds of the way, so two thirds of the way, two thirds of halfway, would put 00:17:02.612 --> 00:17:19.141 2 pi over 3 approximately here. Ok, 9 pi over 4. Well 9 pi over 4, it's a really big angle compared to the other two 00:17:19.143 --> 00:17:27.636 we looked at. In fact, let's think of what if we went a complete revolution? We know that that is 2 pi radians, but 00:17:27.638 --> 00:17:38.911 how much is that in pi over 4s? How many pi over 4s would that be? It would be 8 pi over 4. So, keeping this in mind, 00:17:38.913 --> 00:17:49.917 when we look at the 9 pi over 4, it's 8 pi over 4 plus another pi over 4. That means it's a complete revolution 00:17:49.917 --> 00:17:59.999 in the positive direction plus another pi over 4. Now, that pi over 4 is half of pi over 2. 00:18:00.001 --> 00:18:09.680 So here's 9 pi over 4 in standard position. 00:18:09.682 --> 00:18:16.695 Negative 6 pi. So first of all I notice that I need to go in the negative direction. If I go around one complete revolution 00:18:16.697 --> 00:18:26.949 in the negative direction, that's negative 2 pi. This would make negative 4 pi, so negative 6 pi is three complete revolutions 00:18:26.951 --> 00:18:36.217 in the negative direction. So let's look at negative 7 pi over 12. It's going to go in the negative direction. And how far? 00:18:36.219 --> 00:18:44.725 Well, halfway around would be negative 12 pi over 12, so we're not nearly that far. Is it a quarter of the way around? 00:18:44.727 --> 00:18:56.485 So that would be negative 6 pi over 12. It's just a little bit more than that. 00:18:56.487 --> 00:19:02.481 Later on in this video, I will show you how to convert between radian and degree measure, but you'll notice that 00:19:02.483 --> 00:19:08.247 I did all of these without ever doing that conversion. It's in your best interest to learn to think in radians and 00:19:08.249 --> 00:19:15.498 be able to appropriately place an angle in standard position without having to convert everything back to degrees. 00:19:18.000 --> 00:19:25.772 Let's look at finding coterminal angles using angles given in radian measure. We've previously seen that 00:19:25.774 --> 00:19:34.039 coterminal angles differ by one the number of revolutions or the direction of those revolutions, so 00:19:34.041 --> 00:19:38.293 in degrees, we added or subtracted multiples of 360 to find coterminal angles. 00:19:38.295 --> 00:19:44.040 So what should we do when we're faced with angles in radian measure? 00:19:44.042 --> 00:19:53.057 Right, we're going to add or subtract multiples of 2 pi. So let's start with this angle, 3 pi over 4. It's already positive, 00:19:53.059 --> 00:20:02.065 so one way to get positive angles coterminal with it would be to add multiples of 2 pi. So we start with 3 pi over 4, 00:20:02.067 --> 00:20:08.564 and we add 2 pi. And we're going to need to think of this with, you know, a common denominator, so 2 pi is the same 00:20:08.566 --> 00:20:26.593 as 8 pi over 4, so this is 11 pi over 4. Ok, if we take 3 pi over 4 and we add, instead of 2 pi, we're going to add two 00:20:26.595 --> 00:20:39.356 2 pi's or 4 pi. So again we need to think in terms of a common denominator. This is 4 pi. How many fourths is that? 00:20:39.358 --> 00:20:52.859 16 pi fourths. Ok, so we add this, 19 pi over 4. That takes care of the two positive angles that I needed. Now let's look 00:20:52.861 --> 00:21:04.625 for some negative angles. We can take our 3 pi over 4, and let's try subtracting 2 pi and see what happens. 00:21:04.627 --> 00:21:14.386 Now remember we said 2 pi can be written as 8 pi over 4, so that's what we need to be able to do the subtraction. 00:21:14.388 --> 00:21:23.391 This gives me negative 5 pi over 4. So I subtracted 2 pi one time, and I have one of my negative angles that's 00:21:23.393 --> 00:21:40.399 coterminal with 3 pi over 4. Let's subtract another 2 pi. So that's 4 pi here again, that's the same as 16 pi over 4, 00:21:40.401 --> 00:21:52.424 and a second negative angle coterminal with 3 pi over 4 would be negative 13 pi over 4. Let's look at this next angle, 00:21:52.426 --> 00:22:05.939 negative 20 pi over 3. Ok, well it's already a negative angle. It's more than one revolution in the negative direction. 00:22:05.941 --> 00:22:12.190 Isn't it? So we're going to have to work a little bit harder on this one. Let's go ahead and start by trying to get to some 00:22:12.192 --> 00:22:23.216 positive angles. I'm going to add 2 pi, see what happens. Ok, now well this is thirds, so I want negative 20 pi over 3 00:22:23.218 --> 00:22:36.218 plus 6 pi over 3, so that's negative 14 pi over 3. That'll take care of one of those negative angles, and we're going to 00:22:36.220 --> 00:22:53.238 just keep adding multiples of 2 pi. Ok, so this is 4 pi. How can I write that with a denominator of 3? 00:22:53.240 --> 00:23:03.999 Right, 12 pi over 3. Ok, negative 20 pi over 3 plus 12 pi over 3 is negative 8 pi over 3. It's yet another negative angle 00:23:04.001 --> 00:23:20.261 that's coterminal with my given angle, and I'm still in search of positive angles. So we're going to add another 2 pi. 00:23:20.263 --> 00:23:33.021 Alright, this is 6 pi, so how many, how's that written with a denominator 3. Ok, 18 pi over 3. One thing you should 00:23:33.023 --> 00:23:39.534 notice at this point is that sometimes it's not obvious how far you have to go, but the more you practice with radians, 00:23:39.536 --> 00:23:44.035 right we're still looking for a positive angle and we're not there yet. The more you work with radians, the more you 00:23:44.037 --> 00:23:52.292 might be able to sort of skip some of these steps and go a little faster. Ok, it's getting better. This is less than one 00:23:52.294 --> 00:24:08.059 revolution, so the next time I add another 2 pi to that I should come up with a positive angle. So 8 pi. I'm going to 00:24:08.061 --> 00:24:19.559 have to write that with a denominator of 3. That's going to be 24 pi over 3. Ok, finally I've found a positive angle 00:24:19.561 --> 00:24:28.563 coterminal with the given angle. Remember earlier we talked about theta c, the angle of least non-negative measure 00:24:28.565 --> 00:24:34.559 that is coterminal with our given angle. And we're going to have to practice finding that using radian measures as well, 00:24:34.561 --> 00:24:54.605 so this is the first one we've seen. Ok, if I add 2 pi one more time, this is 10 pi, so written with the denominator of 00:24:54.607 --> 00:25:07.122 3 that's 30 pi over 3, and this is 10 pi over 3. So when I go to type in or write down my answers I found two positive 00:25:07.124 --> 00:25:17.625 angles that are coterminal with my given angle. And then I found three, so pick two of these and there are your answers. 00:25:19.124 --> 00:25:25.877 In this example we're asked to find the angle, theta c, of least non-negative measure that is coterminal with the given 00:25:25.879 --> 00:25:37.123 angle. So we'll start with negative 5 pi over 3. Since it's a negative angle, we're going to try adding 2 pi 00:25:37.125 --> 00:25:53.667 and see if that will give us our theta c. Ok, so 2 pi written with a denominator of 3 is 6 pi over 3. Negative 5 pi 00:25:53.667 --> 00:26:02.928 over 3 plus 6 pi over 3 is pi over 3. That's certainly a positive angle, and it's the smallest one that is coterminal with 00:26:02.930 --> 00:26:16.930 negative [5] pi over 3, so we can call this our theta c. Now let's look at 29 pi over 6. It's a positive angle, so we 00:26:16.932 --> 00:26:29.692 should probably start by trying to subtract some multiples of 2 pi. 2 pi is the same as 12 pi over 6. So just subtracting 00:26:29.694 --> 00:26:40.188 2 pi one time won't do it. In fact, I should be able to subtract it twice. Let's see where that gets us. 29 pi over 6 minus 00:26:40.190 --> 00:27:02.987 24 pi over 6, remember this is 2 pi twice or 4 pi. So when I do the subtraction, I get 5 pi over 6. 00:27:02.989 --> 00:27:11.487 That is a positive angle. It's less than one revolution. In fact, it is the smallest non-negative angle coterminal with 00:27:11.489 --> 00:27:17.480 the given angle, so let's label that theta c. 00:27:18.840 --> 00:27:25.087 Let's look at how we can convert between radian measure and degree measure. So previously we've said that 00:27:25.089 --> 00:27:43.090 one complete revolution is 360 degrees. We've also said that one revolution is 2 pi radians. So that means that 00:27:43.092 --> 00:27:54.833 2 pi radians equals 360 degrees, or if we divide both sides by 2, we find that pi radians is 180 degrees. Notice that 00:27:54.835 --> 00:28:01.590 on the pi I did not put any sort of symbol. That's because typically radians don't have a symbol, and it's understood 00:28:01.592 --> 00:28:06.376 to be radians if it's not written there. Therefore, it's very important if your angle is in degrees that you actually use that 00:28:06.378 --> 00:28:14.000 degree symbol. Ok, this is all we need to know to be able to do the conversions, so let's look at some examples. 00:28:15.376 --> 00:28:25.900 Let's convert each of the following from degree measure to radian measure. We'll start with 30 degrees. Since 00:28:25.902 --> 00:28:35.656 pi radians is 180 degrees, we can multiply by the fraction pi over 180 degrees. This is equal to 1, so we're not changing 00:28:35.658 --> 00:28:44.157 the value. And the degree units will sort of go away and leave us with radian units. We want to reduce the fraction. 00:28:44.159 --> 00:29:02.657 30 goes into 180 six times, so this is pi over 6 radians. Negative 240 degrees. So again, pi radians is equal to 180 00:29:02.659 --> 00:29:16.201 degrees, so we can write this ratio, multiply. It's a little bit trickier to simplify this one. This is negative 240 pi over 180, 00:29:16.203 --> 00:29:30.198 and you need to reduce that to lowest terms. So, let's see. You divide by 6. That would be negative 4 pi over 3. 00:29:30.200 --> 00:29:40.222 The last one is 160 degrees. So again, we're converting degrees to radians. We're going to multiply by pi over 00:29:40.224 --> 00:29:48.980 180 degrees. We can divide the numerator and the denominator both by 10 to get 16 pi over 18, and then 00:29:48.982 --> 00:29:55.233 of course we can reduce that a little bit further to get 8 pi over 9 radians. 00:29:58.000 --> 00:30:04.249 Let's convert each of the following to degree measure. This is 13 pi over 6 radians, and I know it's radians 00:30:04.251 --> 00:30:14.256 because no unit is given and I'm told to go to degree measure. Ok, so what should I multiply by? 00:30:14.258 --> 00:30:21.765 It's still true that pi radians is 180 degrees, but if I multiply by pi over 180 degrees, it won't make any sense. Will it? 00:30:21.767 --> 00:30:28.518 Because I won't end up with the units of degrees that I want in the end. So instead we want to multiply by 00:30:28.520 --> 00:30:38.200 180 degrees over pi. We're going to end up with degrees in our answer. Alright, the nice thing is that our pi's divide out. 00:30:38.202 --> 00:30:48.789 And what do we have left? Well, we have to multiply 13 times 180 and then divide it by 6. Ok, if you have your 00:30:48.791 --> 00:30:54.547 calculator this is very easy to do, and you get a nice number. You're going to get 390 degrees. Don't forget the 00:30:54.549 --> 00:31:07.559 units on your answer. Let's try negative 5 pi over 8. Again we're trying to convert to degrees, so we multiply by 00:31:07.561 --> 00:31:23.560 180 degrees over pi radians. So this is negative 5 times 180 degrees divided by 8. You want to reduce this as far 00:31:23.562 --> 00:31:29.313 as you can. You want a nice, reasonable number, but in this case if you divide it on your calculator, it's exactly equal 00:31:29.315 --> 00:31:40.098 to negative 112.5 degrees. In our last example, the angle is 2.4 radians. Notice that it doesn't have a pi in it. 00:31:40.100 --> 00:31:51.853 Just because it's in radians doesn't mean there will be a pi in the angle. The procedure is the same. Multiply by 00:31:51.855 --> 00:32:02.607 180 degrees over pi. Then, you want your answer to be in degrees, so be careful on this one that you don't get 00:32:02.609 --> 00:32:11.377 confused without the pi in there. And here you'll have to use your calculator. Multiply 2.4 times 180 degrees, 00:32:11.379 --> 00:32:23.124 divide by pi, and then round off to one decimal place or as required. It's about 137.5 degrees. 00:32:23.126 --> 00:32:30.372 Now, you've learned a lot of things in this video about drawing angles in standard position. You've learned about 00:32:30.374 --> 00:32:37.650 radian measure, and you've learned about converting between the two. You need to practice finding and 00:32:37.652 --> 00:32:42.903 drawing angles in standard position, practice finding coterminal angles, and while it's important to be able to 00:32:42.905 --> 00:32:47.907 convert your degrees and radians, don't use that as a crutch and learn to think in radians. 00:32:47.909 --> 00:32:51.411 It will help you throughout the remainder of your trigonometry course.