WEBVTT
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In our study of trigonometry, one of the first things we'll need to be comfortable with is drawing angles
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in standard position. We're going to start by learning what standard position is, and we'll go ahead and
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start with degree measure which is something we should be familiar with from geometry.
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So we're going to draw a coordinate system. An angle that's drawn in standard position will have its vertex
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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
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whatever the size of the angle is, and this will be the terminal side of the angle.
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Notice that I've drawn this angle in a counter-clockwise direction. This angle, which we'll call theta,
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has a positive measure. If I start with my initial side on the x-axis and I rotate in a clockwise direction,
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then my angle will have a negative measure. So let's draw a couple of angles in standard position.
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We're going to start with 120 degrees. So this is 0 degrees. If I rotate the positive x-axis one complete revolution,
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how far around would that be? 360 degrees, right. Ok, so if I do half of a revolution it would be 180 degrees,
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and I think I'm going to label this for you on our original picture. So all the way around is 360 degrees.
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Halfway around would be 180, so a quarter of the way around 90 degrees. Ok, so this must be 270 degrees.
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That mental picture is going to help us to decide where to draw 120 degrees. So we start on the positive x-axis.
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If we go this far, we've gone 90 degrees. We need to go about 30 degrees more, so this is my angle theta,
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and notice how the terminal side lands in quadrant number two. Now let's try another angle.
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Let's go with 400 degrees. Ok, so what do you notice first? It's more than 360 degrees, right?
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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.
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How much is left to get to 400? Right, 40 more degrees. So this is 400 degrees drawn in standard position.
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Now let's try one that's a negative angle. Let's do theta equals negative 200 degrees. Negative angles go in
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we want to rotate the positive x-axis in a clockwise direction. If I go halfway around that's negative 180 degrees,
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so it needs to go just about 20 degrees more and would be about right here.
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Let's look at the angle theta equals negative 270, in degrees. So we start with the initial side along the positive
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x-axis, and we're going to rotate three-quarters of a revolution. And this is our angle negative 270 degrees.
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Notice that there are a lot of angles whose terminal side would be in exactly the same place. In particular,
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there's an easy one to spot. 90 degrees in the positive direction has its terminal side in exactly the same place.
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Are there other angles? Yes. For instance, suppose that we went in the negative direction completely around,
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that would be negative 360, and then went negative 270 more. That's going to be negative 270 minus the 360.
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That would be 630 degrees, negative. That angle, all the way around plus the negative 270 also has the same
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terminal side. Ok, and you could see, I could go around two complete negative revolutions and then stop at
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negative 270, and I'm going to get yet another angle with exactly the same terminal side.
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Don't like negative angles? We can go the other direction. Suppose that I start with, I have the 90, I could make
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a complete revolution all the way around. 360 degrees plus another 90. I'm going to stop in the same place.
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Or I can make 5 revolutions around and then 90 degrees more, and again I'm going to get a terminal side in
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exactly the same place. These are all examples of coterminal angles. They have exactly the same terminal side,
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the same starting point, but they have different measures.
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You need to practice finding coterminal angles to help us with our study of trig.
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Often it will be helpful to be able to find specific coterminal angles for a given angle. Let's look at some examples.
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We're asked to find two positive angles and two negative angles that are coterminal with a given angle. Let's start
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with 210 degrees. This angle is already positive. I know for sure that if I add 360 degrees it's adding one revolution
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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
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a positive angle that is coterminal with 210 degrees. If I add another 360 to this, then I'll get one more revolution on
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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.
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And yes, you could have just added 360 to this one. You get to the same place. Well I have enough positive angles,
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so let's try to get some negative angles. What happens if I subtract 360 degrees from this angle?
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Ok, so I subtract this, it's negative 150 degrees. There's one negative angle that is coterminal with my given angle.
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How do I find a second one? Right, subtract another 360. You can subtract 360 from the negative 150, or you can
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210 degrees minus 2 times 360 will give you the same result.
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Ok. Let's look at negative 100 degrees. What happens if I add 360 to this? You remember coterminal angles just
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differ by multiples of 360 degrees. So let's take negative 100 degrees and add 360 degrees to it, 260 degrees.
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That's one of the positive angles that I need.
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Now let's get another positive angle by adding 360 degrees one more time, and you can see that if I do
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negative 100 degrees plus 3 times 360 I'm just going to get yet another bigger, positive angle and that's not
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what I'm asked to do. I need to look for some negative angles. So now, let's go back to our negative 100
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and subtract. We'll subtract 360 degrees one time, this is negative 460, and do that. Subtract 360 again,
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and I've satisfied the requirements of the problem. I've found two positive angles that are coterminal
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with negative 100 degrees and two negative angles. I'd like for you to put a little star by this one though because
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as it turns out this 260 degree angle is very important to us. It is the angle of least non-negative measure that is
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coterminal with the given angle. It's so important we're going to give this a name, theta c.
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Alright, we have one more example, 700 degrees. What do you notice about this one? It's bigger than 360. Ok.
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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
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positive angle there too. So let's go ahead and start with that. It's 340 degrees. So there's one positive angle
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coterminal with my given angle. Let's subtract 360 again, so we're subtracting two 360's. That's negative 20 degrees.
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Ok, so to get another angle you can see that I'm going to need to subtract another 360 from that,
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and I'm going the wrong directions now. Alright, the instructions were two positive angles and two negative angles
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that are coterminal with my given angle. So how am I going to find that other positive angle? So we can go back
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here to this 700 and add 360.
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Often we'd like to be able the find the angle of least non-negative measure that's coterminal with our given angle,
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and remember we've called this theta c. Let's try that for negative 765 degrees. The first thing we notice is we're
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looking for a non-negative measure, and we have a negative angle. Coterminal angles, remember, differ by
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multiples of 360 degrees, so we can simply start by adding 360 degrees to this and see what happens.
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What size angle do we get? Negative 405 degrees. That's more than one revolution, right, in the negative direction,
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so that's not enough. Let's take negative 765 degrees plus 2 times 360, so that would be negative 765 degrees
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plus 720 degrees. That's negative 45 degrees. Unfortunately, it's still negative, so we just go again.
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Ok, so negative 765 degrees plus 3 times 360 degrees. That's going to be 360 degrees more than this angle,
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and that would be 315 degrees. So this is theta c, the angle of least non-negative measure coterminal with our
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given angle. Alright, let's try this again for 1180 degrees. This is a positive angle. A really big positive angle, right?
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So we need to subtract multiples of 360. Now I could start with subtracting 360 degrees, but I can tell that's not
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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
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still more than one revolution, so that wasn't enough. Let's try 1180 degrees minus 3 times 360. That's going to be
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360 less than that. It's 100 degrees. And that will be our theta c,
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the angle of least non-negative measure coterminal with our given angle.
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We're all pretty comfortable with degree measure for angles, but in our study of trigonometry we'll find another
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measure for angles that's even more useful, and that's radian measure. So let's start by defining what a radian is.
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So we start with a circle of radius r, and we've drawn a central angle theta that intercepts an arc of length r.
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Alright so that means that the arc is the same length as the radius of the circle. In that case, theta is an angle
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of measure 1 radian. It would be helpful to know how many radians are in one complete revolution
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just like we know there are 360 degrees in a complete revolution. So how do we figure this out?
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One complete revolution is the same length as the circumference of the circle. Do you remember the formula
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for the circumference of a circle? Yes, 2 pi times the radius. Ok, so how many r's are there as you go around
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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,
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you notice that if you want to know a quarter of a revolution, a half of a revolution, three quarters of a revolution,
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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.
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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
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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
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your coordinate system because our next step is to draw angles in standard position given a radian measure.
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Let's look at drawing angles in standard position when the angle is given in radian measure.
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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,
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and 3 pi over 2 radians. In standard position we start with the initial side on the positive x-axis, and
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counterclockwise is for positive angles, clockwise rotation for negative angles. That hasn't changed.
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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,
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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.
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Well, it's not quite all the way halfway around because halfway around would be 3 pi over 3. We haven't quite
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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
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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
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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
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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,
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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
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in the positive direction plus another pi over 4. Now, that pi over 4 is half of pi over 2.
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So here's 9 pi over 4 in standard position.
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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
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in the negative direction, that's negative 2 pi. This would make negative 4 pi, so negative 6 pi is three complete revolutions
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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?
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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?
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So that would be negative 6 pi over 12. It's just a little bit more than that.
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Later on in this video, I will show you how to convert between radian and degree measure, but you'll notice that
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I did all of these without ever doing that conversion. It's in your best interest to learn to think in radians and
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be able to appropriately place an angle in standard position without having to convert everything back to degrees.
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Let's look at finding coterminal angles using angles given in radian measure. We've previously seen that
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coterminal angles differ by one the number of revolutions or the direction of those revolutions, so
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in degrees, we added or subtracted multiples of 360 to find coterminal angles.
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So what should we do when we're faced with angles in radian measure?
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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,
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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,
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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
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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
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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?
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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
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for some negative angles. We can take our 3 pi over 4, and let's try subtracting 2 pi and see what happens.
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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.
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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
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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,
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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,
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negative 20 pi over 3. Ok, well it's already a negative angle. It's more than one revolution in the negative direction.
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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
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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
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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
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just keep adding multiples of 2 pi. Ok, so this is 4 pi. How can I write that with a denominator of 3?
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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
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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.
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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
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notice at this point is that sometimes it's not obvious how far you have to go, but the more you practice with radians,
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right we're still looking for a positive angle and we're not there yet. The more you work with radians, the more you
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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
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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
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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
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coterminal with the given angle. Remember earlier we talked about theta c, the angle of least non-negative measure
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that is coterminal with our given angle. And we're going to have to practice finding that using radian measures as well,
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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
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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
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angles that are coterminal with my given angle. And then I found three, so pick two of these and there are your answers.
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In this example we're asked to find the angle, theta c, of least non-negative measure that is coterminal with the given
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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
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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
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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
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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
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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
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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
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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.
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That is a positive angle. It's less than one revolution. In fact, it is the smallest non-negative angle coterminal with
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the given angle, so let's label that theta c.
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Let's look at how we can convert between radian measure and degree measure. So previously we've said that
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one complete revolution is 360 degrees. We've also said that one revolution is 2 pi radians. So that means that
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2 pi radians equals 360 degrees, or if we divide both sides by 2, we find that pi radians is 180 degrees. Notice that
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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
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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
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degree symbol. Ok, this is all we need to know to be able to do the conversions, so let's look at some examples.
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Let's convert each of the following from degree measure to radian measure. We'll start with 30 degrees. Since
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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
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the value. And the degree units will sort of go away and leave us with radian units. We want to reduce the fraction.
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30 goes into 180 six times, so this is pi over 6 radians. Negative 240 degrees. So again, pi radians is equal to 180
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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,
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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.
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The last one is 160 degrees. So again, we're converting degrees to radians. We're going to multiply by pi over
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180 degrees. We can divide the numerator and the denominator both by 10 to get 16 pi over 18, and then
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of course we can reduce that a little bit further to get 8 pi over 9 radians.
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Let's convert each of the following to degree measure. This is 13 pi over 6 radians, and I know it's radians
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because no unit is given and I'm told to go to degree measure. Ok, so what should I multiply by?
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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?
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Because I won't end up with the units of degrees that I want in the end. So instead we want to multiply by
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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.
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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
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calculator this is very easy to do, and you get a nice number. You're going to get 390 degrees. Don't forget the
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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
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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.
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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
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confused without the pi in there. And here you'll have to use your calculator. Multiply 2.4 times 180 degrees,
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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
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radian measure, and you've learned about converting between the two. You need to practice finding and
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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.
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It will help you throughout the remainder of your trigonometry course.