WEBVTT mathematics/trigonometry/murray
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Hi, This is Will Murray and I'm going to be giving the trigonometry lectures for educator.com.
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We're very excited about the trigonometry series.
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In particular, for me, trigonometry is the class that got me excited about math.
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I'm really looking forward to working with you on learning some trigonometry.
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We're going to start right away here, learning about angles.
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The first thing you have to understand is that there's two different ways to measure angles.
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People use degrees which you probably already heard of, and radians which you may not hear about until you start to take your first trigonometry class.
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They're just two different ways in measuring.
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You can use either one but you really need to know how to use both, and convert back and forth.
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That's what I'll be covering in this first lecture.
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We'll start with degrees.
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Degrees are unit of measurement in which a circle gets divided into 360 degrees
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If you have a full circle, the whole thing is 360 degrees.
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That's 360 degrees.
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Then if you have just a piece of a circle, then it gets broken up into smaller chunks.
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For example, an angle that's half of a circle here that's 180 degrees, because that's half of 360, a quarter of a circle which is a right angle, that would be 90 degrees, then so on.
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You can take a 90-degree angle and break it up into two equal pieces.
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Then each one of those pieces would be a 45-degree angle.
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Or you could break up a 90-degree angle into three equal pieces, and each one of those would be a 30-degree angle.
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We'll be studying trigonometric functions of these different angles.
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In the meantime, it's important just to get comfortable with measuring angles in terms of degrees.
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The second unit of measurement we're going to use to measure angles is called radians.
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That's a little bit more complicated.
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You probably haven't learned about this until you start to study trigonometry.
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The idea is that, you take a circle, and remember that the circumference of a circle is equal to 2π times the radius, that's 2π r, it's one of those formulas that you learned in geometry.
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What you do with radians is, you break the circle up, and you say the entire circle is 2π radians.
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2π radians.
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What that means is that a one-radian angle, well, if the entire circle is 2π radians, then 1 radian, use a little r to specify the radians, cuts off an arc that is 1 over 2π of the whole circle.
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One radian, an angle that is 1 radian cuts of a fraction of the circle that is 1 over 2π.
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Since the radius is 2π times r, sorry, the circumference is 2π times r, if you have 1 over 2π of the whole circumference, what you get is exactly the length of the radius.
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That's why they're called radians is because if you take one-radian angle, it cuts of an arc length that is exactly equal to the radius.
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That's the definition of radians.
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It takes a little bit of getting used to.
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What you have to remember that's important is that the whole circle is 2π radians.
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That means a half circle, a 180-degree angle, is π radians.
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A right angle, a 90-degree angle, or a quarter circle is π over 2 radians, and so on.
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Then you can break that down into the even smaller angles like we talked about before.
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If you take a right angle and you cut it in half, so that was a 45-degree angle before, that's π over 4 radians because it's half of π over 2.
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If you take a right angle and you cut it into three equal pieces, so those are 30-degree angles before, in terms of radians, that's π over 2 divided by 3, so that's π over 6 radians.
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You want to practice going back and fort between degrees and radians, and kind of getting and into the feel of how big angles are in terms of degrees and radians.
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We'll practice some of that here in this lecture.
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Remember that π is about 3.14, so 2π is about 6.28.
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That means we're breaking an entire circle up in the 2π radians, so the circle gets broken up into about 6.28 radians.
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That means one radian is about one-sixth of a circle.
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What I've shown up here is pretty accurate that one radian is about one-sixth of a circle.
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It's about 60 degrees but it's not exact there because it's not exactly 6 it's 6.28 something.
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That's about roughly a radian is, about 60 degrees, but we really don't usually talked about whole numbers or radians.
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People almost always talk about radians in multiples of π the same way I was doing here, where I said the circle is 2π radians, the half circle is π radians, the right angle is π over 2.
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People almost always talk about radians in multiples of π and degrees in terms of whole numbers.
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Sometimes, they don't even bother to write the little r.
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It's just understood that if you're using a multiple of π , then you're probably talking about radians.
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Let's practice going back and forth between degrees and radians.
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Remember that 360 degrees is a whole circle.
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That's 2π radians.
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What that means is that π radians is equal to 360 degrees over 2, which is 180 degrees.
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π radians is 180 degrees.
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That gives you the formula to convert back and forth between degree measurement and radian measurement.
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If you know the measurement in degrees, you multiply by π over 180 and that tells you the measurement in radians.
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If you know the measurement in radians, you just multiply by 180 over π , and that tells you the measurement in degrees.
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We'll practice that in some of the examples later on.
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We got a few more definitions here.
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Coterminal angles, what that means is that their angles that differ from each other by a multiple of 2π radians,
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remember that's a whole circle, or if you think about it in degrees, 360 degrees.
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For example, if you take a 45-degree angle, and then you add on 360 degrees, that would count as a 360 plus 45 is 405 degrees.
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Forty-five and 405 degrees are coterminal.
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In the language of radians, 45 degrees is π over 4.
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If you add on 2π radians, if you add on a whole circle to that, you would get...
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This should end up here.
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If you add on 2π plus π over 4, well 2π is 8π over 4, so you get 9π over 4.
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Then π over 4 and 9π over 4 are coterminal angles.
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The reason they're called coterminal angles is because we often draw angles starting with one side on the positive x-axis.
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We start with one side on the positive x-axis.
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I'll draw this in blue.
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Then we draw the other side of the angle just wherever it ends up.
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Coterminal angles are angles that will end up at the same place, that's why they're called coterminal.
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If they differ from each other, if one is 2π more than the other one, or 360 degrees more than the other one, or maybe 720 degrees more than the other one, then we call them coterminal because they really end up on the same terminal line here.
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Couple other definitions we need to learn.
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Complementary angles are angles that add up to being a right angle, in other words, 90 degrees or π over 2.
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If you have two angles, like this two angles right here , that add up to being a right angle, 90 degrees or π over 2, those are complementary.
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Supplementary angles are angles that add up to being a straight line, in other words, π radians or 180 degrees.
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Those two angles right there are supplementary.
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That's all the vocabulary that you need to learn about angles, but we'll go through and we'll do some examples of each one to give you some practice.
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Here's our first example.
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If a circle is divided into 18 equal angles, how big is each one in degrees and radians?
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Let me try drawing this.
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We've got this circle and it's divided into a whole bunch of little angles but each one is the same.
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We want to figure out how big each one is, in terms of degrees and radians.
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Let's solve this in degrees first.
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Remember that a circle is 360 degrees.
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If it's divided into 18 parts, then each part will be 20 degrees.
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Each one of those angles will be 20 degrees.
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We've done the degree one, how about radians?
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Remember that an entire circle is 2π radians.
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If we divide that by 18, then we get π over 9 radians will be the size of each one of those little angles.
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You can measure this angle either way, we say 20 degrees is equal to π over 9 radians.
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Second example here, we want to convert back and forth between degrees and radians.
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Let's practice that.
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We want to convert 27 degrees into radians.
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Well, let's remember the formula here, the conversion formula, is π over 180.
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So we do 27 times π over 180.
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That's our conversion formula from degrees into radians.
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I'm just going to leave the π because it doesn't really cancelled anything.
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The 27 over 180 does simplify.
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I could take a 9 out of each ones.
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That would be 3.
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If we take a 9 out of 180, then there'd be 20.
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What we end up with is 3π over 20 radians, as our answer there.
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Converting back and forth between degrees and radians is just a matter of remembering this conversion factor, π over 180 gets you from degrees into radians.
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For the second part of this example, we're given a radian angle measurement, 5π over 12.
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We want to convert that into degrees, 5π over 12 radians.
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We just multiply by the opposite conversion factor, 180 over π.
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Let's see here.
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The pis cancel.
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One hundred eighty over 12 is 15.
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That's 5 times 15 degrees.
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That gives us 75 degrees.
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The same angle that you would measure, in radians is being 5π over 12, will come out to be a 75-degree angle.
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Converting back and forth there is just a matter of remembering the π and the 180, and multiplying by one over the other to convert back and forth.
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Third example is some practice with coterminal angles.
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In each case, what we want to do is, we're given an angle and we want to find out what quadrant it's in.
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That's assuming that all the angles are drawn in the standard position with their starting side on the positive x-axis.
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We want to start on the positive x-axis.
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We want to see which one of the four quadrants the angle ends up in.
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Then we want to try to simplify these angles down by finding a coterminal angle that's between 0 and 360, or between 0 and 2π radians.
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Let's start out with 1000 degrees.
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A thousand degrees is going to be, that's way bigger than 360.
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Let me just start subtracting multiples of 360 from that.
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If I take off 360 degrees, what I'm left with is 640 degrees.
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That's still way bigger than 360 degrees.
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I'll subtract off another 360 degrees and what I'm left with is 280 degrees.
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That's between 0 and 360.
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I found my coterminal angle there.
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I wanna figure which quadrant it's gonna end up in.
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Now, remember, if we start with 0 degrees being on the x-axis, that would make 90 degrees being on the positive y-axis.
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Then over here on the negative x-axis, we'd have 180 degrees.
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Down here is 270 degrees, because that's 180 plus 90.
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Then 360 degrees would be back here at 0 degrees.
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Two hundred and eighty degrees would be just past 270 degrees.
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That's a little bit bigger than 270 degrees.
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It would be about right there.
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That's 280.
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That puts it in the fourth quadrant.
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Next one's a radian problem.
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We have -19π over 6 radians.
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That's one, I'll do this one red.
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That's one that goes in the negative direction.
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We start on the positive x-axis but now we go in the negative direction.
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Instead of going up around past the positive y-axis, we go down in the negative direction and we go -19π over 6.
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If you think about it, 19π over 6 is bigger than 2π.
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Let me start with 19π over 6 and subtract off a 2π there.
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Well, 2π is 12π over 6, so that gives us 7π over 6.
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If we do, -19π over 6 plus 2π, that will give us, -19π over 6 plus 12π over 6 is -7π over 6.
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That's still not in the range that we want, because we want it to be between 0 and 2π radians.
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Let me add on another 2π plus 2π gives us positive 5π over 6.
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The trick with finding these coterminal angles with degrees, it was just a matter of adding or subtracting 360 degrees at a time.
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With radians, it's a matter of adding or subtracting 2π radians at a time.
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Remember, 2π is a whole circle.
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We end up with 5π over 6.
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That is between 0 and 2π so we're done with that part but we still have to figure out what quadrant it's in.
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Well now 5π over 6, where would that be?
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Well if we map out our quadrants here, 0 is right there on the positive x-axis just as we had before, 90 degrees is π over 2 radians, 180 degrees is π radians, and 270 degrees is 3π over 2 radians.
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Then 360 degrees is 2π radians, a full circle.
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Where does 5π over 6 land?
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Well that's bigger than π over 2, it's less than π, so, 5π over 6 lands about right there.
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That's in the second quadrant.
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OK, we have another degree one.
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Negative 586 degrees, and what are we going to do with that?
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It's going in the negative direction so it's going down south from the x-axis.
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Negative 586 degrees.
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Well, 586 is way outside our range of 0 and 360.
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Let's try adding 360 degrees to that.
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That gives you -226 degrees, which is still outside of our range.
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Let's add another 360 degrees.
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We're adding and subtracting multiples of a full circle 360 degrees.
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That gives us positive 134 degrees.
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Now, positive 134 degrees, that is in our allowed range between 0 and 360.
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So we finished that part of the problem.
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Where would that land in terms of quadrants?
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Let me redraw my axis because those are getting a little messy.
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That's 0, 90, 180, 270 and 360.
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Where's 134 going to be?
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A hundred and thirty-four is going to be between 90 and 180, almost exactly halfway between.
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It's about right there.
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That's in the second quadrant.
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The answer to that one is that that's in the quadrant number two there.
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Finally, we have 22π over 7, again given in radians.
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The question is, is that between 0 and 2π?
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It's not, it's too big.
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It's bigger than 2π.
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Let me subtract off a multiple of 2π.
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I'll subtract off just 2π, which is 14π over 7.
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That simplifies down to 22π over 7 minus 14π over 7, is 8π over 7.
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Now, 8π over 7 is between 0 and 2π.
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We found our coterminal angle.
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Where will that land on the axis?
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Well, remember 0 degrees is 0 radians, 90 degrees is π over 2 radians, 180 degrees is π radians, and 270 degrees is 3π over 2 radians, and finally, 360 degrees is 2π radians.
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Eight π over 7 is just a little bit bigger than 1.
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That's a little bit, or 8 over 7 is a little bigger than 1.
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Eight π over 7 is just a little bit bigger than π.
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Let's going to put it about right there which will put it in the third quadrant.
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Let's recap how we found this coterminal angles.
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Basically, you're given some angle and you check first whether it's in the correct range, whether it's in between 0 and 2π radians,
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or if it's given in degrees, whether it's between 0 and 360 degrees.
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If it's not already in the correct range, if it's negative or if it's too big, then what you do is you add and subtract multiples of 360 degrees or 2π radians until you'll get it into the correct range,
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the range between 0 and 360 degrees or 0 and 2π radians.
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Once you get it in that range, if you want to figure out what quadrant it's in, well in degrees, it's a matter of checking 0, 90, 180, 270, and 360;
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in radians,it's a matter of checking 0, π over 2, π, 3π over 2, 2π.
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Which one of those ranges does it fall into?
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That tells you what quadrant it's in.