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Trigonometry: Finding Quadrant Where an Angle Lies

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About this Lesson

  • Type: Video Tutorial
  • Length: 5:43
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 61 MB
  • Posted: 07/02/2009

This lesson is part of the following series:

Trigonometry: Full Course (152 lessons, $148.50)
Trigonometry: Trigonometric Functions (28 lessons, $26.73)
Trigonometry: Angles and Radian Measure (5 lessons, $5.94)

Taught by Professor Edward Burger, this lesson comes from a comprehensive course, Trigonometry. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/trigonometry. The full course covers trigonometric functions, trigonometric identities, application of trig, complex numbers, polar coordinates, exponential functions, logarithmic functions, conic sections, and more.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Finding the Quadrant in which an Angle Lies
Okay, well now I thought we would begin to start thinking about angles. And we've been measuring everything and basically in sort of lengths, and x's and y's have always been lengths. When we've been graphing things, we've been sort of going over and going up and so forth. But, really, there's a whole world of angles out there and I want us to start thinking about angles and then moving from angles to looking at measuring angles in various ways, which of course will lead us to trig, an extremely powerful and very valuable sort of genre in and of itself.
So let's take a look. Let me first begin by sort of telling you or reminding you about how we just measure angles. So, if you look down here and you think of the axes in the standard position, when we measure angles in the plane like this, we always start on the positive x-axis. So, right here is always going to be sort of our terminal or starting edge of the angle. And then what we do, is we always move in a counter-clockwise direction to measure a positive angle. So, for example, I would start to move up like this and that would actually produce an angle right in here. The angle between these two red rays, and that would be the angle Theta. So, for example, you can imagine this angle right here would be 45 degrees. It's right in between these two. This would be a 90 degree angle and so forth. This would be a really big angle. This would be 180 degrees. This one is called a straight angle, in fact. You can come down here and so forth, and if you come all the way back around, that's actually going to be 360 degrees, one full time around, and it sort of looks like zero degrees and, in fact, 360 degrees is the same thing as zero degrees. Because as you spin around you see, what happens is, even if you spin around a lot, you end up in the same place that you started, so that would be zero degrees. Now, in fact, we give names to each of these four regions here. This is referred to as Region 1, and this is referred to as Region 2, and then Region 3, and Region 4. So, we count them counter-clockwise, just like we measured the angles. So, Region 1, Region 2, Region 3, Region 4. Now, angles that are pretty small, angles that are between zero and 90 degrees, so any angle that is basically in the first Region, those are actually called cute angles - they're so cute, they're so tiny - they're called acute, because they're small. This angle, as you already know, is called a right angle. It has 90 degrees, and if you have an angle that is actually greater than 90 degrees, but smaller than 180 degrees, so you basically live in the second quadrant, then those things are called obtuse, those angles are huge. And remember, you always measure from here. Don't think that you start measuring from this line. You actually always measure from the positive x and zoom up to here and then you see these angles would be called obtuse. And this is something that is called a straight angle. It's 180 degrees so it forms a straight line.
So, that's pretty cool. Now, there's another issue which is, in fact, you could actually make your angle by going downwards, but that would be considered a negative angle. So, if I go down in a clockwise direction, that's sort of bucking the tide, so that angle right here, I would call that a negative angle. So, for example, this would be right here, negative 45 degrees. And this angle right here, that I swept out from here to here would be negative 90 degrees. Now, you may notice, in fact, that negative 90 degrees looks a lot like if I would have gone all the way around this way. And that actually is 270 degrees. So, in fact, if you go around this way, that's 270; if you go this way, it's minus 90. Those two angles are actually called co-terminal, which just means that you sort of end up in the same place. There's two ways of getting to everything. So, in fact, that's all that's going on here.
Okay, so to sort of make sure you sort of follow this quadrant stuff here, let me just give you an angle and let's see if we can figure out exactly what quadrant we'd be in. So, I suppose I say to you, 235 degrees; well, what quadrant would we be in? Well, from zero to 90 is this quadrant, quadrant 1; from 90 to 180 would be this quadrant right here, which would be quadrant 2; 180 to and this would be 270 - 90, 90 and 90 would be 270 - would be here, and 270 to 360 would be in this range. So, if I said to you, 235, what you would have is something that would be sort of right around here. That would be in quadrant 3. What if I said to you, minus 570? If I said to you minus 570, how would you figure that out? Well, you sort of have to spin around a whole bunch and in a clockwise direction, remember, negative means clockwise. So, you would have to spin around a whole bunch, so if you went around sort of once around, that would be minus 360. Every time you go 360, you make a complete cycle. But now you have to keep going because you're minus 570, so you have to go another negative 210. And what happens if you do that? Well, this is negative 90, this is negative 180, and that would be negative 270, so that's too much, so it's negative 210 - it's right about there. In fact, this little angle right here you can compute, it's just 30 degrees. That little teeny thing right in there is 30 degrees. So, in that case, minus 570 would be in the second quadrant. And you can see how to figure these out. Where would 100 degrees be? Well, 100 degrees, you would go just a little bit past 90 and so plainly that's going to also give you the second quadrant, but a little higher up.
So, you get a sense of the quadrants and how we measure angles. We always start from the positive x-axis, and for positive angles, we sweep in a counter-clockwise direction, and for negative angles, we sweep in a clockwise direction. So, those are the very, very basics. This axis would be sort of either zero degrees or 360 degrees, they would be the same. This axis would be at the 90 degrees. This axis would be considered sort of 180 degrees. This axis here would be considered 270 degrees. So, this gives you a sense of how we're going to cut up the plane and we're going to measure our angles. Up next, we'll start taking a look at different types of angles, how to combine them and how to look at them together. We'll start of slowly, but we're going to be building pretty fast. So, stick with us.

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