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Pre-Calculus: Finding Coterminal Angles


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About this Lesson

  • Type: Video Tutorial
  • Length: 3:39
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 39 MB
  • Posted: 01/22/2009

This lesson is part of the following series:

Trigonometry: Full Course (152 lessons, $148.50)
Pre-Calculus Review (31 lessons, $61.38)
Trigonometry: Trigonometric Functions (28 lessons, $26.73)
Trigonometry: Angles and Radian Measure (5 lessons, $5.94)

Coterminal angles are angles that have their terminating rays in the exact same location. By definition, any two angles whose difference is some multiple of 360 degrees. In this lesson, Professor Burger will show you what coterminal angles look like, how to determine if two angles are coterminal using simple math and then he will review several examples of coterminal angles. He will also demonstrate visually how coterminal angles behave and why the definition is appropriate for these types of angles.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, 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.

About this Author

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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit or visit Thinkwell's Video Lesson Store at

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...


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The Trigonometric Functions
Angles and Radian Measure
Finding Coterminal Angles Page [1 of 1]
Okay, so co-terminal angles are just any two angles that basically land at the same spot in terms of where their
terminating ray is. So, for example, this is an angle right here, but we could have also gotten there by doing this.
Now, watch this. Spinning around and then going up there. Or we could have also gotten to this point by starting over
here and going backwards all the way to here. So, in fact those are all considered co-terminal angles. So, what’s the
actual definition of a co-terminal angle? Well, this would be any two angles whose difference is some multiple of 360
degrees, because that would mean that I spun around some number of times but ended up in the exact same spot.
So, for example, let’s just see if the following are co-terminal angles or not.
Suppose I have 360 degrees, and I want to see if that is co-terminal with zero degrees. Well, all I have to do is say,
“Is the difference a multiple of 360?” Well, plainly the answer is, yes, because 360 – 0 = 360, and that’s a multiple of
360. So, what I’m observing here is that basically, the angle that starts and stops here is the exact same angle as if I
would have gone once around and then just stopped. It looks the same at the end. What about minus 570 versus
150? Well, let’s see. All I have to do is take the difference, so I take minus 570 and subtract 150 and let’s see what I
get. If I subtract, I would see –well, I see negative and negative, so I just am basically adding these things, which will
be more negative. So seven and five is two, and five, six and one, is seven. So I see minus 720. Now is that a
multiple of 360? And the answer is, it sure is, because that’s just negative 2 times 360. So, it means these are, in
fact, co-terminal, which means that if you were to sweep out minus 570, what would that look like? Well, minus
means I’m going to go in a clockwise direction, 570. Well, that means I go once around, that’s minus 360, and then I
have to keep going to get to 570. So I go sort of all the way up to here. But it turns out that’s the exact same thing as
going just starting from here and going 150 this way. See, it’s the same angle. So, those are co-terminal angles.
They end in the same spot. One last one. How about minus 800 compared to minus 80? Are those co-terminal or
not? Well, it’s a subtract and see. So, what I do is I take minus 800 and I subtract off minus 80. So, that means I
have –800° + 80, and what does that equal? That equals minus 720 again. And we know that actually is a multiple of
360. So, in fact, these are co-terminal, which means that when I spin around here, where I land here is the same
place I’m going to land here. Now, different angles are measured. This is a huge angle with a negative sign in front.
This is a much more modest angle with a negative sign in front. So they’re different angles but they land in the same
Well, we’ll actually use this idea of coterminal angles, just the angles where in fact, it’s sort of like a spinner. When
you spin around, if there’s no memory of it, the question is, where do you land? Well, if I land there, that’s the same
thing I would have gotten if I would have spun for a long, long time, and then finally land there. Those are co-terminal.
Their differences would have to be a multiple of 360. So, I just get rid of all the spinning and I see where I land. That
happens if they’re co-terminal.

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