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Pre-Calculus: Solving Trig Equations by Factoring

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About this Lesson

  • Type: Video Tutorial
  • Length: 6:03
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 65 MB
  • Posted: 01/23/2009

This lesson is part of the following series:

Trigonometry: Full Course (152 lessons, $148.50)
Pre-Calculus Review (31 lessons, $61.38)
Trigonometry: Trigonometric Identities (23 lessons, $26.73)
Trigonometry: Solving Trigonometric Equations (5 lessons, $7.92)
Pre Cal: Solving Trigonometric Equations (5 lessons, $7.92)

Professor Burger teaches how to solve more complicated equations (tanx * sin^2x = tan x) involving trigonometric functions in this lesson. Solving these types of problems involve use of trig identities, factoring, etc and how to find all of the viable solutions for these types of problems. In the problem listed above, Professor Burger will show you how to factor the equation in order to help simplify and then solve it. Professor Burger also gives a warning about cancelling out in equations that involve trig functions. By canceling, you risk missing valid solutions and solution sets.

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 http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

About Professor Edward Burger:
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

Thinkwell
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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 www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

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Trigonometric Identities
Solving Trigonometric Equations
Solving Trigonometric Equations by Factoring Page [1 of 2]
Let’s take a look at solving some more of these trigonometric equations. Here’s one let’s examine:
tan x sin2x= tanx. So let’s solve this and see what values of x will satisfy this. Now there’s a great temptation here
to cancel away the tan x on both sides. That would simplify things. But that would actually, though, lose a whole
bunch of solutions because, in fact, you can’t cancel unless you make sure those things aren’t zero. And if these
values are actually zero, that solution would be zero equals zero. So in fact, if you look at it carefully you just can’t
cancel things away blindly. What’s better is to factor. So what I’ll do is I’m going to bring everything over to the lefthand
side and make this thing equal zero.
If I do that, what I would see is: tanxsin2x?tanx=0 . And now I see a common factor of tan x here and here,
and I can factor that out. Let’s do that. If I factor that out, I see: tanx(sin2x?1)=0 , right? Because if I distribute,
that gives me the tan x here. That -1 is necessary. Okay, now I have a product of two things equals zero. Well, that
means we have two possibilities: either this equals zero—so either tan x equals zero—or this thing equals zero,
which means what? It means that sin2 x =1 , right? That’s what it means for this to equal zero.
Okay, well either tangent equals zero, so we’ve got to find all the places where tangent equals zero or—and now I
guess I can take square roots but I’ve got to be careful; I’ve got to take plus or minus square roots. And so I see
sin x = ± 1 , which is 1. Okay, so I see there are a whole bunch of possible solutions and now I’ve got to figure out
exactly where those things occur.
So where does tangent equal zero? Well, a little graph of the tangent function should be revealing. A little graph of
the tangent function should be really fast. So this is 2? , this is
2
?
, that’s 2, this is 2? here, ?2? , ?? , and so forth.
And then what I see is we have the asymptote at the half multiples—oh, shoot, I screwed it up. I put the dotted line in
the wrong spot. That’s okay, no problem. See, the great thing about graphing is if you make a mistake, you just do it
over and it’s not a big deal. Okay, here we go. If I cut it in half, that’s ? . Half again, half again. And the asymptotes
are going to be here at
2
?? , at
2
?
,
3
2
?
,
3
2
? ? . Okay, now we’re in good shape. Let’s start to graph this function.
And all I care about is when it equals zero. Asymptotes here.
Well, where does it equal zero? Notice it equals zero at zero and then at ? , and then at 2? , and then at ?? , and
then at ?2? . It’s all the integer multiples of ? . So in fact, this solution here tells me that x =? n , where
n=0,±1,±2... and so forth. Any solution will make, in fact, the tangent function zero, which will make this whole
thing zero and satisfy the equation. So there’s one solution.
But what about where sin x = ±1? Well, let’s look at the sine function and ask, where does that equal either ±1?
Well, it equals +1 at
2
?
. It equals -1 at
3
2
?
, and then it will equal a +1 again where? Well, that will be at the next
2
?
that we get, which would be
5
2
?
and so forth. So what we see here is that we have, as a solution—how could I write
this? I could say x here will equal any multiple that is an odd multiple, so let’s just say it this way: (2 1)
2
n + ? , where
n=0,±1,±2... and so forth. Because notice that when I plug in any value for n here, that will give me all the odd
numbers. If n were to be zero, I’d get just 1. If n were to be 1, I would get
3
2
?
. If n were to be 2, I would get
5
2
?
.
And with the negative numbers, I’d get those negative odd
2
?
. So in fact, the solution to this equation is all the x’s
Trigonometric Identities
Solving Trigonometric Equations
Solving Trigonometric Equations by Factoring Page [2 of 2]
that satisfy this or the x’s that satisfy this. So it’s a huge solution set. And see how I found it? I just factored like we
used to always do, and then solved each of these things separately, just thinking about where were all the x’s that
satisfied each of these individually.
Okay, there’s a solution for this particular equation that involved trig functions. Let’s take a look at some more
solutions to more trig functions up next.

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