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Pre-Calculus: Inverse Trig Function Equations

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About this Lesson

  • Type: Video Tutorial
  • Length: 4:50
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 52 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: Inverse Trigonometric Functions (5 lessons, $7.92)

An inverse function asks the question ""What is the angle whose function is X."" In this lesson, you will learn to solve equations that include an inverse function (arc sine, arc cosine, arc tangent, etc). Professor Burger first shows you how to untangle the equation, re-writing it so that you can understand for what you are solving. He will also show you examples when there may be an infinite numbers of solutions, and how you will need to correctly denote this answer. Finally, he suggests that you check your answers by graphing, and shows you how. This lesson will include several examples of evaluating problems involving arc sin, arc cos, etc. You will begin by seeing how to approach and solve a problem like 'inverse cosine of cosine x = pi/4' While it would seem that the cosine and inverse cosine here would cancel, you will learn in this lesson why this is not the case and how you can correctly solve for the answer.

This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

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.

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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The Trigonometric Functions
Inverse Trigonometric Functions
Solving an Equation Involving an Inverse Trig Function Page [1 of 2]
Let’s just try a couple of more issues involving these inverse functions and let’s, in fact, take a look at how we could
actually solve equations that have, in fact, inverse functions sort of sitting inside of them. And the thing to do of
course, the cardinal rule is take it very slowly, so let’s say inverse cosine of cosine x =
4
?
. Now, again, there’s this
great temptation to say, “Oh, inverse cosine of cosine, they sort of cancel each other out, and so x =
4
?
.” You have to
be a little bit careful here and make sure that’s okay. What is this saying? This is saying that this is the angle whose
cosine equals that. So, if you just translate that and decode that, inverse cosine means, this is the angle whose
cosine is that. So, if you untangle that, that says cosine
4
?
= this, which is cosine x. So, this statement leads to this
statement. Now cosine
4
?
I happen to know, that’s the
2
2
= cosine x. So now the question is, what are all the x
values that have cosine equals the
2
2
? Well, actually, there are infinitely many. Right? Think of the graph of
cosine and let me put in some intervals here. This is 2? , this is 4? , this -2? , okay, and here’s 1, and -1, and if you
graph in the cosine function, you see this basic shape. Now, you want to know, for which values of x do you get
2
2
? Well, you know you get that at 45°. You know you get that at
4
?
. Of course, that’s what we already knew.
So, at
4
?
, we get the right thing, but look, we get the right thing at a lot of places. You see wherever, in fact, that red
curve hits this line -- for example, right here -- and that’s where? Well, that’s at -
4
?
. So there’s another solution.
And, in fact, there’s a solution right here. And what solution is that? Well, that solution is actually the solution we get
if we take 2? and subtract off
4
?
. So, what would that be? Well, that would be 2? and then subtract off -- you see
how the symmetry is? This is
4
?
, so this distance right here is
4
?
. So, I take 2? and take off
4
?
. I get a common
denominator. I have
8
4
?
-
4
?
=
7
4
?
. So, at
7
4
?
, I also get a solution to this. And you can see any multiple of 2?
plus this number, will give me an answer, like this one, this one, this one, and so forth. And similarly, any 2? multiple
added to this will give me a solution. So, in fact there’s a lot of solutions. The solutions would be x =
4
?
+ 2? n,
where is any number at all, like any integer. So, 0, 1, 2, 3, 4, 5, -1, -2, -3, -4, and so forth. And those are going to
give you these solutions. Watch. This one, this one, and then the one that would come right here, this one, and so
on. To get these other solutions, the other possibility is that x could equal
7
4
?
+ 2? n. And that’s going to give you
these solutions, and these solutions, and these solutions, and so on. So, in fact, they’re infinitely many solutions to
this, even though you might have guessed only 1, you’ve got to take it very carefully. First, untangle this,
remembering that arc cosine means angle whose cosine is this. So,
4
?
is the angle whose cosine is this. This I was
able to evaluate and then basically what you want to do is take a look at the cosine function, draw a horizontal line at
height
2
2
and ask, where does it cross it? It crosses it at all these points right here, which I was able to find just
The Trigonometric Functions
Inverse Trigonometric Functions
Solving an Equation Involving an Inverse Trig Function Page [2 of 2]
using a little teeny bit of, I guess, it’s more sort of just the geometry of this curve. Just looking at the geometry and
saying, that’s
4
?
, so that’s -
4
?
, and by symmetry, this must be 2? -
4
?
. And so on.
Really tricky business, but if you take your time and work at it, you will be able to actually solve these kind of
equations. So, just remember to decode it and then look for all the x’s that give you this, and you’ve got it.

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