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Pre-Calculus: Evaluating Inverse Trig Functions

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About this Lesson

  • Type: Video Tutorial
  • Length: 8:24
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 90 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)

Professor Burger shows you how to evaluate inverse trigonometric functions, which include arc sine, arc cosine, arc tangent, etc. He reminds you that inverse functions are asking "What is the angle whose function is X?" Thus, the output for any inverse trig function should be an angle for which, if you apply the indicated trig function, you get the indicated value as a result. Then he walks through finding the inverse of sine, the inverse of cosine, and the inverse of tangent. Finally, Prof Burger shows you how to interpret the presence of a negative sign and how to evaluate inverse trig functions using a calculator (indicated in radians).

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.

About this Author

Thinkwell
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Recent Reviews

Nopic_gry
clear and easy inverse sine and cosine
06/04/2009
~ joyherber

This was great. My son was stuck on some homework and it had been years since I had studied sine and cosine. Once he watched the video and reread the assignment it all be came so simple. Thanks.

Nopic_gry
clear and easy inverse sine and cosine
06/04/2009
~ joyherber

This was great. My son was stuck on some homework and it had been years since I had studied sine and cosine. Once he watched the video and reread the assignment it all be came so simple. Thanks.

The Trigonometric Functions
Inverse Trigonometric Functions
Evaluating Inverse Trig Functions Page [1 of 3]
Let’s really try to get a sense of what these inverse function things are really all about. Let’s just delve into actually
some numerical examples and see if we can actually evaluate the inverse functions. Let’s begin with looking at the
arc sine of
3
2
. Now, let’s think about what this means. What this is asking us is, what is the angle whose sine is
3
2
? So, the output for an inverse sine function, or for any inverse trig function is an angle. And it’s the angle so that
if I take the sine of it, I get
3
2
. So what angle? Now you have to remember what the sine function looks like. The
sine function -- well, we know what it looks like. But remember, where it’s defined for the inverse purposes, so how
does the sine function work? Well, for inverse purposes, we only look at a little window of it. Look at the window from
2
?
to -
2
?
. That’s the only window we’re allowed to look at for the inverse function.
So, if I say, what angle in between here and here, has the sine =
3
2
. That’s sort of like around here somewhere
maybe. Well, you remember the chart and you should be able to sort of think back. What angle has sine
3
2
? Well,
it turns out that’s 60°, or as we would say it in radians,
3
?
.
Okay, so there’s that one. Let’s try another one. How about this one? How about sine inverse, so inverse sine of -
1
2
? Now that’s a little tricky, because where is -
1
2
? Well, -
1
2
is here. So this is the angle whose sine is -
1
2
. So we
have to find that so it’s going to be a negative angle, notice. So, let’s see, what angle gives me just the half? Let’s do
that first. What angle gives me
1
2
? Well, that’s going to be 30°, or
6
?
. But I want it to be a negative, so I want -
6
?
.
So, look what I’m doing here. I have to see sort of where in this abridged picture of the sine function do I have y = -
1
2
? It’s right here, so I’ve got to know figure out what angle that is. That’s a negative angle. And so what I do is I jut
find it’s positive angle first. So I just look for +
1
2
, which is
6
?
, and then put a negative sign in front of it, and I’m down
here. It’s a little tricky.
All right, let’s try some with cosine. Let’s try the following with cosine. How about cosine inverse, or arc cosine, or
inverse cosine, of -
1
2
? Now, this is really sort of tricks, because I have to remind you what we look at. The standard
thing we look at for cosine is starting at zero and going down to ? , which gets -1. So, where’s -
1
2
for y here? It’s
right around here. So, now it’s this value that I want. I want that angle right here because that angle will have cosine
equal to -
1
2
. How am I going to find that out? Well, first of all, let’s just find out what angle gives me +
1
2
. That one
is a little more easier. That angle is going to be 60°,which is
3
?
. I memorized that from the chart. So, I know that just
The Trigonometric Functions
Inverse Trigonometric Functions
Evaluating Inverse Trig Functions Page [2 of 3]
getting a half is 60°. Well, I don’t want 60°. I sort of want the same thing on the other side. Let me draw actually
maybe a larger picture of this. Because this gets so tricky-wicky. So, I have this picture right here. Very symmetric,
you can see. This is ? . This is
2
?
. And what I know is, here at
1
2
, this happens at 60°, which is
3
?
. That I know
from just my chart. What I want is -
1
2
, which just on this piece, this the only piece we’re allowed to look at for inverse
functions, it would sort of be in the corresponding spot, you see? Now how would I find that? Well, let’s see, if this is
3
?
here, then by symmetry, this must be
3
?
here. You see how this is sort of the same exact thing, except it’s flipped
a little bit? So, in fact, this length must be
3
?
. So, if I want to find this point, what should I do? Well, I should just
take this point, ? , and subtract off
3
?
. So what I should do to find this exact value, is take ? and subtract off this
little region, which is the same as this by symmetry. So I subtract
3
?
and that gives me
2
3
?
. So this angle must be
2
3
?
. So this equals 2
3
?
.
It’s really important to know what part of the graph you’re allowed to look at, and then year-old have to sort of figure
out how to solve these things. By first finding the positive solution and then seeing how the negative solution sort of
plays into that.
All right, let’s try one more of these computing things. Let’s try tangent, inverse, or arc tangent of -1. What does that
equal? Well, we just look at the graph of this to start with. Now, the only place we’re allowed to look here is for -
2
?
to
2
?
, and there we see this tangent thing that looks like this. Where’s -1? Well, -1 for a tangent is over here. So the
question is, what angle gives you that? So, it’s this angle here. So, it’s going to be a negative angle. So this answer
will be negative because it’s in a negative range. You see, the angle is negative. And what angle gives me tangent
just of one now? Well, what’s the angle whose tangent is 1? That means that sine and cosine have to be the same.
That happens at 45° or
4
?
. But the negative sign means I’m on this side. So, in fact that negative sign tells me,
looking at this picture, that I’m going to be over here, negative and then what has arc tan 1? Well, that’s
4
?
, so I have
-
4
?
, looking at this picture. Okay, really tricky business.
Let me just show you that, in fact, you can do these things just using a calculator. Suppose you wanted to find arc
sine of .362? All you do is just use the calculator. I have it set for radians. And you just use the -- there’s a sine with
a -1 on top symbol, so that’s inverse sine -- and just type in .362. Type equals. And the answer is .3704 radians.
What that means is, if you now compute the sine of that, you’re going to see exactly this number in here.
Let’s try one more. Tangent inverse of -12.6. So you take inverse tan of -12.6 -- it’s going to give me in radians -- and
this equals -1.491 something. And if you look, that makes at least a teeny-weeny bit of sense because where is -12.6
in the tangent thing on the y? It’s way down here, way, way down here and so that means that this value that should
The Trigonometric Functions
Inverse Trigonometric Functions
Evaluating Inverse Trig Functions Page [3 of 3]
give me that should be negative, and this is
2
?
and ? is around three something, so if
3
2
is like, you know, one and
a half something, and this is just a little bit before that, or after that, I guess, actually. You can see, it’s right here and
so it’s -1.49. So at least it makes a little bit of sense, even visually. But a calculator will spit out these values for you if
you don’t want to spit yourself. And if you don’t want to spit, I can’t blame you. These inverse trig function things are
tricky. Good luck, you can do it.

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