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Welcome to an introduction to the
inverse functions for cosecant, secant and cotangent.
Just as we did when we found the inverses
for sine, cosine and tangent.
In order to determine the inverses of cosecant, secant,
and cotangent, the domains must be restricted
to make the functions one to one.
Remember this is necessary,
in order to have an inverse function.
And we will restrict the domains as follows.
For y equals cosecant x,
we'll restrict it on the closed interval
from negative pi over two to pi over two.
And of course x can't equal zero cause it's undefined there.
Y equals secant x on the closed interval from zero to pi,
x not equal to pi over two.
And for y equals cotangent x, the domain will be
on the open interval from zero to pi.
So the inverse function for y equals cosecant x
is given as y equals inverse cosecant of x
or y equals arc cosecant x.
There's two ways to represent the inverse
of y equals cosecant x.
Down below here I graphed y equals cosecant x
and y equals inverse cosecant x to do a comparison.
Here's our restricted domain.
And notice that the domain of y equals cosecant x
becomes the range of the inverse function.
And the range of the cosecant function
becomes a domain of the inverse cosecant function.
And that's important because you need to recognize
when you have an inverse cosecant function,
the x or the input of this function is actually
the cosecant function value and the output
or y is the angle that produces that function value.
Another property that we've discussed
about a function and its inverse,
is that it's symmetrical across the line y equals x.
On this screen, you can see very clearly
these two functions are symmetrical
across the line y equals x.
The inverse function of y equals secant x
is given as y equals inverse secant x,
or y equals arc secant x.
And again I graphed y equals secant x here
and y equals inverse secant x here.
And again, the important part to remember here
is that this restricted domain for y equals secant x
becomes the range for the inverse function
and the range of the function becomes
a domain of the inverse.
So when we're looking at y equals inverse secant x,
x is the secant function value
and y is the angle that produces that secant function value.
And if we graph these two functions on the coordinate plane
with y equals x, we can see they do share
the symmetry across that line y equals x.
And lastly, the inverse of y equals cotangent x is
y equals inverse cotangent x or y equals arc cotangent x.
And again I graphed y equals cotangent x here,
and y equals inverse cotangent here.
Because these equations were obtained
by interchanging the x and y variables,
the domain and ranges have also interchanged
the domain of the function becomes a range of inverse
and the range of the function
becomes a domain of the inverse.
So the input x is actually the cotangent function value,
and y is the angle that produces that
cotangent function value.
And again if we graph them on the coordinate plane
with y equals x, we'll see their reflections
across that line y equals x.
Let's go and take a look at some problems.
I've done some preliminary work for us.
In red on the left, I've listed the possible
outputs for these functions.
I've also listed the reference triangles down below,
and this will help us find the exact angle
that produces these given function values.
So when we look at number one, arc secant of negative two,
it'll be helpful to write this as a fraction
or a ratio of negative two over one.
So this is asking us to find the angle that has a cosecant
function value of negative two over one.
Well remember that cosecant is a reciprocal of sign,
so this angle would also have a sign of negative one-half.
And that tells us that our reference angle
is going to be a 30 degree angle because
a 30 degree angle has a sign of one-half
or a cosecant of two over one.
Next, because the function value is negative,
we're going to have to be in the fourth quadrant.
So we're going to sketch a 30 degree reference angle
in the fourth quadrant.
So the angle we're looking for in this interval
would be negative 30 degrees, or negative
pi over six radians.
So notice how even though we have arc cosecant,
it's often helpful to look at the reciprocal ratio
because those are the ones we're more familiar with.
On number two, we have arc secant of two square root
three over three.
This is pretty difficult to determine
which angle would have a secant function value of this.
So it might be helpful if we take a look
at arc cosine instead.
Since cosine is a reciprocal of secant,
our cosine would be three over two square root three.
Still not easy to recognize but I think
if we rationalize this, we might be able
to recognize it in one of these reference triangles,
so let's go ahead and do that.
We'll take three divided by two square root three
and rationalize it.
So we'll have three square root three over six,
which does simplify nicely to square root three over two.
So our reference angle will be an angle
that has a cosine ratio of square root three over two.
Look at this triangle here.
Three degrees has a cosine ratio of square root
three over two, so that's our reference angle.
And in this interval, we're going to be in
the first quadrant.
And this is the angle we're looking for,
so arc secant of two square root three divided by three
equals three degrees, or pi over six radians.
Let's go and take a look at one more.
Number three, we have arc cotangent
negative square root three.
Might be helpful to write this as negative
square root three over one.
So we're looking for an angle that has a cotangent
function value of negative square root three over one.
Going over to our reference triangles,
we're going to look for an angle that has
an adjacent over opposite ratio of
square root three over one.
If you look at three degrees again,
adjacent over opposite is square root three over one.
So that tells us our reference angle will be 30 degrees.
But since the cotangent function value is negative,
and here are the possible outputs,
we're going to have a 30 degree
reference angle in quadrant two.
So a three degree reference angle would actually be
an angle that measures 150 degrees,
which is equal to five pi six radians.
While we're here let's go and check a couple of these
on our graphing calculator.
So for number one, since there's no arc cosecant key,
we can use arc sign.
So second sign, but then we have to take
the reciprocal ratio so it'll be
negative one over two.
And this verifies our output of negative 30 degrees.
For number two, instead of arc secant
we'll use arc cosine, second cosine
and then we'll take the reciprocal of this,
which you'll have listed here.
Three divided by two square root three,
and that one looks good.
On this last one we're going to have
a little bit of a problem, let's see what happens.
We're going to use arc tangent,
because we don't have an arc cotangent key.
And if the arc cotangent is negative square root
three over one, we would think that the arc tangent
would be negative one divided by square root three.
And notice, we have negative 30 degrees,
which does not match your answer.
This does not mean that we are incorrect
because remember that the output for arc tangent
is between negative pi over two and pi over two,
and the output for arc cotangent is between
zero and pi radians.
So we can notice that we have a
30 degree reference angle from this,
but it doesn't really help us evaluate this.
So what this tells us is that we have to be very careful
when we use our calculator to evaluate some of these.
This is the correct answer here,
but there's no way of really verifying it
on the graphing calculator because of the restrictions
that we made in the domain to come up with these inverse
trick functions.
Okay, I hope you found this video helpful.
Thank you for watching.
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