We will evaluate the given
in cotangent function values
using the reference triangles shown here below
where each inverse cotangent function value
is equal to an angle on the open interval
from zero to pi radians,
which means in standard position,
again, the angle will be from zero to pi radians
not including the two end points,
which is this interval here.
Since it's cotangent theta's equal
to X divided by Y on the coordinate plane,
notice cotangent is positive in the first quadrant
and negative in the second quadrant.
To begin we have inverse cotangent of negative one,
which is equal to the angle in this interval
that has a cotangent function value of negative one.
Using the reference triangles, we begin
by ignoring the negative sign for a moment
and determine which angle has
a cotangent function value of positive one.
Using right triangle trigonometry,
cotangent theta is equal to the ratio
of the adjacent side to the opposite side.
Referring to the reference triangles,
notice how the cotangent of 45 degrees
or pi divided by four radians
again is equal to the ratio of the adjacent side
to the opposite side, which gives us one
divided by one, which is positive one,
and since we are looking for the angle
that has a cotangent function value of negative one,
we are not looking for the angle of 45 degrees
or pi divided by four radians,
but this is the reference angle
that we use in the second quadrant
to determine the angle in this interval
that does have a cotangent function value of negative one.
The next step is to sketch a reference angle
of 45 degrees in the second quadrant,
which means this is the terminal side
of the angle we are looking for, again,
where the reference angle is 45 degrees
or pi divided by four radians.
Let's go ahead and sketch that reference triangle.
Because we are in the second quadrant
where X is negative and Y is positive,
we would label the adjacent leg negative one,
the opposite leg one,
and the hypotenuse square root of two.
Notice how this reference triangle
does give us a cotangent function value
of negative one divided by one,
which is negative one,
which means the angle we are looking for
in the open interval from zero to pi radians
has its initial side along the positive x-axis
and the terminal side along this ray,
and therefore the angle we are looking for
in the interval that has a cotangent function value
of negative one is 135 degrees
or 3/4 pi radians,
which we could determine by taking 180 degrees
and subtracting the reference angle of 45 degrees,
or pi radians, and subtracting pi over four,
or 1/4 pi radians,
and therefore inverse cotangent of negative one
is equal to 135 degrees,
or in radians, 3/4 pi radians.
Next we have inverse cotangent
of the square root of three,
which, if it's helpful, we could write
square root of three as a fraction
with the denominator of one.
This is equal to inverse cotangent
of square root of three over one,
and now we go back to the reference triangles
and determine which angle has a cotangent function value
of square root of three divided by one.
Looking at the 30 60 90 reference triangle,
notice how the cotangent of 30 degrees
or pi divided by six radians,
which again is equal to the ratio of the adjacent side
to the opposite side,
is square root of three divided by one,
which is the given cotangent function value,
and therefore the angle we are looking for
is 30 degrees, or in radians, pi over six or 1/6 pi radians.
So notice how when the given cotangent function value
is positive, we can use the reference triangle as is,
and we don't have to sketch the reference angle
or the reference triangle.
Let's go ahead and show the reference angle
and triangle anyway on the coordinate plane,
which, because the cotangent function value is positive,
would be in the first quadrant.
So let's sketch a reference angle of 30 degrees
in the first quadrant,
which means the terminal side would be here
where the reference angle again is 30 degrees
or pi divided by six radians,
and now let's sketch the reference triangle.
We are in the first quadrant
where both X and Y are positive,
and therefore the opposite side is one,
hypotenuse is two,
and the longer leg is positive square root of three.
Again, notice how the ratio of the adjacent side
to opposite side is square root of three,
and therefore the angle we are looking for
has its initial side along the positive x-axis,
terminal side along this ray,
and notice how the angle is the same
as the reference angle of 30 degrees,
or in radians, pi over six or 1/6 pi radians.
Next we have inverse cotangent
of negative square root of three divided by three.
Because of the square root of three,
we should be able to recognize
we will be using the 30 60 90 reference triangle,
but to make it easier to recognize the angle
we will use to evaluate this,
it may be helpful to rationalize the numerator.
So let's go ahead and show that.
We have negative square root of three
divided by three and want to rationalize the numerator.
We multiply the numerator and the denominator
by the square root of three,
which gives us negative three
divided by three square root of three.
Three divided by three simplifies to one,
and therefore this simplifies to negative one
divided by a square of three.
We can rewrite this as inverse cotangent
of negative one divided by a square root of three.
Ignoring the negative sign for a moment,
going back to the reference triangles,
notice how the cotangent of 60 degrees
or pi divided by three radians
is equal to the ratio of the adjacent side
to the opposite side, which is one
divided by a square root of three,
and because we want the angle
that has a cotangent function value
of negative one divided by a square root of three,
the angle is not 60 degrees
or pi divided by three radians,
but this is the reference angle we will use
in the second quadrant to determine the angle
we are looking for.
So we now sketch a reference angle of 60 degrees
or pi divided by three radians in the second quadrant,
and therefore the terminal side is here,
the angle we are looking for
where this reference angle is again 60 degrees,
or pi divided by three radians.
Let's sketch the reference triangle.
We're in the second quadrant
where X is negative and Y is positive.
We'll label this leg negative one,
the hypotenuse two,
and this leg positive square root of three.
Again, notice how this reference triangle
does give a cotangent function value
of negative one divided by square root of three,
which means the angle we are looking for
in the open interval from zero to pi radians
that has a cotangent function value of negative one
divided by the square root of three
has its initial side along the positive x-axis,
terminal side along this ray.
Rotating counter clockwise
this is going to be 120 degrees or 2/3 pi radians.
If we can determine by taking 180 degrees
and subtracting the reference angle of 60 degrees
or pi radians and subtracting pi over three
or 1/3 pi radians,
which gives us the angle in degrees and radians.
This is equal to 120 degrees.
or in radians, 2/3 pi radians,
and then finally we have inverse cotangent of zero.
I wanted to include this one
because we will not be able to use
the reference triangles to evaluate this.
We will have to use either the unit circle
or our knowledge of the graph
of the cotangent function.
Now on the unit circle, cotangent theta is equal
to X divided by Y, which means we evaluate this
using the unit circle.
We need to find the point on the unit circle
in this interval where X divided by Y is equal to zero.
X divided by Y is equal to zero at this point
on the unit circle in the given interval
where the x-coordinate is zero
and the y-coordinate is positive one.
Notice X divided by Y is zero divided by one,
which is zero,
which means the terminal side of the angle
we are looking for
that has a cotangent function value of zero
in this interval has the terminal side
along the positive y-axis,
the initial side along the positive x-axis,
and therefore the angle is 90 degrees,
or in radians, pi over two or 1/2 pi radians.
Another way to evaluate this
would be to take a look at the graph
of the cotangent function
and determine the angle in the open interval
from zero to pi radians
that gives a cotangent function value of zero.
So here is the graph of the cotangent function
only on the open interval from zero to pi radians,
and notice how the cotangent function value is zero
at this point where the angle is pi over two radians
or 90 degrees.
I hope you found this helpful.