We will evaluate the inverse cosine function
values using the reference triangles shown here below.
For each inverse cosine function value is equal
to an angle on the closed interval from zero to pi radians,
which means in standard position,
the angle would be in this interval here
from zero to pi radians,
including both endpoints.
The first expression is inverse cosine
of negative square root three divided by two,
which again is equal to the angle in this interval
that has a cosine function value of negative square root
of three divided by two.
Looking at the reference triangles below,
notice the cosine of 30 degrees,
or pi over six radians,
is equal to the ratio of the adjacent side
to the hypotenuse,
which gives us positive square root three divided two.
We're looking for the angle in this interval that has
a cosine function value of negative square root three
divided by two.
So the angle is not 30 degrees, or pi over six radians,
but this angle is the reference angle
in the second quadrant,
where cosine is negative.
So the next step, we sketch a reference angle,
and corresponding reference triangle
using 30 degrees or pi over six radians,
which means the terminal side of the angle
we are looking for would be this right here,
where again the reference angle,
this angle here is 30 degrees or pi over six radians.
And now sketch the reference triangle.
Because we are in the second quadrant,
where x is negative and y is positive,
we label the opposite side positive one,
the adjacent side negative square root three,
again because x is negative,
and the hypotenuse positive two.
Notice how this reference triangle does give
a cosine function value of negative
square root three divided by two,
and then for the angle we are looking for has
the initial side along the positive x-axis,
the terminal side along this ray,
with a counter-clockwise rotation,
the angle is 150 degrees, or in radians,
5/6 pi radians,
which we can determine by taking 180 degrees
and subtracting the reference angle of 30 degrees,
which gives us 150 degrees,
or pi radians minus 1/6 pi radians
which gives us 5/6 pi radians.
Next we have inverse cosine of square root
two divided by two,
which is equal to the angle in this interval
that has a cosine function value of
square root two divided by two.
This cosine function value should remind us
of the 45-45-90 reference triangle shown here,
but it could also be helpful to rationalize
the numerator of this cosine function value,
so let's also do that.
If we have the square root of two divided by two,
to rationalize the numerator,
we multiply the numerator and denominator
by the square root of two.
This gives us two divided by two square root two,
two divided by two simplifies to one,
leaving us with one divided by square root of two.
So we can also rewrite this as
inverse cosine of one divided by square root of two.
Now looking at the 45-45-90 reference triangle,
notice the cosine of 45 degrees or pi over four radians
is equal to the ratio of the adjacent side
to the hypotenuse,
which does give us one divided by square root two,
and because the given cosine function value
is also positive,
the angle we are looking for is 45 degrees,
or pi over four radians.
So this is equal to 45 degrees or pi over four radians
or 1/4 pi radians.
So because the given cosine function value is positive,
the angle had to be in the first quadrant,
and therefore we did not have to sketch the reference angle,
or the reference triangle.
Let's go ahead and do it anyway.
We would sketch a 45 degree reference angle
in the first quadrant.
This would be the terminal side of the angle
we are looking for.
Where the reference angle, this angle here,
is 45 degrees or pi over four radians,
and this would be the reference triangle.
Because we are in the first quadrant,
both x and y are positive one,
and therefore both legs are positive one,
hypotenuse is square root two,
notice how this reference triangle does give
a cosine function value of one
divided by square root two,
which means the angle we are looking for
in the first quadrant has its initial side
along the positive x-axis,
terminal side along this ray,
with a counter-clockwise rotation,
notice how the angle is the same as the reference angle,
of 45 degrees, or radians, pi over four, or 1/4 pi radians.
Next we have inverse cosine of negative 1/2,
going back to the 30-60-90 reference triangle,
notice how the cosine of 60 degrees,
or pi over three radians, is equal to one half,
but we are given a cosine function value of
negative one half,
which means 60 degrees, or pi over three radians,
is the reference angle in the second quadrant,
for the angle we are looking for,
since cosine is negative in the second quadrant.
Now we sketch a reference angle of
60 degrees or pi over three radians in the second quadrant,
and therefore this would be the terminal side
of the angle we are looking for,
where the reference angle, this angle here,
is 60 degrees or pi over three radians.
Let's sketch the reference triangle,
again we are in quadrant II,
where x is negative and y is positive,
the opposite leg would be positive square root three,
the adjacent leg would be negative one,
hypotenuse positive two.
Again notice how this reference triangle does give
a cosine function value of negative one half,
which means the angle we are looking for
has initial side along the positive x-axis,
terminal side along this ray,
with a counter-clockwise rotation,
the angle is 120 degrees, or 2/3 pi radians,
which we can determine by taking 180 degrees
and subtracting the reference angle of 60 degrees,
or pi radians minus 1/3 pi radians,
which gives us 2/3 pi radians.
The last expression is inverse cosine of negative one.
We will not be able to evaluate this using
the reference triangles,
we will have to use the unit circle
or knowledge of the graph of the cosine function.
This is equal to the angle in the closed interval from
zero to pi radians that has a cosine function
value of negative one.
On the unit circle, x equals cosine theta,
and therefore if this is the unit circle,
this is the point where the x-coordinate is negative one.
The ordered pair for this point would be
negative one comma zero,
which means this is the terminal side
of the angle we are looking for,
this is the initial side,
with a counter-clockwise rotation,
the angle is pi radians, or 180 degrees.
Or if we analyze the graph of the cosine function,
only over the interval from zero to pi radians,
which is this yellow interval here,
notice how the cosine function value is negative one,
at this point here where the angle is pi radians,
or 180 degrees.
I hope you found this helpful.