We want to solve the equation
cosine x equals 0.45 on the given interval,
which means we're trying to find the angles x
that have a cosine function value of 0.45,
again on the given interval.
The first thing we might recognize
is that we're not going to find angles
that have a cosine function value of 0.45
using the unit circle or reference triangles,
so we'll have to use the calculator
in order to solve this equation.
And also, in order to solve this for x,
we will have to use the inverse cosine
or arccosine function.
So for a quick review, if we have y equals
arccosine x, x are all of the possible
cosine function values, which means the domain
would be the closed interval from negative one to one,
and the output or y of the arccosine function
would be an angle on the closed interval
from zero to pi radians.
So in order to solve the equation, cosine x equals 0.45,
we want to solve the equation for x,
which means to undo cosine, we'll have to take
the arccosine or inverse cosine
of both sides of the equation.
So we'd have inverse cosine of the left,
as well as inverse cosine of the right.
So we'd have inverse cosine, cosine x,
equals inverse cosine 0.45.
So inverse cosine of cosine x simplifies to just x,
so we have x equals inverse cosine of 0.45.
So now we'll go to the calculator
to determine the angle on the closed interval
from zero to pi radians
that has a cosine function value of 0.45.
We first want to verify that we're on radian mode,
so we'll press the mode key.
Notice how radian is highlighted,
so we'll go back to the home screen,
and now we'll press second cosine for inverse cosine,
0.45, close parenthesis, and enter.
So notice how the calculator is giving us
the angle 1.1040 radians if we round to four decimal places.
So we know one solution is x equals 1.1040 radians,
which again, is the angle on the closed interval
from zero to pi radians
that has a cosine function value of 0.45,
and this would be the angle in the first quadrant.
Let's go ahead and sketch it in the standard position.
So we'd have the positive x-axis here.
Rotate counterclockwise, 1.1040 radians,
to approximately here.
So if we sketch the reference triangle,
it would look like this.
But this is not the only solution on the given interval.
Remember cosine theta is equal to x divided by r,
and because x is positive in the first
and the fourth quadrants, there's a second angle
in this interval that has the same cosine function value.
It would be the angle in the fourth quadrant
that has the same reference angle.
So if we sketch an angle in the fourth quadrant
that has the same reference angle,
the terminal side might look something like this.
Again, where this reference angle is 1.1040 radians.
So the reference triangle would look something like this.
And notice how these two right triangles
do share this side along the positive x-axis,
which would be x.
So to determine what this angle would be,
so to determine this angle,
you would start along the positive x-axis
and rotate counterclockwise to here.
So to find this angle, we would take two pi radians
and subtract a reference angle of 1.1040 radians.
So, the second solution on the given interval
would be x equals two pi radians
minus 1.1040 radians.
Now we did round this angle here,
so if we wanted to be more precise,
we could say two pi minus inverse cosine of 0.45.
Let's show both of these on the calculator
to see if there's much of an error
if we round to four decimal places.
So we'll take two pi
and subtract 1.1040,
and let's go ahead and compare this value to two pi
minus inverse cosine of 0.45.
Again, if we round to four decimal places,
we would have the same approximation of 5.1792 radians.
Which again is this angle here
that terminates in the fourth quadrant.
So we'll say x sub one is approximately 1.1040 radians
and x sub two is approximately 5.1792 radians.
I hope you found this helpful.