We're asked to find an angle
between zero and two pi radians
that is coterminal with the given angle.
Number one, we have 11 pi divided by two radians.
Number two, we have 78 pi divided by nine radians.
To determine coterminal angles to a given angle in radians,
we add or subtract multiples of two pi radians
to the given angle, which means all the coterminal angles
for 11 pi divided by two, can be expressed
as 11 pi divided by two plus two pi K,
where K is some integer.
Again, this expression shows the coterminal angles are found
by adding or subtracting multiples of two pi
to 11 pi divided by two radians,
again where K is some integer.
If K is positive, we add multiples of two pi.
If K is negative, we subtract multiples of two pi.
Now we need to determine the value of K
so that the coterminal angle
is between zero and two pi radians.
Well, first two pi radians is the same
as two pi divided by one radians.
And because our denominator is two,
we can multiply the numerator and denominator by two
and two pi is equal to four pi divided by two.
This will help us determine when we find a value of K
that gives us a coterminal angle
between zero and two pi radians.
For example, when K is equal to negative one,
we have 11 pi divided by two plus two pi
times negative one, which is equal to 11 pi
divided by two minus two pi over one.
And we know two pi over one
is equal to four pi divided by two,
which gives us a denominator of two.
And the numerator is now 11 pi minus four pi,
which is seven pi.
Well, seven pi divided by two is a coterminal angle,
but it's not between zero and two pi radians
or zero and four pi divided by two radians.
Let's try K equals negative two.
When K is equal to negative two,
we have 11 pi divided by two plus two pi
times negative two, which is equal to 11 pi
divided by two minus four pi over one.
Four pi over one is equal to eight pi over two.
We have a common denominator of two
and the numerator is now 11 pi minus eight pi,
which is three pi.
And three pi divided by two
is between zero and two pi radians,
or zero and four pi divided by two radians.
This is the coterminal angle we are looking for
between zero and two pi radians.
Now, number two is a little more challenging
because we have a much larger angle.
We have 78 pi divided by nine radians,
but all the coterminal angles can be found
by evaluating the expression 78 pi divided by nine
plus two pi K where K is some integer.
With this example, though,
let's use the help of Desmos to determine the value of K
that will give us the coterminal angle
between zero and two pi radian.
So, first in cell one if we enter two pi,
this will remind us, we're looking for an angle
between zero and approximately 6.28 radians.
And then in cell two, we'll enter the expression
78 pi divided by nine,
right arrow, plus two pi times K.
Next, we create a slider for K.
We know K has to be negative
in order to determine a coterminal angle
between zero and two pi radians.
Let's click in cell three.
Let's change the interval for K
from let's say negative 20 to zero.
The step has to be one because K is an integer.
And now we click outside cell three, close the keyboard.
And now what we're going to do is determine the value of K
that will give us a coterminal angle
between zero and approximately 6.28 radians.
Notice when K is equal to zero,
we have the decimal approximation
for the given angle of 78 pi divided by nine.
And I'm moving the slider to the left.
Again, we're looking for an angle
between zero and approximately 6.28 radians.
And we found the angle when K is equal to negative four.
So, this is the decimal approximation
of the coterminal angle,
but we went the exact coterminal angle.
So now we'll perform the calculation
when K is equal to negative four.
So when K is equal to negative four, we have 78 pi
divided by nine plus two pi times negative four,
which is equal to 78 pi divided by nine
minus eight pi over one.
The common denominator is nine
and the numerator is 78 pi minus 72 pi, which is six pi.
Well, six pi divided by nine does simplify.
There's a common factor of three between six and nine.
This gives us a coterminal angle of two pi divided by three.
Two pi divided by three is the coterminal angle
to 78 pi divided by nine
that is between zero and two pi radians.
And if we go back to the calculator one last time,
we can check this.
If we enter two pi divided by three in cell four,
notice that we do get the same decimal
that we have above for the coterminal angle
between zero and approximately 2.68 radians.
I hope you found this helpful.