We're asked to find an angle between zero
and 2 pi radians that is coterminal with the given angle.
Number one, we have negative 14 pi divided by 5 radians.
To determine coterminal angles to an angle given in radians,
we add or subtract multiples of 2 pi radians
to the given angle.
Which means all the coterminal angles
for number one would be in the form
of negative 14 pi divided
by 5 radians plus 2 pi radians times k,
where k is some integer.
And now we just need to find the value of k
that will give us a coterminal angle between zero
and 2 pi radians.
Because the given angle has a denominator of 5,
it might be helpful to recall that 2 radians,
or 2 over 1 radians, is equivalent to 10 pi divided
by 5 radians.
And because we're given a negative angle,
we'll have to be adding multiples of 2 pi.
So, let's see what angle we get when k's equal
to positive 1.
We would have negative 14 pi divided by 5 plus 2 pi,
or 2 pi over 1 radians, and we already know 2 pi is equal
to 10 pi divided by 5.
We have a common denominator of 5,
and the numerator is now negative 14 pi plus 10 pi,
which is negative 4 pi.
So, negative 4 pi divided by 5 is coterminal
to a negative 14 pi divided by 5 radians,
but it's not in the interval from zero to 2 pi radians,
and therefore, let's try k equals 2.
When k's equal to 2, we have negative 14 pi divided
by 5 plus 2 pi times 2, which would be 4 pi, or 4 pi over 1,
and again, the common denominator is going to be equal to 5.
Adding, we have a common denominator of 5,
and the numerator is now negative 14 pi plus 20 pi,
which is 6 pi, and 6 pi divided by 5 is in the interval
from zero to 2 pi radians, or zero to 10 pi divided
by 5 radians, and therefore,
this is the coterminal angle we are looking for.
Number two's going to be a little more challenging
because the angle is much smaller.
We have negative 125 pi divided by 7 radians.
But again, to find the coterminal angles,
we add or subtract multiples of 2 pi radians,
which means all the coterminal angles will be in the form
of negative 125 pi divided by 7 plus 2 pi k,
where again k is some integer.
To help us determine the value of k
that will give us the coterminal angle between zero
and 2 pi radians, let's use desmos.com.
To set this up,
recall that 2 pi is approximately 2.68 radians.
And then in cell two, enter the expression,
negative 125 pi divided by 7, right arrow, plus 2 pi k.
To enter k, press the ABC button, and then the variable k.
Next, we want to create a slider by clicking the k,
and now we need to change the slider by clicking
in cell three.
We know we're going to be adding multiples of 2 pi
to get a coterminal angle between zero and 2 pi radians.
So, let's let the interval for k be from zero to,
let's say, 40.
We can always come back and change this,
but because k is an integer, the step must always be 1,
and now let's go ahead and close the keypad,
click out of cell three, and now using the slider,
if we let k equal zero, we have the decimal approximation
with a given angle, and now we can change the value
of k until we have a coterminal angle that's between zero
and approximately 2.68 radians.
So notice here, when k's equal to 9,
we do have a coterminal angle that's between zero
and 6.28 radians.
But this is just the decimal approximation
for the coterminal angle that we want,
we want the exact value.
So now, we'll take the value of k equals 9,
and determine the exact angle.
So, when k is equal to 9, we have negative 125 pi divided
by 7 plus 2 pi times 9,
which is equal to negative 125 pi divided
by 7 plus 18 pi, or 18 pi over 1.
The common denominator is 7.
This gives us negative 125 pi divided
by 7 plus 18 pi times 7 is equal to 126 pi,
and the denominator is 7.
When adding, the denominator remains 7,
and the numerator is now negative 125 pi plus 126 pi,
which is 1 pi, or just pi.
The coterminal angle to negative 125 pi divided by 7 that is
between zero and 2 pi radians is pi divided by 7 radians.
If we go back to the calculator one more time,
we can check this.
Let's make sure that when we enter pi divided
by 7 we do get the same decimal approximation of the angle
above, which we found when k is equal to 9,
and we can see that we do.
I hope you found this helpful.