Given an angle,
find the corresponding point on the unit circle.
To find the corresponding point,
we sketch the angle in standard position
and determine where the terminal side of the angle
intersects the unit circle.
This is an important skill to learn
because you can use the points on the unit circle
to determine the six trig function values
as defined here below.
The sketch the angle in standard position
the initial side is always along the positive access,
and then if the angle is positive,
we rotate counter-clockwise.
If the angle is negative,
we rotate clockwise.
And most unit circles do have the angles labeled
from zero to two pi radians,
but normally doesn't have negative angles labeled.
The first angle is three pi divided by four radians,
or 3/4 pi radians.
So starting along the positive x axis,
need to rotate counter-clockwise 3/4 pi radians,
and, therefore, the terminal side is here.
Again, rotating counter-clockwise from the initial side.
Notice how the unit circle also shows the angle and degrees.
The corresponding point on the unit circle
for 3/4 pi radians
is negative square root of two divided by two,
positive square root of two divided by two.
Now that we know this point,
we could find these six trig function values
using this point
when theta is 3/4 pi radians.
The next angle is 4/3 pi radians,
the initial side is along the positive x-axis
and now we rotate counter-clockwise 4/3 pi radians to here,
which notice is the same as 240 degrees.
So the corresponding point on the unit circle
for 4/3 pi radians is -1/2,
negative square root three divided by two.
The next angle is 11/6 pi radians,
or 11 pi divided by 6 radians.
We have the initial side rotating counter-clockwise,
the terminal side is going to be here.
The corresponding point on the unit circle
is going to be positive square root three divided by two,
1/2.
And now let's take a look at three negative angles.
Here we have negative pi divided by two radians,
or -1/2 pi radians.
You'll notice how positive pi divided by two radians
is equal to 90 degrees and,
therefore, because this angle is negative,
we need to rotate clockwise 90 degrees,
or clockwise pi over two radians.
So initial side is here.
To rotate clockwise pi over two radians,
the terminal side would be along the negative y-axis,
again, with a clockwise rotation of pi over two radians.
So this is the angle, negative pi over two radians,
and in standard position,
the corresponding point on the unit circle
is the point 0, -1.
Next we have -5/3 pi radians.
Notice how +5/3 pi radians
is equal to 300 degrees
and, therefore, to sketch this angle in standard position,
need to rotate clockwise 300 degrees,
or clockwise 5/3 pi radians.
The initial side is here.
To rotate clockwise, 5/3 pi radians,
or 300 degrees, then the terminal side is going to be here.
Notice -5/3 pi radians is co-terminal
to pi divided by three radians.
Or we could say -300 degrees is co-terminal to 60 degrees.
Either way, the terminal side intersects the unit circle
at the point 1/2, square root of three divided by two.
The last angle is -9/4 pi radians,
which is more challenging.
If we needed to, we can always convert to degrees.
9/4 pi radians is equal to -405 degrees,
which is equal to -360 degrees minus 45 degrees.
So one way to graph the angle
would be to rotate one complete rotation clockwise
for -360 degrees,
and then for -45 degrees, or minus 45 degrees,
we would rotate 45 degrees in the clockwise direction.
Let's also show how to do this using radians.
Let's write -9/4 pi radians
as -8/4 pi radians,
minus 1/4 pi radians.
Notice this is still -9/4 pi radians,
but -8/4 pi is equal to -2 pi
and, therefore, we can write this as -2 pi radians
minus 1/4 pi radians.
You'll notice that -2 pi is the same as -360 degrees,
and the minus 1/4 pi, or minus pi over four,
is the same as minus 45 degrees.
So either way, to graph the angle in standard position,
the initial side is along the positive x-axis,
and we have one complete rotation in the clockwise direction
for the negative two pi, or -360 degrees,
and then another rotation of 1/4 pi radians
in the clockwise direction
for -1/4 pi radians,
or 45 degrees in the clockwise direction for -45 degrees.
Either way, the terminal side is here for the given angle,
and, therefore, the corresponding point on the unit circle
is positive square root two divided by two,
negative square root of two divided by two.
I hope you found this helpful.