Determine the angle graphed in standard position.
Each angle is multiple of 30 degrees or 45 degrees.
When our angle is in standard position,
the initial side is along the positive X axis.
If the terminal side is rotated counterclockwise,
as we see for these three examples,
the angle is positive.
If the terminal side is rotated clockwise,
as we see in these three examples,
the angle is negative.
So for a quick review,
the initial side is along the positive X axis,
which is here,
and if we rotate counterclockwise,
then the angle is positive.
So rotating counterclockwise,
we have 30 degrees, 45 degrees, 60 degrees,
90 degrees and so on.
One complete rotation counterclockwise is equal
to 360 degrees.
And again, if the rotation is clockwise,
the angle is negative.
So if the initial side is here,
and the the terminal side is rotated clockwise,
we have negative 30 degrees,
negative 45 degrees,
negative 60 degrees, negative 90 degrees,
one complete rotation clockwise is equal
to negative 360 degrees.
Let's go back to our examples.
Looking at the first three examples,
again, because their rotation indicated
by the arrow is counterclockwise,
these three angles are positive.
So for this first angle.
Again, the initial side is along the positive X axis,
which is here.
The terminal side in the second quadrant is here,
and because the rotation is counterclockwise,
we know the angle is positive,
so we have 30 degrees, 45 degrees,
60 degrees, 90 degrees.
We need to rotate another 60 degrees,
90 plus 60 is equal to 150,
giving us 150 degrees.
So the next angle is standard position.
The initial side is along the positive X axis,
the terminal side is here in the third quadrant.
So the terminal side is here.
Because the rotation is counterclockwise,
the angle is positive,
so we have 90 degrees, 180 degrees,
and then we rotate another 45 degrees.
180 degrees plus 45 degrees is equal to 225 degrees.
And for the last positive angle,
the initial side is here along the positive X axis
and now the terminal side is in the fourth quadrant here.
Again, rotating counterclockwise,
we have 90 degrees, 180 degrees,
270 degrees, and then we rotate another 30 degrees,
270 degrees plus 30 degrees is equal to 300 degrees.
And now, let's take a look at three negative angles.
Again, we know the angles are negative
because the rotation is clockwise.
So with the first example,
the initial side is here along the positive X axis.
The terminal side is in the fourth quadrant here,
but now the rotation is clockwise
and therefore the angle is negative.
So we have negative 30, negative 45, negative 60 degrees.
For the next angle.
Again, initial side is here along the positive X axis.
The terminal side is in the first quadrant here
and the rotation is clockwise.
So we have negative 90 degrees
and negative 180 degrees,
negative 270 degrees.
We need to rotate another 30 degrees clockwise,
negative 270 degrees plus negative 30 degrees
equals negative 300 degrees.
And finally, for the last angle.
We have the initial side here, along the positive X axis,
the terminal side is now in the third quadrant here.
Rotating clockwise, we have negative 90 degrees,
and then we rotate another 45 degrees clockwise,
negative 90 plus negative 45 is equal
to negative 135 degrees.
I hope you found this helpful.