We draw the angle -120°.
Since it's negative we draw the rotation clockwise
120° from the right side of the x-axis as indicated
by the arc below. But we call it -120° because the
rotation is clockwise. The line segment, called the
"radius vector" is of arbitrary length but we will
give it a specific length later.
Next we calculate the reference angle of -120°,
which is the smallest angle taken positive between
the radius vector and the x-axis.
We have measured 120° clockwise so as to get
the -120° angle, and we know that the rotation all
the way to the left side of the x-axis is 180°. So we
subtract 180°-120° and get that the reference angle
is 60°, which I have indicated below with a red
arc:
Now 60° happens to be one of the special angles. For
this special angle, We must have memorized the 30°-60°-90°
right triangle which has shorter leg 1, hypotenuse 2,
and longer leg 
. So we use the radius vector
as the hypotenuse, and so we will give it the length 2 units.
We will label it r, as shown below.
We will draw a perpendicular up to the x-axis, shown in green
below, which is units long. We call it y
because it is parallel to the y-axis. However we will give it
a negative sign and make it 
because it goes down
below the x-axis.
We will call the horizontal distance from the origin
to the top of the green line the short leg of the 30-60-90
triangle. We label that upper leg of the triangle x because it's
on the x-axis. It is 1 unit long, but we will give it a negative
sign (-1) because it goes to the left of the origin.
Now we want the cosecant of the angle -120°.
The cosecant is the hypotenuse over the opposite.
In this case it is 
Therefore 
If we rationalize the denominator we will have
Edwin
