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