3 - i

This is the vector from the origin (0,0) to point . 
It has length r, which is always a positive number.

So we draw the vector:



Then we draw the perpendicular to the x-axis:




Then the green line is the same as the y-coordinate ,
negative because it goes down from the x-axis. and the horizontal
line from the origin to the green line is the x-coordinate 3.




We calculate r which is the length of the vector by the Pythagorean
theorem:

r² = x² + y²

r² = 3² + ()²

r² = 9 + 3

r² = 12

r = 

r = 

r = 

Next we calculate , which is the angle from
the right side of the x-axis to the vector, as indicated
by the red arc below:



First we calculate the reference angle, indicated by the blue arc, the
smallest angle between the vector and the x-axis:



We can calculate it using any trig function, let's pick the sine:

sin(reference angle) =  =  

So the reference angle is 30° and therefore  = 360° - 30° = 330°

So the trig form (or polar form) of the complex number

3 - i

is

r(cos + i·sin) or

2[cos(330°) + i·sin(330°)]

Edwin