Convert each to trig form:

First we convert -3-3i to trig form:

1.  The complex number x+yi is represented by the radius vector (line segment)
    from the origin (0,0) to the point (x,y).  So draw the radius vector, and
    the perpendicular from the point (x,y) to the x-axis.
2.  Calculate the length of that vector r, using the Pythagorean theorem:
    r²=x²+y².  That value is called the modulus of the complex number.
3.  Calculate the angle q from the right side of the axis around to the
    radius vector.  To do this you may use any of the trig ratios involving
    x, y and r.  This angle is called the argument. 
4.  Write the trig form as r(cosq + i·sinq)

First we convert -3-3i to trig form:

1.  We draw the radius vector connecting the origin to the point (-3,-3) and
    the perpendicular from that point to the to the x-axis.  We label the
    perpendicular to the x-axis as y=-3 and the segment from the origin to
    the perpendicular.  We label the length of the radius vector r, and
    indicate the argument q with a counter-clockwise
    arc from the right side of the x-axis around to the radius vector:       

    

2.  We calculate the length of that vector r, using the Pythagorean theorem:
    r²=x²+y².  
    r²=(-3)²+(-3)²
    r²=9+9
    r²=18
     r=
     r=
     r = 3 

    

3.  We calculate the angle q, by realizing that
    the right-triangle is a 45°-45°-90° with a reference angle of 45°, and
    the actual angle q = 180°+45° = 225°

4.  We write the trig form as r(cosq + i·sinq), or 
    
    (cos225° + i·sin225°).

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Next we convert 2+2i to trig form:

1.  We draw the radius vector connecting the origin to the point (2,2) and
    the perpendicular from that point to the to the x-axis.  We label the
    perpendicular to the x-axis as y=2 and the segment from the origin to
    the perpendicular x=2.  We label the length of the radius vector r, and
    indicate the argument q with a counter-clockwise
    arc from the right side of the x-axis around to the radius vector:       

    

    

2.  We calculate the length of that vector r, using the Pythagorean theorem:
    r²=x²+y².  
    r²=(2)²+(2)²
    r²=4+4
    r²=8
     r=
     r=
     r = 2 

  

 
3.  We calculate the angle q, by realizing that
    the right-triangle is also a 45°-45°-90° which is an angle of 45°

4.  We write the trig form as r(cosq + i·sinq), or 
    
    (cos45° + i·sin45°).

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Now we use the formula for multiplying complex numbers in trig form:

  r1(cosq1 + i·sinq1)·r2(cosq2 + i·sinq2) =   r1r2[cos(q1+q2) + i·sin(q1+q2)]. 
 
So we have:

(-3-3i)(2+2i) = (cos225° + i·sin225°)·(cos45° + i·sin45°) =

[cos(225°+45°) + i·sin(225°+45°)] = 6·2[cos270° + i·sin270°] =

12(cos270° + i·sin270°).

Edwin