Question 577079
(-3-3i)(2+2i)
<pre>
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 <font face="symbol">q</font> 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(cos<font face="symbol">q</font> + i·sin<font face="symbol">q</font>)

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 <font face="symbol">q</font> with a counter-clockwise
    arc from the right side of the x-axis around to the radius vector:       

    {{{drawing(400,400,-5,5,-5,5,graph(400,400,-5,5,-5,5),line(0,0,-3,-3),
       line(-3,-3,-3,0),locate(-2.3,.6,x=-3),locate(-4.2,-1.4,y=-3),
       red(arc(0,0,1.8,-1.8,0,225)), locate(-1.7,-1.7,r),locate(-.6,1.2,theta) )}}}

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={{{sqrt(18)}}}
     r={{{sqrt(9*2)}}}
     r = 3{{{sqrt(2)}}} 

    {{{drawing(400,400,-5,5,-5,5,graph(400,400,-5,5,-5,5),line(0,0,-3,-3),
       line(-3,-3,-3,0),locate(-2.3,.6,x=-3),locate(-4.2,-1.4,y=-3),
       red(arc(0,0,1.8,-1.8,0,225)), locate(-1.7,-1.7,r=3sqrt(2)),locate(-.6,1.2,theta) )}}}

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

4.  We write the trig form as r(cos<font face="symbol">q</font> + i·sin<font face="symbol">q</font>), or 
    
    {{{3sqrt(2)}}}(cos225° + i·sin225°).

---------------

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 <font face="symbol">q</font> with a counter-clockwise
    arc from the right side of the x-axis around to the radius vector:       

    

    {{{drawing(400,400,-5,5,-5,5,graph(400,400,-5,5,-5,5),line(0,0,2,2),
       line(2,2,2,0),locate(1.1,.6,x=2),locate(2.2,1,y=2),
       red(arc(0,0,1.8,-1.8,0,45)), locate(.7,1.4,r),locate(-.6,1.2,theta) )}}}

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={{{sqrt(8)}}}
     r={{{sqrt(4*2)}}}
     r = 2{{{sqrt(2)}}} 

  {{{drawing(400,400,-5,5,-5,5,graph(400,400,-5,5,-5,5),line(0,0,2,2),
       line(2,2,2,0),locate(1.1,.6,x=2),locate(2.2,1,y=2),
       red(arc(0,0,1.8,-1.8,0,45)), locate(.1,1.4,r=2sqrt(2)),locate(-.6,1.2,theta) )}}}

 
3.  We calculate the angle <font face="symbol">q</font>, 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(cos<font face="symbol">q</font> + i·sin<font face="symbol">q</font>), or 
    
    {{{2sqrt(2)}}}(cos45° + i·sin45°).

-----------------

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

  r<sub>1</sub>(cos<font face="symbol">q</font><sub>1</sub> + i·sin<font face="symbol">q</font><sub>1</sub>)·r<sub>2</sub>(cos<font face="symbol">q</font><sub>2</sub> + i·sin<font face="symbol">q</font><sub>2</sub>) =   r<sub>1</sub>r<sub>2</sub>[cos(<font face="symbol">q</font><sub>1</sub>+<font face="symbol">q</font><sub>2</sub>) + i·sin(<font face="symbol">q</font><sub>1</sub>+<font face="symbol">q</font><sub>2</sub>)]. 
 
So we have:

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

{{{2sqrt(2)*3sqrt(2)}}}[cos(225°+45°) + i·sin(225°+45°)] = 6·2[cos270° + i·sin270°] =

12(cos270° + i·sin270°).

Edwin</pre>