Question 867773
Write the complex number in rectangular form.
<pre>
1.) {{{3(cos(pi/3) + i*sin(pi/3))}}}

Just evaluate it:

    {{{3(1/2+i* expr(sqrt(3)/2)) }}}

    {{{3/2+expr(3sqrt(3)/2)i }}}




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</pre>write the complex number in polar form. Express the argument theta in degrees, with 0 less than or equal to theta less than or equal to 360 degrees.<pre>
3&#8730;<span style="text-decoration: overline">3</span> - 3i = x + iy,

where  P(x,y) =  P(3&#8730;<span style="text-decoration: overline">3</span>,-3),

Plot the point P(x,y) = P(3&#8730;<span style="text-decoration: overline">3</span>,-3),
draw a line from P to the origin, 
and another line from P perpendicular to the x-axis.  
Label the sides of the resulting right triangle x,y, and r,
Label the counter-clockwise angle <font face="symbol">q</font>.

{{{drawing(300,300,-6,6,-6,6,graph(300,300,-6,6,-6,6),
green(triangle(0,0,3sqrt(3),0,3sqrt(3),-3)),
locate(2.6,.8,x),locate(2.6,-1.7,r), locate(5.2,-1.2,y),
red(arc(0,0,3,-3,0,330),locate(-1.5,1.4,theta)),
locate(3,-3,P(3*sqrt(3),-3))


 )}}}


Since the point P is P(x,y) =  P(3&#8730;<span style="text-decoration: overline">3</span>,-3), x = 3&#8730;<span style="text-decoration: overline">3</span> and y = -3.

We calculate r:

r² = x² + y²
r² = (3&#8730;<span style="text-decoration: overline">3</span>)² + (-3)².
r² = 9(3) + 9
r² = 27 + 9
r² = 36
 r = &#8730;<span style="text-decoration: overline">36</span>
 r = 6

{{{sin(theta)=y/r=(-3)/6=-1/2}}}

Therefore <font face="symbol">q</font> = 330°,

since it is in Q4, with a reference angle of 30°.

    Use {{{x/r=cos(theta)}}} and {{{y/r=sin(theta)}}}.
Solve them for x and y

    {{{x=r*cos(theta)}}} and {{{y=r*sin(theta)}}}

    x = 6*cos(330°) and y = 6*sin(330°)

So
 
3&#8730;<span style="text-decoration: overline">3</span> - 3i = x + iy =  6*cos(330°) + i*6sin(330°) = 6(cos(330° + isin(330°)  

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convert the polar equation to rectangular form.
3.) r = 5

That is a circle at the origin (pole) with a radius of 5.

r² = x² + y²
5² = x² + y²
x² + y² = 25

Edwin</pre>