Question 1124676
The polar form of a complex number is another way to represent a complex number. The form {{{z=a+bi}}} is called the rectangular coordinate form of a complex number. 
The horizontal axis is the real axis and the vertical axis is the imaginary axis. We find the real and complex components in terms of {{{r}}} and {{{theta}}} where {{{r}}} is the length of the vector and {{{theta}}} is the angle made with the real axis.


From Pythagorean Theorem :

{{{r^2=a^2+b^2}}}

By using the basic trigonometric ratios : 

{{{cos(theta)=a/r}}} and {{{sin(theta)=b/r}}}

Multiplying each side by {{{r}}}

{{{a=r*cos(theta)}}} and  {{{b=r*sin(theta)}}}


The rectangular form of a complex number is given by:

{{{z=a+bi}}}

Substitute the values of {{{a}}} and {{{b}}}

{{{z=r*cos(theta)+(r*sin(theta))i   }}} 
{{{z=r(cos(theta)+i*sin(theta))}}}


you have:
{{{ z=-34/5-3i}}}  
{{{ z= -6.8 -3i}}} where {{{a=-6.8}}} and {{{b=-3}}}


to write it in polar form, first find the absolute value of {{{r}}}


{{{r=abs(z)=sqrt(a^2+b^2)}}} if {{{a=-6.8}}} and {{{b=-3}}}, we have

{{{r=sqrt((-6.8)^2+(-3)^2)}}} 

{{{r=sqrt(46.24+9)}}} 

{{{r=sqrt(55.24)}}} 

{{{r }}} » {{{7.43236}}}

then

{{{z=7.43236(cos(theta)+i*sin(theta))}}}

Now find the argument {{{theta}}}

 To find argument {{{theta}}} we will use one of the following formulas:

{{{theta=arctan(b/a)}}} if {{{a>0}}}
{{{theta=arctan(b/a)+180}}}° if {{{a<0}}}



so you can use the formula {{{theta=arctan(b/a)+180}}}° since you have {{{a<0}}}

{{{theta=arctan(-3/-6.8)+180}}}° 

{{{theta=arctan(3/6.8)+180}}}° 
{{{theta=24°+180}}}° 
{{{theta=204}}}° 


polar form is:

{{{z=7.43236(cos(204)+i*sin(204))}}}