Question 773999
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To find the argument {{{theta}}},

{{{IMAGINARY_PART/REAL_PART}}}{{{""=""}}}{{{tan(theta)}}}{{{""=""}}}{{{(sin(A)-cos(A))/(cos(A)+sin(A))}}}

Write {{{tan(theta)}}} as {{{sin(theta)/cos(theta)}}}

{{{sin(theta)/cos(theta)}}}{{{""=""}}}{{{(sin(A)-cos(A))/(cos(A)+sin(A))}}}

Cross-multiply

{{{sin(theta)(cos(A)+sin(A))}}}{{{""=""}}}{{{cos(theta)(sin(A)-cos(A))}}} 

{{{sin(theta)cos(A)+sin(theta)sin(A)}}}{{{""=""}}}{{{cos(theta)sin(A)-cos(theta)cos(A)}}} 

Rearrange the equation:

{{{sin(theta)cos(A)-cos(theta)sin(A)}}}{{{""=""}}}{{{-sin(theta)sin(A)-cos(theta)cos(A)}}} 

{{{sin(theta)cos(A)-cos(theta)sin(A)}}}{{{""=""}}}{{{-(sin(theta)sin(A)+cos(theta)cos(A))}}} 

{{{sin(theta)cos(A)-cos(theta)sin(A)}}}{{{""=""}}}{{{-(cos(theta)cos(A)+sin(theta)sin(A))}}}  

{{{sin(theta-A)}}}{{{""=""}}}{{{-cos(theta-A)}}}

Divide both sides by {{{cos(theta-A)}}}

{{{sin(theta-A)/cos(theta-A)}}}{{{""=""}}}{{{-1}}}

{{{tan(theta-A)}}}{{{""=""}}}{{{-1}}}

{{{theta-A}}}{{{""=""}}}{{{3pi/4}}}, {{{7pi/4}}}

{{{theta}}}{{{""=""}}}{{{3pi/4+A}}}, {{{7pi/4+A}}}

To find the modulus or absolute value r:

{{{r^2}}}{{{""=""}}}{{{(REAL_PART)^2+(IMAGINARY_PART)^2}}}

{{{r^2}}}{{{""=""}}}{{{(cos(A)+sin(A))^2}}}{{{""+""}}}{{{(sin(A)-cos(A))^2}}}

{{{r^2}}}{{{""=""}}}{{{cos^2(A)+2cos(A)sin(A)+sin^2(A)+sin^2(A)-2sin(A)cos(A)+cos^2(A)}}}

Using the identity cosineČ + sineČ = 1, the right side is just 2

rČ = 2

r = &#8730;<span style="text-decoration: overline">2</span> 

Polar forms:

&#8730;<span style="text-decoration: overline">2</span>[cos({{{3pi/4+A}}}) + i·sin({{{3pi/4+A}}})]

&#8730;<span style="text-decoration: overline">2</span>[cos({{{7pi/4+A}}}) + i·sin({{{7pi/4+A}}})]

Edwin</pre>
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