How to Convert Complex Numbers from Rectangular to Exponential Form?

[elcotufa]

Homework Statement

Trying to write
<br /> <br /> -8\pi - 8\pi\sqrt3 j <br />

in exponential

I got the coefficient as 16 pi

but to get the theta in top of the exponential I have to do

the inverse tangent of <br /> \frac{-8\pi} {-8\pi\sqrt3 j} <br />


I know it is pi over 3, but what is the easiest way to find the multiplicative factor of pi? it is four I just want to know how to find it for other harder examples



answer <br /> <br /> 16e^{\frac{j4\pi} 3}<br />


Input appreciated

 

[HallsofIvy]

Homework Statement

Trying to write
<br /> <br /> -8\pi - 8\pi\sqrt3 j <br />

in exponential

I got the coefficient as 16 pi

but to get the theta in top of the exponential I have to do

the inverse tangent of <br /> \frac{-8\pi} {-8\pi\sqrt3 j} <br />

The "argument", or \theta for x+ jy is arctan(y/x), not arctan(x/y) and certainly not arctan(x/jy)! You want
arctan(\frac{-8\pi\sqrt{3}}{-8\pi}= arctan(\sqrt{3})
because, of course, the "-8" terms cancel. Now you could use a calculator or, perhaps better, imagine a right triangle with opposite side of length \sqrt{3} and near side 1 (because tan= opposite side/near side). By the Pythagorean theorem, the hypotenuse has length \sqrt{(\sqrt{3})^2+ 1}= 2. That is, you are looking for an angle that has sine (opposite side divided by hypotenuse) \frac{-\sqrt{3}}{2}[/itex] and cosine \frac{-1}{2}. The negatives are because the real and imaginary parts of the number you give are both - so the point is in the fourth quadrant, not the first.<br /> sin(\theta)= \sqrt{3}/2 and cos(\theta)= 1/2 should be among the &quot;special angles&quot; you learned long ago.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I know it is pi over 3, but what is the easiest way to find the multiplicative factor of pi? it is four I just want to know how to find it for other harder examples </div> </div> </blockquote> No, you do NOT know &quot;it is pi over 3&quot; because pi/3 is in the first quadrant and the value you want is in the fourth. <b>4</b>pi/3 does happen to be in the fourth quadrant.<br /> <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> answer &lt;br /&gt; &lt;br /&gt; 16e^{\frac{j4\pi} 3}&lt;br /&gt;<br /> <br /> <br /> Input appreciated </div> </div> </blockquote>