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Meredith's question at Yahoo! Answers regarding converting from trigonometric to rectangular form

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MarkFL

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Feb 24, 2012
13,775
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MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Hello Meredith,

We are given:

\(\displaystyle z=7\sqrt{5}\text{cis}\left(\tan^{-1}(2) \right)=7\sqrt{5}\left(\cos\left(\tan^{-1}(2) \right)+i\sin\left(\tan^{-1}(2) \right) \right)\)

In order to evaluate the trig functions, consider the following diagram:

meredith.jpg

As you can see:

\(\displaystyle \tan(\theta)=\frac{2}{1}=2\,\therefore\,\theta= \tan^{-1}(2)\)

and so:

\(\displaystyle \sin(\theta)=\frac{2}{\sqrt{5}}\)

\(\displaystyle \cos(\theta)=\frac{1}{\sqrt{5}}\)

and so we have:

\(\displaystyle z=7\sqrt{5}\left(\frac{1}{\sqrt{5}}+i\frac{2}{ \sqrt{5}} \right)=7+14i\)

Hence:

\(\displaystyle a=7,\,b=14\)

To Meredith and any other guests viewing this topic, I invite and encourage you to post other complex number problems in our Pre-Calculus forum.

Best Regards,

Mark.