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 September 24th, 2013, 04:41 AM #1 Senior Member   Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0 Rectangular Form Hi, does anyone know how to write $e^{\frac{2-i \pi}{4}}$ in rectagular form?
 September 24th, 2013, 04:53 AM #2 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Rectangular Form You need to use Euler's formula: $e^{ix}=cos(x)+isin(x)$
 September 24th, 2013, 05:08 AM #3 Senior Member   Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0 Re: Rectangular Form so if I write it $e^{\frac{2-i\pi}{4}}= e^{\frac{2}{4}-{\frac{i\pi}{4}}} = e^{\frac{2}{4}} e^{{\frac{-i\pi}{4}}$ I have $e^{\frac{1}{2}} cos(\frac{\pi}{4}) - i e^{\frac{1}{2}} sin(\frac{\pi}{4})$ but then I am lost there is that already rectangular form? I think it is bc the real part (cosine) is not in terms of z.
 September 24th, 2013, 07:23 AM #4 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Rectangular Form Do you not know what "rectangular form" means? You have a+ bi with $a= e^{1/2}cos(\pi/4)$ and $b= -e^{1/2}sin(\pi/4)$. I don't know what you mean by "the real part (cosine) is not in terms of z". There is no "z" in anything you write.
 September 24th, 2013, 07:40 AM #5 Senior Member   Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0 Re: Rectangular Form Hey, I'm sorry if that is a stupid question. This is my first exposure to complex variables. I do not have a firm footing yet. In class the professor said to beware $e^{iz}= cos(z) + isin(z)$ was not in rectangular form. That's what I meant. He said cos is the real part and sin is the real part.
September 24th, 2013, 08:41 AM   #6
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Re: Rectangular Form

Quote:
 Originally Posted by aaron-math In class the professor said to beware $e^{iz}= cos(z) + isin(z)$ was not in rectangular form. That's what I meant. He said cos is the real part and sin is the real part.
He may have meant that you are still expected to simplify the cos and sin, if you can. In this case:

$cos(\frac{\pi}{4})=sin(\frac{\pi}{4})=\frac{1}{\sq rt{2}$

So, you can simpify your answer and get rid of the sin and cosine functions.

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