[SOLVED] Euler Expression

**ronaldo_07** · January 16th 2009, 12:01 PM

Using Euler’s formula, express in terms of $sin\alpha$ and $cos\alpha$ for :

$sin2\alpha$ and $cos2\alpha$

I know that Euler's formula is :

$e^{ix}$ = $cos\alpha$ + $isin\alpha$

So I have come to the conclusion that:

$cos2\alpha =Re[e^{2i\alpha}]$ and $sin2\alpha =Im[e^{2i\alpha}]$

Is this correct and to answer this question do I leave it in this form?

**running-gag** · January 16th 2009, 12:20 PM

Hi

You must express $sin2\alpha$ and $cos2\alpha$ in terms of $sin\alpha$ and $cos\alpha$

$cos2\alpha =Re[e^{2i\alpha}]$ and $e^{2i\alpha} = \left(e^{i\alpha}\right)^2 = (cos\alpha + i\:sin\alpha)^2$

**DeMath** · January 16th 2009, 12:38 PM

Originally Posted by ronaldo_07

Using Euler’s formula, express in terms of $sin\alpha$ and $cos\alpha$ for :

$sin2\alpha$ and $cos2\alpha$

I know that Euler's formula is :

$e^{ix}$ = $cos\alpha$ + $isin\alpha$

So I have come to the conclusion that:

$cos2\alpha =Re[e^{2i\alpha}]$ and $sin2\alpha =Im[e^{2i\alpha}]$

Is this correct and to answer this question do I leave it in this form?

I think you need to use these Euler's formulas: $\sin \alpha = \frac{{{e^{ia}} - {e^{ - ia}}}}{{2i}}$ and $\cos \alpha = \frac{{{e^{ia}} + {e^{ - ia}}}}{2}$ .

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