Given this complex number,
$$e^{9ix/2} \frac{\sin 4x }{ (\sin (x/2) }$$
The real part of this complex number can be worked out easily, by replacing the $e^{9ix/2}$ with $\cos(9x/2)$
However if I'm given the complex number,
$$\frac{3} {3 – e^{ix} }$$
I cannot work out the real part by replacing the $e^{ix}$ with $\cos x $.
I want to understand why I can do this replacement in the first example and why I can't do it in the second example; and what I should look for when doing practice questions myself. And how I would actually go about working out the real part of the second example.
Thanks, any help would be appreciated.
Best Answer
Replacement is a good strategy for both, but you have to do additional steps for the second problem.
Note that the first one is of the form $$ (a + ib)c $$ where $a,b$ and $c$ are all real numbers, so you just have to do one multiplication to get a result of the form $x + iy$ from which it is easy to determine the real (or imaginary) part.
In the second case, you should still go ahead and expand the exponential, you'll have something of the form $$ \frac{a}{b + ic}. $$ This cannot be easily written as $x + iy$, but try multiplying by $\frac{b-ic}{b-ic}$ and see if that helps.