How can I go from sine to cosine using exponential numbers?

[amama]

$cos(\omega)$ is

$$\frac{e^{j \omega } + e^{-j \omega }}{2}$$

$sin(\omega)$ is

$$\frac{e^{j \omega } - e^{-j \omega }}{2 j}$$

I also know that $cos(\omega - \pi / 2) = sin(\omega)$.

I've been trying to show this using exponentials, but I can't seem to manipulate one form into the other. Specifically, I can't figure out how to manipulate

$$\frac{e^{j (\omega - \pi / 2)} + e^{-j( \omega - \pi / 2)}}{2}$$

into

$$\frac{e^{j \omega } - e^{-j \omega }}{2 j}$$

or the other way around.

I seem to remember from back in school that there was some trick to this, but I can't remember it. I keep getting stuck on the fact that one has a plus between the exponential terms, and one has a minus.

 

[mfb]

Write the exponentials as $e^{j\omega} \cdot e^{...}$ and then evaluate the second term.

 

[amama]

Write the exponentials as $e^{j\omega} \cdot e^{...}$ and then evaluate the second term.

So simple... thanks! I kept trying to convert everything to exponentials, but after evaluating $e^{\pi /2}$ and $e^{-\pi /2}$ to $j$ and $-j$ it all worked out.

 

[dextercioby]

Just for future reference. The circular trigonometric functions have the neat LaTex codes \sin, \cos, \tan, \cot. So compare $sin x$ or $sin(x)$ to $\sin x$. The brackets to denote the variable are superfluous.