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I can't figure out, how to express $\cos(5t)$ in the form $e^{j\omega t}$. I don't even know the right answer. How would you deal with this task?

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up vote 3 down vote accepted

Because $$e^{ix} = \cos x + i\sin x$$ $$e^{-ix} = \cos x - i\sin x$$

it follows that

$$\cos x = \frac{e^{ix}+e^{-ix}}{2}$$

and therefore

$$\cos (5t) = \frac{e^{5it}+e^{-5it}}{2}$$

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Just use that $\text{Re}(z)=\dfrac{z+\bar{z}}{2}$

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