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Convert the following to Cartesian (rectangular) form and provide a graph.

$$e^{i7\pi /2}$$

The problem comes after a long series of similar problems. However, the noticeable difference with this one is that it is 1.75x the value of $2\pi$. I'm supposed to understand something here, but I'm not sure what.

My guess: I would say that there is something to be learned about how there is a principle range of $theta$ values, $(\pi,\pi]$, but then after that everything is just a $+2\pi$ or $-2\pi$ multiple of the principle angle. Am I on the right track?

whatwhatwhat
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1 Answers1

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All you need is to exploit Euler's formula:

$e^{i\theta} = \cos\theta + i\sin\theta$.

In this case, $\cos \frac{7\pi}{2} = 0$ and $\sin \frac{7\pi}{2} = -1$, giving the answer $z = -i$.

Angles don't change when you add or subtract whole multiples of $2\pi$. So you're right in that here $\frac{7\pi}{2}$ is exactly the same angle as $\frac{3\pi}{2}$. But even without recognising this, you can immediately get the correct answer as I've shown.

You should be able to plot $z$ on an Argand diagram very easily now.

Deepak
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