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Write in the form $x+jy=\sqrt{32} (cos(\frac{\pi}{4}) + jsin(\frac{\pi}{4}))$.

I'm confused with what the question is asking for and my book doesn't give any examples, help would be much appreciated :)

sarah ryan
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1 Answers1

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You have a number in polar form: $Z = \sqrt{32}(\cos \pi/4 + i \sin \pi/4)$.

The general form for this is $r(\cos \theta + i \sin \theta)$. Here $r$ is the length of the vector, and $\theta$ is the direction it's pointing, going counterclockwise from the positive $x$ axis.

To get this number to the form $x+yi$, you split the number into a sum of a real part ($x$, which is not multiplied by $i$) and an imaginary part ($y$, which is multiplied by $i$):

$$Z = (\sqrt{32} \cos \pi/4) + (\sqrt{32} \sin \pi/4)i.$$

You're almost there. Then since $\cos \pi/4 = \sin \pi/4 = \sqrt{2}/2,$

$$Z = 4 + 4i.$$

Now, if you wanted to go back the other way, you would use these formulas:

$$r = \sqrt{x^2 + y^2}; \theta = \arctan \frac{y}{x}.$$

John
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