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Is the polar form of $-5j$ equal to

$5(\cos\frac{3}{2}\pi+j\sin\frac{3}{2}\pi)$

or

$5(\cos-\frac{\pi}{2}+j\sin-\frac{\pi}{2})$

I'm confused as to which way to go when calculating the argument.

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  • $\begingroup$ What's the difference between the angles $3\pi/2$ and $-\pi/2$? $\endgroup$ – Neal Oct 6 '16 at 1:33
  • $\begingroup$ Both forms are correct. The second form uses the principal value of the argument. $\endgroup$ – dxiv Oct 6 '16 at 1:36
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This is like asking "is $2$ equal to $4/2$ or $6/3$?" You have come across two equivalent ways of phrasing the same number. They're the same because $\cos(3\pi/2) = \cos(-\pi/2)$ and $\sin(3\pi/2) = \sin(-\pi/2)$.

Think of this as two sets of directions to the same address ($-5j$). The representation $-5j = 5(\cos(3\pi/2) + j\sin(3\pi/2))$ says "start at $1$ and walk counter-clockwise three quarters of the way around the unit circle. Then face away from the origin and walk five steps." The representation $-5j = 5(\cos(-\pi/2) + j\sin(-\pi/2))$ says "start at $1$ and walk clockwise one quarter of the way around the unit circle. Then face away from the origin and walk five steps."

Same number, two more ways of writing it.

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Both are equivalent.

$$\frac{3}{2}\pi - (- \frac{1}{2}\pi)2 =2\pi$$

and trigonometry function has a period of $2 \pi$.

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