# Polar form of complex number

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.

• What's the difference between the angles $3\pi/2$ and $-\pi/2$? – Neal Oct 6 '16 at 1:33
• Both forms are correct. The second form uses the principal value of the argument. – dxiv Oct 6 '16 at 1:36

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."
$$\frac{3}{2}\pi - (- \frac{1}{2}\pi)2 =2\pi$$
and trigonometry function has a period of $2 \pi$.