MHB Quickest way to calculate argument of a complex number

[Guest2]

What's the quickest way to calculate the argument of $\displaystyle \pi e^{-\frac{3i\pi}{2}}$?

 

[Guest2]

To be sure, I know that $|\displaystyle \pi e^{-\frac{3i\pi}{2}}| = \pi.$

 

[Opalg]

What's the quickest way to calculate the argument of $\displaystyle \pi e^{-\frac{3i\pi}{2}}$?

The argument of $\displaystyle \pi e^{-\frac{3i\pi}{2}}$ can take any value of the form $-\dfrac{3\pi}2 + 2k\pi$, where $k$ is an integer. If you want the principal value of the argument then you need to choose $k$ so as to get a value in the range $(-\pi,\pi]$ (or maybe $[0,2\pi)$, depending on which definition you are using for the principal range). In this example, you would want $k = 1$, giving the principal value of the argument as $\dfrac\pi2.$

 

[Guest2]

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Thank you very much! :D