MHB Quickest way to calculate argument of a complex number

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The quickest way to calculate the argument of the complex number πe^{-3iπ/2} is to recognize that its argument can be expressed as -3π/2 + 2kπ, where k is an integer. For the principal value, k should be chosen to keep the result within the range of (-π, π] or [0, 2π), depending on the definition used. In this case, selecting k = 1 yields the principal value of the argument as π/2. The magnitude of the complex number is confirmed to be π. This method efficiently determines the argument while adhering to the principal value constraints.
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What's the quickest way to calculate the argument of $\displaystyle \pi e^{-\frac{3i\pi}{2}}$?
 
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To be sure, I know that $|\displaystyle \pi e^{-\frac{3i\pi}{2}}| = \pi.$
 
Guest said:
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 $\dfrac\pi2.$
 
Opalg said:
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Thank you very much! :D
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

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