The number of complex numbers p such that |p| | vedclass

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(C) Let $p = e^{i	heta} = \cos 	heta + i \sin 	heta$, where $|p| = 1$. Then $p^{4} = e^{i4	heta} = \cos(4	heta) + i \sin(4	heta)$. The imaginary

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$8$

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(C) Let $p = e^{i	heta} = \cos 	heta + i \sin 	heta$, where $|p| = 1$. Then $p^{4} = e^{i4	heta} = \cos(4	heta) + i \sin(4	heta)$. The imaginary part of $p^{4}$ is $\sin(4	heta)$. We are given that $	ext{Im}(p^{4}) = 0$, so $\sin(4	heta) = 0$. This implies $4	heta = n\pi$ for any integer $n$, or $	heta = \frac{n\pi}{4}$. For $p$ to be distinct values on the unit circle, we consider $	heta \in [0, 2\pi)$. The possible values for $	heta$ are $0, \frac{\pi}{4}, \frac{2\pi}{4}, \frac{3\pi}{4}, \frac{4\pi}{4}, \frac{5\pi}{4}, \frac{6\pi}{4}, \frac{7\pi}{4}$. There are $8$ such values of $	heta$, corresponding to $8$ distinct complex numbers $p$.

The number of complex numbers $p$ such that $|p|=1$ and the imaginary part of $p^{4}$ is $0$ is:

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