Respuesta :
The equivalent complex expression of [tex](\frac{1}{2}(cos(\frac{\pi}{5} )+i~sin(\frac{\pi}{5} )) )^5[/tex] is
[tex]\frac{1}{32} (cos(\pi)+i~sin(\pi))[/tex]
The correct answer is an option (b)
What is complex number?
"The number of the form a + ib, where a, b are real numbers and [tex]i=\sqrt{-1}[/tex]"
What is De Moivre's theorem?
"This theorem gives a formula for computing powers of complex numbers.
[tex](r(cos\theta+i~sin\theta))^n=r^n~(cos(n\theta)+i~sin(n\theta))[/tex] "
For given question,
We have been given a complex expression [tex](\frac{1}{2}(cos(\frac{\pi}{5} )+i~sin(\frac{\pi}{5} )) )^5[/tex]
Using the DeMoivre's theorem,
[tex](\frac{1}{2}(cos(\frac{\pi}{5} )+i~sin(\frac{\pi}{5} )) )^5\\\\=(\frac{1}{2})^5~((cos(\frac{\pi}{5} )+i~sin(\frac{\pi}{5} )) )^5\\\\=\frac{1}{32} (cos(\frac{5\pi}{5} )+i~sin(\frac{5\pi}{5} )) )\\\\= \frac{1}{32} (cos(\pi)+i~sin(\pi))[/tex]
Therefore, the equivalent complex expression of [tex](\frac{1}{2}(cos(\frac{\pi}{5} )+i~sin(\frac{\pi}{5} )) )^5[/tex] is
[tex]\frac{1}{32} (cos(\pi)+i~sin(\pi))[/tex]
The correct answer is an option (b)
Learn more about the De Moivre's theorem here:
https://brainly.com/question/17211848
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