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8_murik_8 [283]
2 years ago
8

Use DeMoivre's theorem to evaluate the expression

Mathematics
1 answer:
natima [27]2 years ago
7 0

Answer:

B. \dfrac{9}{2}-i\dfrac{9\sqrt{3}}{2}

Step-by-step explanation:

DeMoivre's theorem:

If the complex number z=r(\cos \alpha+i\sin\alpha), then for natural number n

z^n=r^n(\cos n\alpha+i\sin n\alpha).

In your case,

z=\sqrt{3}\left(\cos\dfrac{5\pi}{12}+i\sin\dfrac{5\pi}{12}\right)

and you have to evaluate z^4.

By DeMoivre's theorem,

z^4=(\sqrt{3})^4\left(\cos 4\cdot \dfrac{5\pi}{12}+i\sin 4\cdot \dfrac{5\pi}{12}\right)=9\left(\cos \dfrac{5\pi}{3}+i\sin\dfrac{5\pi}{3}\right)=9\left(\dfrac{1}{2}-i\dfrac{\sqrt{3}}{2}\right)=\dfrac{9}{2}-i\dfrac{9\sqrt{3}}{2}.

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