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Question Number 146467 by mathdanisur last updated on 13/Jul/21

$$if\frac{\sqrt{3}}{2}+\frac{1}{2}\mathit{i}find{\mathit{z}}^{11}=?$$

Answered by gsk2684 last updated on 13/Jul/21

$$z^{11}={\left(cis\frac{\mathrm{\Pi }}{6}\right)}^{11}=cis\left(\frac{11\mathrm{\Pi }}{6}\right)$$$$=cis(2\mathrm{\Pi }-\frac{\mathrm{\Pi }}{6})$$$$=cis(-\frac{\mathrm{\Pi }}{6})$$$$=\frac{\sqrt{3}}{2}-\frac{i}{2}$$

Commented by mathdanisur last updated on 13/Jul/21

$$thanksSer,butcis.?cos.?$$

Commented by gsk2684 last updated on 13/Jul/21

$$cis\theta =\cos \theta +i\sin \theta$$

Commented by mathdanisur last updated on 13/Jul/21

$$thanksSer$$

Answered by puissant last updated on 13/Jul/21

$$\mathrm{Z}=\cos \frac{\pi }{6}+\mathrm{i}\sin \frac{\pi }{6}={\mathrm{e}}^{\mathrm{i}\frac{\pi }{6}}$$$${\mathrm{Z}}^{11}={\left({\mathrm{e}}^{\mathrm{i}\frac{\pi }{6}}\right)}^{11}={\mathrm{e}}^{\mathrm{i}\frac{11\pi }{6}}=\cos \frac{11\pi }{6}+\mathrm{i}\sin \frac{11\pi }{6}$$$$\Rightarrow \mathrm{Z}=\frac{\sqrt{3}}{2}-\frac{1}{2}\mathrm{i}$$

Commented by mathdanisur last updated on 13/Jul/21

$$thankyouSer$$$$cos\frac{11\pi }{6}+isin\frac{11\pi }{6}\Rightarrow z^{11}=\frac{\sqrt{3}}{2}-\frac{1}{2}i.?$$

Commented by mathdanisur last updated on 13/Jul/21

$$thanksSer$$

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