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Express \(\left(\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i\right)^{2015}\) in the form a + bi.

 Jun 23, 2020
 #1
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Express
\(\left(\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i\right)^{2015}\)
in the form a + bi.

 

\(\begin{array}{|rcll|} \hline \varphi &=& \arctan\left( \dfrac{\dfrac{\sqrt{3}}{2}}{\dfrac{1}{2}} \right) \\ \varphi &=& \arctan(\sqrt{3}) \\ \mathbf{\varphi} &=& \mathbf{60^\circ} \\\\ r &=& \sqrt{ \left( \dfrac{1}{2} \right) ^2 + \left( \dfrac{\sqrt{3}}{2} \right) ^2 } \\ r &=& \sqrt{ \dfrac{1}{4} + \dfrac{3}{4} } \\ \mathbf{r} &=& \mathbf{1} \\ \hline \end{array} \)

 

\(\begin{array}{|rcll|} \hline && \mathbf{\left(\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i\right)^{2015}} \\ &=& \Big(\cos(60^\circ)+i\sin(60^\circ) \Big)^{2015} \\ &=& \cos(2015*60^\circ)+i\sin(2015*60^\circ) \\ &=& \cos(120900^\circ)+i\sin(120900^\circ) \\ &=& \cos(300^\circ)+i\sin(300^\circ) \\ &=& \mathbf{\dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i} \\ \hline \end{array}\)

 

laugh

 Jun 24, 2020

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