Question

Write the complex number z = 1-i/(cos π/3 + i sin π/3) .

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Answer 1

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$\to \frac{2}{4} - \frac{2i}{4}$

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$\to {(i)}^{2} = - 1$

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$\to \frac{1 - i}{ (\cos( \frac{\pi}{3} )+ i \sin( \frac{\pi}{3} ) ) } \\ \\ = \frac{1 - i}{ \frac{ 1 }{2} + i \frac{ \sqrt{3} }{2} } \\ \\ = \frac{1 - i}{ \frac{1 + i \sqrt{ 3} }{2} } \\ \\ = \frac{2(1 - i)}{1 + i \sqrt{3} } \\ \\ = \frac{2 - 2i}{1 + i \sqrt{3} } \\ \\ = \frac{(2 - 2i)(1 - i \sqrt{3} )}{(1 + i \sqrt{3} )(1 - i \sqrt{3} )} \\ \\ = \frac{2 - 2i \sqrt{3} - 2i + 2i \sqrt{3} }{ {(1)}^{2} - {(i \sqrt{3} )}^{2} } \\ \\ = \frac{2 - 2i}{1 - ( - 1)3} \\ \\ = \frac{2 - 2i}{1 + 3} \\ \\ = \frac{2 - 2i}{4} \\ \\ = \frac{2}{4} - \frac{2i}{4}$