Express the following complex number in polar form and exponential form: $- 2 + 2 i$

2 \sqrt{2} (c o s \frac{3 π}{4} + i s i n \frac{3 π}{4}); 2 \sqrt{2} e^{i ⎛ ⎝ \frac{3 π}{4} ⎞ ⎠}

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2 \sqrt{2} (c o s \frac{3 π}{4} - i s i n \frac{3 π}{4}); 2 \sqrt{2} e^{i ⎛ ⎝ \frac{3 π}{4} ⎞ ⎠}

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2 \sqrt{2} (c o s \frac{3 π}{4} - i s i n \frac{3 π}{4}); 2 \sqrt{2} e^{i ⎛ ⎝ \frac{- 3 π}{4} ⎞ ⎠}

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2 \sqrt{2} (c o s \frac{3 π}{4} + i s i n \frac{3 π}{4}); 2 \sqrt{2} e^{i ⎛ ⎝ \frac{- 3 π}{4} ⎞ ⎠}

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Solution

The correct option is A $2 \sqrt{2} (c o s \frac{3 π}{4} + i s i n \frac{3 π}{4}); 2 \sqrt{2} e^{i ⎛ ⎝ \frac{3 π}{4} ⎞ ⎠}$
$z = - 2 + 2 i$
$= 2 (- 1 + i)$
$= 2 \sqrt{2} (\frac{- 1 + i}{\sqrt{2}})$
$∴ a r g (z) = π - \frac{π}{4} = \frac{3 π}{4}$
$= 2 \sqrt{2} (c o s (\frac{3 π}{4}) + i s i n (\frac{3 π}{4}))$
$= 2 \sqrt{2} e^{\frac{3 π i}{4}}$

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- 3 \sqrt{2} + 3 \sqrt{2} i

The polar form of complex number 1 - i

{⎛ ⎝ \sqrt{{(\sqrt{2} - \frac{3}{2})}^{2}} - \sqrt[3]{(1 - \sqrt{2})^{3}} ⎞ ⎠}^{2} + 2^{- \frac{3}{2}} cos \frac{3 π}{4}

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