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Question
Write −1 + i \[\sqrt{3}\] in polar form .
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Solution
\[\text{Let z }= - 1 + \sqrt{3}i . \text { Then } , \]
\[r = \left| z \right| = \sqrt{\left[ - 1 \right]^2 + \left[ \sqrt{3} \right]^2} = 2\]
\[\text { Let } \tan \alpha = \left| \frac{Im(z)}{Re (z)} \right|\]
\[ = \sqrt{3}\]
\[ \Rightarrow \alpha = \frac{\pi}{3}\]
\[\text { Since the point representing z lies in the second quadrant . Therefore, the argument of z is given by } \]
\[\theta = \pi - \alpha\]
\[ = \pi - \frac{\pi}{3}\]
\[ = \frac{2\pi}{3}\]
\[\text { So, the polar form is } r\left( \cos\theta + i\sin\theta \right)\]
\[ \therefore z = 2\left( \cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3} \right)\]
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