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प्रश्न
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\sqrt{3} + i\]
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उत्तर
\[ z = \sqrt{3} + i\]
\[r = \left| z \right|\]
\[ = \sqrt{3 + 1}\]
\[ = \sqrt{4}\]
\[ = 2\]
\[\text { Let } \tan \alpha = \left| \frac{Im\left( z \right)}{Re\left( z \right)} \right|\]
\[ \Rightarrow \tan \alpha = \left( \frac{1}{\sqrt{3}} \right)\]
\[ \Rightarrow \alpha = \frac{\pi}{6}\]
\[\text { Since point } (\sqrt{3}, 1) \text { lies in the first quadrant, the argument of z is given by } \]
\[\theta = \alpha = \frac{\pi}{6}\]
\[\text { Polar form } = r \left( \cos\theta + i\sin\theta \right) \]
\[ = 2 \left( \cos \frac{\pi}{6} + i\sin\frac{\pi}{6} \right)\]
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