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