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Question
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
Options
\[2 \sin\frac{\theta}{2}\]
\[2 \cos\frac{\theta}{2}\]
\[2\left| \sin\frac{\theta}{2} \right|\]
\[2\left| \cos\frac{\theta}{2} \right|\]
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Solution
\[2\left| \sin\frac{\theta}{2} \right|\]
\[\because z = 1 - \cos\theta + i \sin\theta\]
\[ \Rightarrow \left| z \right| = \sqrt{\left( 1 - \cos\theta \right)^2 + \sin^2 \theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{1 + \cos^2 \theta - 2\cos\theta + \sin^2 \theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{1 + 1 - 2\cos\theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{2\left( 1 - \cos\theta \right)}\]
\[ \Rightarrow \left| z \right| = \sqrt{4 \sin^2 \frac{\theta}{2}}\]
\[ \Rightarrow \left| z \right|=2\left| \sin\frac{\theta}{2} \right|\]
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