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Question
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
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Solution
\[\pi < \theta < 2\pi\]
\[ \frac{\pi}{2} < \frac{\theta}{2} < \pi \left( \text { Dividing by } 2 \right)\]
\[z = 1 + \cos\theta + i sin\theta\]
\[ \Rightarrow \left| z \right| = \sqrt{\left( 1 + \cos\theta \right)^2 + \sin^2 \theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{1 + \cos^2 \theta + 2\cos\theta + \sin^2 \theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{1 + 1 + 2\cos\theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{2\left( 1 + \cos\theta \right)}\]
\[ \Rightarrow \left| z \right| = \sqrt{2 \times 2 \cos^2 \frac{\theta}{2}}\]
\[ \Rightarrow \left| z \right| = 2\sqrt{\cos^2 \frac{\theta}{2}}\]
\[ \Rightarrow \left| z \right| = - 2\cos\frac{\theta}{2} \left[ \text { Since } \frac{\pi}{2} < \frac{\theta}{2} < \pi , \cos\frac{\theta}{2} \text { is negative } \right]\]
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