Question

If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .

Answer 1

\[\pi < \theta < 2\pi\]

\[ \frac{\pi}{2} < \frac{\theta}{2} < \pi \left( \text { Dividing by } 2 \right)\]

\[z = 1 + \cos\theta + i sin\theta\]

\[ \Rightarrow \left| z \right| = \sqrt{\left( 1 + \cos\theta \right)^2 + \sin^2 \theta}\]

\[ \Rightarrow \left| z \right| = \sqrt{1 + \cos^2 \theta + 2\cos\theta + \sin^2 \theta}\]

\[ \Rightarrow \left| z \right| = \sqrt{1 + 1 + 2\cos\theta}\]

\[ \Rightarrow \left| z \right| = \sqrt{2\left( 1 + \cos\theta \right)}\]

\[ \Rightarrow \left| z \right| = \sqrt{2 \times 2 \cos^2 \frac{\theta}{2}}\]

\[ \Rightarrow \left| z \right| = 2\sqrt{\cos^2 \frac{\theta}{2}}\]

\[ \Rightarrow \left| z \right| = - 2\cos\frac{\theta}{2} \left[ \text { Since } \frac{\pi}{2} < \frac{\theta}{2} < \pi , \cos\frac{\theta}{2} \text { is negative } \right]\]