AP EAMCET · Maths · Complex Number
If \(z=1+\cos \theta-i \sin \theta\) and \(0 < \theta < \pi\), then \(\left[|z-1|^2-\frac{|z|^2}{4}\right]^{1 / 2}=\)
- A \(\sqrt{2} \cos \theta\)
- B \(\sqrt{2} \sin \theta\)
- C \(\cos \left(\frac{\theta}{2}\right)\)
- D \(\sin \left(\frac{\theta}{2}\right)\)
Answer & Solution
Correct Answer
(D) \(\sin \left(\frac{\theta}{2}\right)\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text {Since, } Z=1+\cos \theta-i \sin \theta \\ & \Rightarrow z-1=\cos \theta-i \sin \theta \\ & \Rightarrow|z-1|=\left|e^{i \theta}\right|=1 \\ & \text { Now, }|z|=\sqrt{(1+\cos \theta)^2+\sin ^2 \theta}=\sqrt{2+2 \cos \theta} \\ &…
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