MHT CET · Maths · Complex Number
Argument of \(\frac{1-i \sqrt{3}}{1+i \sqrt{3}}\) is
- A \(210^{\circ}\)
- B \(120^{\circ}\)
- C \(240^{\circ}\)
- D \(60^{\circ}\)
Answer & Solution
Correct Answer
(C) \(240^{\circ}\)
Step-by-step Solution
Detailed explanation
\(z=\frac{1-i \sqrt{3}}{1+i \sqrt{3}} \times \frac{1-i \sqrt{3}}{1-i \sqrt{3}}=\frac{-2-2 \sqrt{3} i}{4}=-\frac{1}{2}-\frac{1}{2} \sqrt{3} i\)
Now Arg (Z) \(=\pi+\tan ^{-1}\left|\frac{\operatorname{Im}(Z)}{\operatorname{Re}(Z)}\right|\)
\(=180^{\circ}+\tan ^{-1}(\sqrt{3})=180^{\circ}+60^{\circ}=240^{\circ}\)
Now Arg (Z) \(=\pi+\tan ^{-1}\left|\frac{\operatorname{Im}(Z)}{\operatorname{Re}(Z)}\right|\)
\(=180^{\circ}+\tan ^{-1}(\sqrt{3})=180^{\circ}+60^{\circ}=240^{\circ}\)
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