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Complex Numbers — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers3 Marks
Question
Find the modulus and argument of the following complex numbers and hence express each of them in the polar form: $\frac{1-\text{i}}{1+\text{i}}$
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Answer

$\frac{1-\text{i}}{1+\text{i}}=\frac{(1-\text{i})(1-\text{i})}{(1+\text{i})(1-\text{i})}=\frac{(1-\text{i})^2}{1^2-\text{i}^2}=\frac{1-2\text{i}-1}{1+1}=\frac{-2\text{i}}{2}=\text{-i}$ Modulus, $\Big|\frac{1-\text{i}}{1+\text{i}}\Big|=|\text{-i}|=1$ Argument, $\tan^{-1}\Big(\frac{-1}{0}\Big)=-\frac{\pi}{2}$ Polar form, $\text{z}=\text{r}(\cos\theta+\text{i}\sin\theta)$ $\text{z}=\Big(\cos\frac{\pi}{2}-\text{i}\sin\frac{\pi}{2}\Big)$

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