Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers1 Mark
Question
Write $1 - \text{i}$ in polar form.
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Answer
$\text{z}=1 - \text{i}$
$\text{r}=|\text{z}|$
$=\sqrt{1+1}$
$=\sqrt{2}$
Let $\tan\alpha=\Big|\frac{\text{Im(z)}}{\text{Re(z)}}\Big|$
$\therefore\tan\alpha=\Big|\frac{-1}{1}\Big|$
$=\frac{\pi}{4}$
$\Rightarrow\alpha=\frac{\pi}{4}$
Since point (1,−1) lies in the fourth quadrant, the argument of z is given by $\theta=-\alpha=-\frac{\pi}{4}$
Polar form $=\text{r}(\cos\theta+\text{i}\sin\theta)$
$\sqrt{2}\Big\{\cos\big(-\frac{\pi}{4}\big)+\text{i}\sin\big(-\frac{\pi}{4}\big)\Big\}$
$\sqrt{2}\big(\cos\frac{\pi}{4}-\text{i}\sin\frac{\pi}{4}\big)$
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