Gujarat BoardEnglish MediumSTD 11 ScienceMATHSCOMPLEX NUMBERS AND QUADRATIC EQUATIONS4 Marks
Question
Convert the complex numbers given in Exercises in the polar form: 1 - i.
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Answer
Given: $\text{z}=\text{r}(\cos\theta+\text{i}\sin\theta)=1-\text{i}$ $\therefore\ \text{r}\cos\theta=1$ and $\text{r}\sin\theta=-1$Squaring both sides and adding both the equations, we get
$\text{r}^2(\cos^2\theta+\sin^2\theta)=1+1$ $\Rightarrow\ \text{r}^2=2\Rightarrow\ \text{r}=\sqrt{2}$ $\therefore\ \sqrt{2}\cos\theta=1$ and $\sqrt{2}\sin\theta=-1$$\Rightarrow\ \cos\theta=\frac{1}{\sqrt{2}}$ and $\sin\theta=\frac{-1}{\sqrt{2}}$
$[\theta$ lies in fourth quadrant$]$
$\therefore\ \theta=\frac{-\pi}{4}$
Therefore, Polar form of z is $\sqrt{2}\Big[\cos\Big(\frac{-\pi}{4}\Big)+\text{i}\sin\Big(\frac{-\pi}{4}\Big)\Big].$
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