Gujarat BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers1 Mark
Question
Write $-1 + \text{i}\sqrt{3}$ in polar form.
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Answer
Let $\text{z}=-1 + \text{i}\sqrt{3}.$ Then, $\text{r}=|\text{z}|=\sqrt{[-1]^2+[\sqrt{3}]^2}=2$ Let $\tan\alpha=\Big|\frac{\text{Im(z)}}{\text{Re(z)}}\Big|$ $=\sqrt{3}$ $\Rightarrow\alpha=\frac{\pi}{3}$ Since the point representing z lies in the second quadrant. Therefore, the argument of z is given by $\theta=\pi-\alpha$ $=\pi-\frac{\pi}{3}$ $=\frac{2\pi}{3}$ So, the polar form is $\text{r}(\cos\theta+\text{i}\sin\theta)$ $\therefore\text{z}=2\Big(\cos\frac{2\pi}{3}+\text{i}\sin\frac{2\pi}{3}\Big)$
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