CBSE BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers1 Mark
Question
Write the argument of $(1+\text{i}\sqrt{3})(1+\text{i})(\cos\theta+\text{i}\sin\theta).$
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Answer
Let the argument of $(1+\text{i}\sqrt{3})$ be $\alpha.$ Then,
$\tan\alpha=\frac{\sqrt{3}}{1}=\tan\frac{\pi}{3}$
$\Rightarrow\alpha=\frac{\pi}{3}$
Let the argument of $(1+\text{i})$ be $\beta.$ Then,
$\tan\beta=\frac{1}{1}=\tan\frac{\pi}{4}$
$\Rightarrow\beta=\frac{\pi}{3}$
Let the argument of $(\cos\theta+\text{i}\sin\theta)$ be $\gamma$ Then,
$\tan\gamma=\frac{\sin\theta}{\cos\theta}=\tan\theta$
$\Rightarrow\gamma=\theta$
$\therefore$ The argument of $(1+\text{i}\sqrt{3})(1+\text{i})(\cos\theta+\text{i}\sin\theta)=\alpha+\beta+\gamma=\frac{\pi}{3}+\frac{\pi}{4}+\theta=\frac{7\pi}{12} +\theta$
Hence, the argument of $(1+\text{i}\sqrt{3})(1+\text{i})(\cos\theta+\text{i}\sin\theta)$ is $\frac{7\pi}{12}+\theta.$
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