Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers1 Mark
Question
Write the argument of $(1+\text{i}\sqrt{3})(1+\text{i})(\cos\theta+\text{i}\sin\theta).$
✓
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 $\frac{7\pi}{12}+\theta.$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.