The modulus ($$r$$​) and amplitude $$(\theta)$$ of the complex number $$z=1+i tan \frac {3\pi}{5}$$ are​

Correct option is B

Solution:

$$\textbf{Given:} \quad z = 1 + i\tan\left(\frac{3\pi}{5}\right)\\[8pt]\textbf{Step 1: Rewrite in exponential form}\\[4pt]\tan\left(\frac{3\pi}{5}\right) = \frac{\sin\left(\frac{3\pi}{5}\right)}{\cos\left(\frac{3\pi}{5}\right)} \quad \Rightarrow \quad \\\ \\z = 1 + i \cdot \frac{\sin\left(\frac{3\pi}{5}\right)}{\cos\left(\frac{3\pi}\\\ \\{5}\right)} = \frac{1}{\cos\left(\frac{3\pi}{5}\right)} \cdot \left( \cos\left(\frac{3\pi}{5}\right) + i \sin\left(\frac{3\pi}{5}\right) \right)\\[8pt]\Rightarrow z = \sec\left(\frac{3\pi}{5}\right) \cdot e^{i\frac{3\pi}{5}}\\[10pt]\textbf{Step 2: Modulus}\\[4pt]|z| = \left| \sec\left(\frac{3\pi}{5}\right) \right| = -\sec\left(\frac{3\pi}{5}\right) \quad \text{(since } \cos\left(\frac{3\pi}{5}\right) < 0 \text{)}\\[8pt]\textbf{Step 3: Amplitude}\\[4pt]\theta = \arg(z) = \frac{3\pi}{5} - \pi = -\frac{2\pi}{5} \quad \text{(adjusted because of negative modulus)}\\[8pt]\textbf{Final Answer:} \quad \boxed{r = -\sec\left(\frac{3\pi}{5}\right), \quad \theta = -\frac{2\pi}{5}}$$

​