Let r and respectively be the modulus and amplitude of the complex number z=2-i(2 (5 )/(8)…

Question

Let $r$ and $	heta $ respectively be the modulus and amplitude of the complex number $z=2-i(2	an \frac{5\pi }{8})$, then $(r,	heta )$ is equal to

Accepted Answer

$$\begin{aligned}
& z=2-i\left(2 \tan \frac{5 \pi}{8}\right)=x+i y(\text { let }) \
& r=\sqrt{x^2+y^2} ~& ~\theta=\tan ^{-1} \frac{y}{x} \
& r=\sqrt{(2)^2+\left(2 \tan \frac{5 \pi}{8}\right)^2} \
& =\left|2 \sec \frac{5 \pi}{8}\right|=\left|2 \sec \left(\pi-\frac{3 \pi}{8}\right)\right| \
& =2 \sec \frac{3 \pi}{8} \
& & ~\theta  = {\tan ^{ - 1}}\left( {{{ - 2\tan {{5\pi } \over 8}} \over 2}} \right) \
& =\tan ^{-1}\left(\tan ^2\left(\pi-\frac{5 \pi}{8}\right)\right) \
& =\frac{3 \pi}{8}
\end{aligned}$$

Solutions · 1