Complex Number Question 9
Question 9 - 2024 (29 Jan Shift 2)
Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z=2-i\left(2 \tan \frac{5 \pi}{8}\right)$, then $(r, \theta)$ is equal to
(1) $\left(2 \sec \frac{3 \pi}{8}, \frac{3 \pi}{8}\right)$
(2) $\left(2 \sec \frac{3 \pi}{8}, \frac{5 \pi}{8}\right)$
(3) $\left(2 \sec \frac{5 \pi}{8}, \frac{3 \pi}{8}\right)$
(4) $\left(2 \sec \frac{11 \pi}{8}, \frac{11 \pi}{8}\right)$
Show Answer
Answer (1)
Solution
$z=2-i\left(2 \tan \frac{5 \pi}{8}\right)=x+i y($ let $)$
$r=\sqrt{x^{2}+y^{2}} \& \theta=\tan ^{-1} \frac{y}{x}$
$r=\sqrt{(2)^{2}+\left(2 \tan \frac{5 \pi}{8}\right)^{2}}$
$=\left|2 \sec \frac{5 \pi}{8}\right|=\left|2 \sec \left(\pi-\frac{3 \pi}{8}\right)\right|$
$=2 \sec \frac{3 \pi}{8}$
$\& \theta=\tan ^{-1}\left(\frac{-2 \tan \frac{5 \pi}{8}}{2}\right)$
$=\tan ^{-1}\left(\tan \left(\pi-\frac{5 \pi}{8}\right)\right)$
$=\frac{3 \pi}{8}$
