Question

Express
8ı
in polar form

Answer 1

Answer on Question #58300 – Math – Complex Analysis

Question

Express

8i

in polar form

Solution

$z = a + ib = |z| (\cos \varphi + \sin \varphi)$, where $|z| = \sqrt{a^2 + b^2}$ and

$$\varphi = \left\{ \begin{array}{c} \arccos \left(\frac {a}{| z |}\right), b \geq 0, | z | > 0, \\ - \arccos \left(\frac {a}{| z |}\right), b < 0, | z | > 0, \\ \text{undefined}, | z | = 0, \end{array} \right.$$

or

$$\varphi = \left\{ \begin{array}{c c} \arctan \frac {b}{a}, & a > 0, \\ \frac {\pi}{2}, & a = 0, b > 0, \\ - \frac {\pi}{2}, & a = 0, b < 0, \\ \arctan \frac {b}{a} + \pi , & a < 0, b \geq 0, \\ \arctan \frac {b}{a} - \pi , & a < 0, b < 0. \end{array} \right.$$ $$z = 8i, a = 0, b = 8,$$ $$| z | = \sqrt {0 + 64} = 8; \varphi = \arccos \left(\frac {0}{8}\right) = \frac {\pi}{2}.$$

Then the polar form of $z = 8i$ is

$$z = 8 \left(\cos \frac {\pi}{2} + \sin \frac {\pi}{2}\right).$$

Answer: $8i = 8\left(\cos \frac {\pi}{2} + \sin \frac {\pi}{2}\right)$.

www.AssignmentExpert.com