Question

What is the polar form of the complex number $\left(i^{25}\right)^{3}$ ?

• A. $cos \frac{\pi}{3}-isin \frac{\pi}{3}$
• B. $\left(cos\left(-\frac{\pi}{2}\right)+i\, sin\left(-\frac{\pi}{2}\right)\right)$
• C. $cos \frac{\pi}{4}+ i\, sin \frac{\pi}{4}$
• D. $\left(cos\frac{\pi}{2}+i\, sin \frac{\pi}{2}\right)$

Answer 1

Answer: (B)

Let $z =\left(i^{25}\right)^{3} =\left(i\right)^{75}$
$=i^{4\times18+3}=\left(i^{4}\right)^{18}\left(i\right)^{3}$
$=i^{3}=-i=0-i$
Polar form of $z = r \left(cos\,\theta+isin\,\theta\right)$
$=1\left\{cos\left(-\frac{\pi}{2}\right)+i\, sin \left(-\frac{\pi}{2}\right)\right\}$
$=cos \frac{\pi}{2}-i \, sin \frac{\pi}{2}$