Chapter 9.3, Problem 44AYU

In Problems 43-54, write each expression in the standard form $a+bi$.

${\left[\sqrt{2}\left(\cos \frac{5\pi }{16}+i\sin \frac{5\pi }{16}\right)\right]}^4$

To determine

To write: Each expression in the standard form $a+bi$.

Answer to Problem 44AYU

Solution:

$-2\sqrt{2}-2\sqrt{2}i$

Explanation of Solution

Given:

${\left[\sqrt{2}\left(\cos \frac{5\pi }{16}+i\sin \frac{5\pi }{16}\right)\right]}^4$

Calculation:

${\left[\sqrt{2}\left(\cos \frac{5\pi }{16}+i\sin \frac{5\pi }{16}\right)\right]}^4={\left(\sqrt{2}\right)}^4\left[\cos \left(4\cdot \frac{5\pi }{16}\right)+i\sin \left(4\cdot \frac{5\pi }{16}\right)\right]$ Apply De Moivre’s Theorem.

$=4\left(\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4}\right)$

$=4\left(-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i\right)$

$=-2\sqrt{2}-2\sqrt{2}i$