Question

Use DeMoivre's Theorem to determine the following power of a complex number. Write the answer in form a + bi, where a and b are real numbers and do not involve the use of a trigonometric function. V2 2 =

Answer 1

we write the trignometric form of the complex number as

the magnitude is

2 -)2 + 7 - .2 =1 2

the argument is in second quadrant

$\theta=\tan^{-1}(\frac{\sqrt{2}/2}{-\sqrt{2}/2})=180\degree-45\degree=135\degree$

that is

$(\frac{-\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)^5=(\cos 135\degree+i\sin 135\degree)^5$

by demoivres theorem ,

= cos(5 * 135)° +isin(5 * 135°)

$=\cos (675)\degree+i\sin (675\degree)$

$=\cos (360+315)\degree+i\sin (360+315)\degree$

$=\cos (315)\degree+i\sin (315)\degree$

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