# Complex Numbers

• Feb 6th 2007, 03:36 AM
classicstrings
Complex Numbers
Stuck on a few....

I think this has something to do with the following questions:

sin(theta) + cos(theta)i = cis(pie/2 - (theta))

Simplify

a. (cos(theta)-sin(theta)i)^5

b. (sin(theta) + cos(theta)i)(cos(theta) + sin(theta)i)
• Feb 6th 2007, 04:00 AM
topsquark
Quote:

Originally Posted by classicstrings
Stuck on a few....

I think this has something to do with the following questions:

sin(theta) + cos(theta)i = cis(pie/2 - (theta))

Simplify

a. (cos(theta)-sin(theta)i)^5

b. (sin(theta) + cos(theta)i)(cos(theta) + sin(theta)i)

a) $\displaystyle [cos(\theta) - i \cdot sin(\theta)]^5 = cos(5 \theta) - i \cdot sin(5 \theta)$
(Similar to $\displaystyle [cos(\theta) + i \cdot sin(\theta)]^n = cos(n \theta) + i \cdot sin(n \theta)$)

b) Just expand:
$\displaystyle [sin(\theta) + i \cdot cos(\theta)][cos(\theta) + i \cdot sin(\theta)]$

$\displaystyle = [sin(\theta) cos(\theta) - sin(\theta) cos(\theta)] + i \cdot [sin^2(\theta) + cos^2(\theta)] = i$

-Dan