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