# Multiplication of Polar Forms

#### MuhTheKuh

Hey guys. I am struggling right now with a question where I was asked to multiply two different polar forms with each other (or take one of them to the power of something, or even both).
As you should be able to see on this picture question a) was perfectly fine and understandable for me but in question b) in during the third step he tells me to add cis(2xpi) to my existing (and with 5 multiplied) cis. Basically I would have thought that with step 2 everything is done and i could simply finish evaluating from this step on but in question b) d) e) he somehow adds or takes something to/from my cis, and I have no idea why and where he gets those values from.

#### tonio

Hey guys. I am struggling right now with a question where I was asked to multiply two different polar forms with each other (or take one of them to the power of something, or even both).
As you should be able to see on this picture question a) was perfectly fine and understandable for me but in question b) in during the third step he tells me to add cis(2xpi) to my existing (and with 5 multiplied) cis. Basically I would have thought that with step 2 everything is done and i could simply finish evaluating from this step on but in question b) d) e) he somehow adds or takes something to/from my cis, and I have no idea why and where he gets those values from.

This is pure trigonometry stuff: as $$\displaystyle \cos\left(-\frac{5\pi}{4}\right)=\cos\left(-\frac{5\pi}{4}+2\pi\right)=\cos\left(\frac{3\pi}{4}\right)$$ , and the same for the sine, we get $$\displaystyle cis\left(-\frac{5\pi}{4}\right)=cis\left(\frac{3\pi}{4}\right)$$ , and with the other ones is the same.

In general, $$\displaystyle cis(\phi)=cis(\theta)$$ whenever $$\displaystyle \phi-\theta=2n\pi\,,\,\,n\in\mathbb{Z}$$ , because of the periodicity of cosine and sine.

Tonio

MuhTheKuh

#### MuhTheKuh

Alright, thanks a lot, so basically he basically wants something in the brackets that is smaller than π but bigger than -π?
So my next question would be why he gets 128 as answer for b) while d) and e) are done after that little cosmetic correction?