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<my favorite problem>

A particle is located on the coordinate plane at [unparseable or potentially dangerous latex formula] Define a [unparseable or potentially dangerous latex formula] for the particle as a counterclockwise rotation of [unparseable or potentially dangerous latex formula] radians about the origin followed by a translation of [unparseable or potentially dangerous latex formula] units in the positive x-direction. Given that the particle's position after [unparseable or potentially dangerous latex formula] moves is [unparseable or potentially dangerous latex formula] find the greatest integer less than or equal to [unparseable or potentially dangerous latex formula]

I got fifteen, and I have no idea how to get 19.
Somehow I determined every four moves returns back to (5,0), which has to be wrong considering the answer...
Help?

Consider it on the complex plane (which doesn't change the answer, there is just an i on the y-coordinate)

Recall that [unparseable or potentially dangerous latex formula] and the x-coordinate is [unparseable or potentially dangerous latex formula], y-coordinate [unparseable or potentially dangerous latex formula]
Also recall that [unparseable or potentially dangerous latex formula]
Define [unparseable or potentially dangerous latex formula] where the coordinate [unparseable or potentially dangerous latex formula] is the coordinate after [unparseable or potentially dangerous latex formula] moves.
We have that [unparseable or potentially dangerous latex formula]

Each rotation is the same as multiplying [unparseable or potentially dangerous latex formula] and each shift is the same as adding [unparseable or potentially dangerous latex formula]


[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

Adding these looks like a pretty daunting task, but luckily for us [unparseable or potentially dangerous latex formula] has some nice properties.

The one that I will use is that [unparseable or potentially dangerous latex formula]
The proof of this is fairly straightforward. [unparseable or potentially dangerous latex formula] is simply the sum of the [unparseable or potentially dangerous latex formula] roots of unity, which is simply the polynomial [unparseable or potentially dangerous latex formula] By Vieta, the sum of the roots is 0.

So, going back to the problem.
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

So, [unparseable or potentially dangerous latex formula]
Thus the answer is [unparseable or potentially dangerous latex formula](as seen by collegebookworm's calculations)