The cosine summation

Geometry Level 4

The value of \(\displaystyle\sum_{r=0}^{12} \binom{12}{r} \cos \frac {r\pi}{6}\)

is of the form\(-(x+\sqrt {y})^{z}\), where \(x,y\) and \(z\) are integers, and \(z\) is as large as possible.