Answer
$\theta = \frac{\pi}{6} +2n\pi$ and $n$ is an integer
Work Step by Step
Let $u=\left(\cos \frac{ \pi}{6},\sin \frac{ \pi}{6}\right),\quad v=\left(\cos \frac{5\pi}{6},\sin \frac{5\pi}{6}\right) \quad \langle u, v\rangle=u\cdot v $.
The angle $\theta$ between $u$ and $v$ is given by the formula
$$\cos \theta=\frac{\langle u, v\rangle}{\|u\| \cdot\|v\|}=\frac{\cos \frac{\pi}{6}\cos \frac{ \pi}{6}+\sin \frac{5\pi}{6}\sin \frac{5\pi}{6}}{\sqrt{ \cos^2 \frac{ \pi}{6}+\sin^2 \frac{ \pi}{6}}\sqrt{ \cos^2 \frac{5\pi}{6}+\sin^2 \frac{5\pi}{6}}}.$$
Using the properties of the trignometric functions, we get
$$\cos \theta=\cos\left( \frac{ \pi}{6}- \frac{5\pi}{6}\right)=\cos \frac{-\pi}{6} =\cos \frac{\pi}{6}$$
that is $\theta = \frac{\pi}{6} +2n\pi$ and $n$ is an integer.
