Answer
$\theta = \frac{\pi}{12} +2n\pi$ and $n$ is an integer.
Work Step by Step
Let $u=\left(\cos \frac{3\pi}{4},\sin \frac{3\pi}{4}\right),\quad v=\left(\cos \frac{2\pi}{3},\sin \frac{2\pi}{3}\right) \quad \langle u, v\rangle=u\cdot v $.
The angle $\theta$ between $u$ and $v$ is given by the formula
$$\cos \theta=\frac{\langle u, v\rangle}{\|u\| \cdot\|v\|}=\frac{\cos \frac{3\pi}{4}\cos \frac{2\pi}{3}+\sin \frac{3\pi}{4}\sin \frac{2\pi}{3}}{\sqrt{ \cos^2 \frac{3\pi}{4}+\sin^2 \frac{3\pi}{4}}\sqrt{ \cos^2 \frac{2\pi}{3}+\sin^2 \frac{2\pi}{3}}}.$$
Using the properties of the trignometric functions, we get
$$\cos \theta=\cos\left( \frac{3\pi}{4}- \frac{2\pi}{3}\right)=\cos \frac{\pi}{12} $$
that is $\theta = \frac{\pi}{12} +2n\pi$ and $n$ is an integer.
