Answer
$\arccos\left( \frac{\sqrt{6}}{4}+\frac{\sqrt{2}}{4}\right)$.
Work Step by Step
The angle between the vectors $$\mathbf{u}=\left(\cos \frac{\pi}{3}, \sin \frac{\pi}{3}\right), \quad \mathbf{v}=\left(\cos \frac{\pi}{4}, \sin \frac{\pi}{4}\right)$$
is given by
$$\arccos\left( \frac{u\cdot v}{\|u\| \|v\|}\right)=\arccos\left( \frac{\sqrt{6}}{4}+\frac{\sqrt{2}}{4}\right)$$