The angle θ between the vectors \(\rm \vec{a} = 5\hat{i} - \hat{j} + \hat{k}\) and \(\rm \vec{b} = \hat{i} + \hat{j} - \hat{k}\) equals to
Let $ f: \mathbb{R} \to \mathbb{R} $ be a twice differentiable function such that $$ f''(x)\sin\left(\frac{x}{2}\right) + f'(2x - 2y) = (\cos x)\sin(y + 2x) + f(2x - 2y) $$ for all $ x, y \in \mathbb{R} $. If $ f(0) = 1 $, then the value of $ 24f^{(4)}\left(\frac{5\pi}{3}\right) $ is: