(i)[3]
Write $5\cos \theta - 12\sin \theta$ in the form $R\cos(\theta + \alpha)$, where $R > 0$ and $0^{\circ} < \alpha < 90^{\circ}$, and give $\alpha$ correct to 2 decimal places.
(ii)[4]
Hence solve the equation $5\cos \theta - 12\sin \theta = 8$ for $0^{\circ} < \theta < 360^{\circ}$.
(iii)[4]
Determine the greatest possible value of $7 + 5\cos \frac{1}{2}\phi - 12\sin \frac{1}{2}\phi$ as $\phi$ changes, and also find the smallest positive value of $\phi$ for which this greatest value is attained.