(a)[4]
Express $3\sin x + 2\sqrt{2}\cos\left(x + \frac{1}{4}\pi\right)$ in the form $R\sin(x + \alpha)$, with $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. State the exact value of $R$, and give $\alpha$ correct to $3$ decimal places.
(b)[5]
Hence solve the equation $6\sin\left(\frac{1}{2}\theta\right) + 4\sqrt{2}\cos\left(\frac{1}{2}\theta + \frac{1}{4}\pi\right) = 3$ in the interval $-4\pi < \theta < 4\pi$.