May/June 2026 Paper 21

Express $5\cos \theta +3\sin \theta$ in the form $R\cos (\theta -\alpha )$, where $R>0$ and $0^{\circ }<\alpha <90^{\circ }$. Give the exact value of $R$, and give the value of $\alpha$ correct to 2 decimal places.

Hence, solve the equation $5\cos \theta +3\sin \theta =2$ for $-180^{\circ }<\theta <180^{\circ }$.