Question 5 - A-Level Maths

Express \(\sqrt { 7 } \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
Hence solve the equation \(\sqrt { 7 } \sin 2 \theta + 2 \cos 2 \theta = 1\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).