Question 9 - A-Level Maths

Exam Board	OCR
Module	C3 (Core Mathematics 3)
Year	2011
Session	June
Topic	Addition & Double Angle Formulae

Prove that $\frac { \sin ( \theta - \alpha ) + 3 \sin \theta + \sin ( \theta + \alpha ) } { \cos ( \theta - \alpha ) + 3 \cos \theta + \cos ( \theta + \alpha ) } \equiv \tan \theta$ for all values of $\alpha$.
Find the exact value of $\frac { 4 \sin 149 ^ { \circ } + 12 \sin 150 ^ { \circ } + 4 \sin 151 ^ { \circ } } { 3 \cos 149 ^ { \circ } + 9 \cos 150 ^ { \circ } + 3 \cos 151 ^ { \circ } }$.
It is given that $k$ is a positive constant. Solve, for $0 ^ { \circ } < \theta < 60 ^ { \circ }$ and in terms of $k$, the equation $$\frac { \sin \left( 6 \theta - 15 ^ { \circ } \right) + 3 \sin 6 \theta + \sin \left( 6 \theta + 15 ^ { \circ } \right) } { \cos \left( 6 \theta - 15 ^ { \circ } \right) + 3 \cos 6 \theta + \cos \left( 6 \theta + 15 ^ { \circ } \right) } = k .$$