8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
- Solve, for \(0 < \theta < 360 ^ { \circ }\), the equation
$$3 \sin \left( \theta + 30 ^ { \circ } \right) = 7 \cos \left( \theta + 30 ^ { \circ } \right)$$
giving your answers to one decimal place.
- (a) Show that the equation
$$3 \sin ^ { 3 } x = 5 \sin x - 7 \sin x \cos x$$
can be written in the form
$$\sin x \left( a \cos ^ { 2 } x + b \cos x + c \right) = 0$$
where \(a , b\) and \(c\) are constants to be found.
(b) Hence solve for \(- \frac { \pi } { 2 } \leqslant x \leqslant \frac { \pi } { 2 }\) the equation
$$3 \sin ^ { 3 } x = 5 \sin x - 7 \sin x \cos x$$
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