Standard +0.3 This is a standard C3 question requiring routine application of addition formulae to derive a known identity, followed by solving a trigonometric equation using the derived result. Part (i) is guided proof with a clear method, and part (ii) involves factorising and solving basic trig equations within a given interval—slightly above average due to the multi-step nature but well within typical C3 scope.
3. (i) Use the identity for \(\sin ( A + B )\) to show that
$$\sin 3 x \equiv 3 \sin x - 4 \sin ^ { 3 } x$$
(ii) Hence find, in terms of \(\pi\), the solutions of the equation
$$\sin 3 x - \sin x = 0$$
for \(x\) in the interval \(0 \leq x < 2 \pi\).
3. (i) Use the identity for $\sin ( A + B )$ to show that
$$\sin 3 x \equiv 3 \sin x - 4 \sin ^ { 3 } x$$
(ii) Hence find, in terms of $\pi$, the solutions of the equation
$$\sin 3 x - \sin x = 0$$
for $x$ in the interval $0 \leq x < 2 \pi$.\\
\hfill \mbox{\textit{OCR C3 Q3 [8]}}