| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Topic | Standard trigonometric equations |
| Type | Equation with non-equation preliminary part (sketch/proof/identity) |
| Difficulty | Moderate -0.8 Part (a) is a basic sketch of cos x with standard intercepts (routine recall). Part (b)(i) requires expanding and using the Pythagorean identity to reach cos θ = 1/2, which is straightforward algebraic manipulation. Part (b)(ii) applies this result with a double angle substitution and solving for x in a given interval. All steps are standard techniques with no novel insight required, making this easier than average but not trivial due to the multi-part structure. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $y = \cos x$ in the interval $0 \leq x \leq 2\pi$. State the values of the intercepts with the coordinate axes. [2 marks]
\item \begin{enumerate}[label=(\roman*)]
\item Given that
$$\sin^2 \theta = \cos \theta(2 - \cos \theta)$$
prove that $\cos \theta = \frac{1}{2}$. [2 marks]
\item Hence solve the equation
$$\sin^2 2x = \cos 2x(2 - \cos 2x)$$
in the interval $0 \leq x \leq \pi$ [2 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2022 Q10 [6]}}