| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Topic | Addition & Double Angle Formulae |
| Type | Given sin/cos/tan, find other expressions |
| Difficulty | Standard +0.8 Part (a) is straightforward application of the addition formula sin(x+y) = sin x cos y + cos x sin y, yielding (2√3)/4. Part (b) requires finding individual angles from their sum and difference: students must recognize that subtracting the equations gives sin(x-y), then solve a system to find x and y separately. The constraint that both angles are acute adds complexity, requiring careful consideration of which solutions are valid. This goes beyond routine formula application into problem-solving territory. |
| Spec | 1.05g Exact trigonometric values: for standard angles1.05l Double angle formulae: and compound angle formulae |
15. Two angles, $x$ and $y$, are acute.
$$\begin{aligned}
\sin x \cos y & = \frac { 1 + \sqrt { 3 } } { 4 } \\
\cos x \sin y & = \frac { - 1 + \sqrt { 3 } } { 4 }
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $\sin ( x + y )$.
\item Find all possible pairs of values of $x$ and $y$, giving your answers in terms of $\pi$. Fully justify your answer.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q15 [8]}}