| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | October |
| Marks | 12 |
| Topic | Addition & Double Angle Formulae |
| Type | Given sin/cos/tan, find other expressions |
| Difficulty | Moderate -0.3 Part (a) is a routine double angle formula application requiring finding sin A from cos A in a given quadrant (standard C3/C4 material). Part (b)(i) is a straightforward compound angle expansion to verify an identity. Part (b)(ii) involves differentiating trigonometric functions and simplifying using the result from (b)(i), which is methodical but not demanding. All parts follow standard procedures with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.07k Differentiate trig: sin(kx), cos(kx), tan(kx) |
\begin{enumerate}[label=(\alph*)]
\item Given that $\cos A = \frac{3}{4}$, where $270° < A < 360°$, find the exact value of $\sin 2A$. [5]
\item
\begin{enumerate}[label=(\roman*)]
\item Show that $\cos\left(2x + \frac{\pi}{3}\right) + \cos\left(2x - \frac{\pi}{3}\right) = \cos 2x$. [3]
Given that
$$y = 3\sin^2 x + \cos\left(2x + \frac{\pi}{3}\right) + \cos\left(2x - \frac{\pi}{3}\right),$$
\item show that $\frac{dy}{dx} = \sin 2x$. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q3 [12]}}