| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 7 |
| Topic | Trigonometric equations in context |
| Type | Convert sin/cos ratio to tan |
| Difficulty | Standard +0.8 Part (a) requires expanding compound angle formulas, manipulating to isolate tan x, and rationalizing to reach a non-obvious exact form (3√3 - 4), which demands careful algebraic manipulation beyond routine exercises. Part (b) applies this result through angle substitution and solving for θ in a restricted domain. The multi-step algebraic complexity and the need to work toward a specific exact form elevate this above standard trig equation questions. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
11.
\begin{enumerate}[label=(\alph*)]
\item Given that
$$2 \cos ( x + 30 ) ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$
without using a calculator, show that
$$\tan x ^ { \circ } = 3 \sqrt { 3 } - 4$$
(4)
\item Hence or otherwise solve, for $0 \leqslant \theta < 180$,
$$2 \cos ( 2 \theta + 40 ) ^ { \circ } = \sin ( 2 \theta - 20 ) ^ { \circ }$$
Give your answers to one decimal place.\\
(3)
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q11 [7]}}