| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Show equation reduces to tan form |
| Difficulty | Moderate -0.3 This is a straightforward application of compound angle formulae to show a given result, followed by routine angle-finding. Part (a) requires expanding cos(x-30°) and sin(x+30°) using standard formulae, then algebraic manipulation to isolate tan x—mechanical but requiring careful algebra. Part (b) is trivial once (a) is complete. Slightly easier than average due to being a 'show that' question with the answer provided and standard technique application. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use correct trig expansions and obtain an equation in \(\sin x\) and \(\cos x\) | *M1 | |
| Use correct exact trig ratios for \(30°\) in *their* expansion | B1 FT | e.g. \(\cos x\!\left(\frac{\sqrt{3}}{2}-1\right) = \sin x\!\left(\sqrt{3}-\frac{1}{2}\right)\) |
| Obtain an equation in \(\tan x\) | DM1 | Allow if their error in line 1 was a sign error |
| Obtain \(\tan x = \dfrac{2-\sqrt{3}}{1-2\sqrt{3}}\) from correct working | A1 | AG |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain answer in the given interval, e.g. \(173.8°\) | B1 | Accept \(174°\), \(354°\) or better |
| Obtain a second answer and no other in the given interval, e.g. \(353.8°\) | B1 | Ignore answers outside the given interval. Treat answers in radians (\(3.03\) and \(6.17\)) as a misread. |
| Total | 2 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use correct trig expansions and obtain an equation in $\sin x$ and $\cos x$ | *M1 | |
| Use correct exact trig ratios for $30°$ in *their* expansion | B1 FT | e.g. $\cos x\!\left(\frac{\sqrt{3}}{2}-1\right) = \sin x\!\left(\sqrt{3}-\frac{1}{2}\right)$ |
| Obtain an equation in $\tan x$ | DM1 | Allow if their error in line 1 was a sign error |
| Obtain $\tan x = \dfrac{2-\sqrt{3}}{1-2\sqrt{3}}$ from correct working | A1 | AG |
| **Total** | **4** | |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain answer in the given interval, e.g. $173.8°$ | B1 | Accept $174°$, $354°$ or better |
| Obtain a second answer and no other in the given interval, e.g. $353.8°$ | B1 | Ignore answers outside the given interval. Treat answers in radians ($3.03$ and $6.17$) as a misread. |
| **Total** | **2** | |
---3
\begin{enumerate}[label=(\alph*)]
\item Given that $\cos \left( x - 30 ^ { \circ } \right) = 2 \sin \left( x + 30 ^ { \circ } \right)$, show that $\tan x = \frac { 2 - \sqrt { 3 } } { 1 - 2 \sqrt { 3 } }$.
\item Hence solve the equation
$$\cos \left( x - 30 ^ { \circ } \right) = 2 \sin \left( x + 30 ^ { \circ } \right)$$
for $0 ^ { \circ } < x < 360 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q3 [6]}}