| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation only (no integral) |
| Difficulty | Standard +0.3 This is a straightforward application of addition formulae requiring expansion of sin(θ+60°) and sin(θ+120°), simplification using exact values, then two routine applications. Part (ii)(a) is direct substitution, and (ii)(b) is a standard trigonometric equation. The multi-step nature and exact value work place it slightly above average, but all techniques are standard P2 material with no novel insight required. |
| Spec | 1.05g Exact trigonometric values: for standard angles1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State \(\sin\theta\cos 60 + \cos\theta\sin 60 + \sin\theta\cos 120 + \cos\theta\sin 120\) | *B1 | |
| Use \(\sin 60 = \sin 120 = \frac{1}{2}\sqrt{3}\) and \(\cos 60 = \frac{1}{2}\), \(\cos 120 = -\frac{1}{2}\) | *B1 | |
| Confirm result \(\sqrt{3}\cos\theta\), dependent on *B *B | DB1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\cos 45\) seen | *B1 | |
| State \(\sqrt{\frac{3}{2}}\) or \(\frac{1}{2}\sqrt{6}\) or exact equivalent, dependent *B | DB1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Carry out correct process to find at least one value of \(\theta\) from \(\cos^2\theta = k\) | M1 | |
| Obtain 40.6 | A1 | |
| Obtain 139.4 | A1 | [3] |
## Question 4:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State $\sin\theta\cos 60 + \cos\theta\sin 60 + \sin\theta\cos 120 + \cos\theta\sin 120$ | *B1 | |
| Use $\sin 60 = \sin 120 = \frac{1}{2}\sqrt{3}$ and $\cos 60 = \frac{1}{2}$, $\cos 120 = -\frac{1}{2}$ | *B1 | |
| Confirm result $\sqrt{3}\cos\theta$, dependent on *B *B | DB1 | [3] |
### Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\cos 45$ seen | *B1 | |
| State $\sqrt{\frac{3}{2}}$ or $\frac{1}{2}\sqrt{6}$ or exact equivalent, dependent *B | DB1 | [2] |
### Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Carry out correct process to find at least one value of $\theta$ from $\cos^2\theta = k$ | M1 | |
| Obtain 40.6 | A1 | |
| Obtain 139.4 | A1 | [3] |
---4 (i) Show that $\sin \left( \theta + 60 ^ { \circ } \right) + \sin \left( \theta + 120 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \cos \theta$.\\
(ii) Hence
\begin{enumerate}[label=(\alph*)]
\item find the exact value of $\sin 105 ^ { \circ } + \sin 165 ^ { \circ }$,
\item solve the equation $\sin \left( \theta + 60 ^ { \circ } \right) + \sin \left( \theta + 120 ^ { \circ } \right) = \sec \theta$ for $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2016 Q4 [8]}}