| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2008 |
| Session | November |
| 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 addition formulae followed by routine algebraic manipulation and solving a basic tan equation. Part (i) requires expanding sin(x+30°) and cos(x+60°) using standard formulae, then simplifying—mechanical but with several steps. Part (ii) is immediate once part (i) is done: rearrange to tan x = 1/(3√3) and find solutions in the given range. The question is slightly below average difficulty because it's a standard textbook exercise with clear structure and no novel insight required, though the algebraic manipulation requires care. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use correct \(\sin(A + B)\) and \(\cos(A + B)\) formulae | M1 | |
| Substitute exact values for sin 30° etc. | M1 | |
| Obtain given answer correctly | A1 | [3] |
| (ii) Solve for \(x\) | M1 | |
| Obtain answer \(x = 10.9°\) | A1 | |
| Obtain second answer \(x = -169.1°\) and no others in the range | A1 | [3] |
**(i)** Use correct $\sin(A + B)$ and $\cos(A + B)$ formulae | M1 |
Substitute exact values for sin 30° etc. | M1 |
Obtain given answer correctly | A1 | [3]
**(ii)** Solve for $x$ | M1 |
Obtain answer $x = 10.9°$ | A1 |
Obtain second answer $x = -169.1°$ and no others in the range | A1 | [3]
[Ignore answers outside the given range.]4 (i) Show that the equation
$$\sin \left( x + 30 ^ { \circ } \right) = 2 \cos \left( x + 60 ^ { \circ } \right)$$
can be written in the form
$$( 3 \sqrt { } 3 ) \sin x = \cos x$$
(ii) Hence solve the equation
$$\sin \left( x + 30 ^ { \circ } \right) = 2 \cos \left( x + 60 ^ { \circ } \right)$$
for $- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P2 2008 Q4 [6]}}