| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2009 |
| 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 the compound angle formula for sin(A-B), followed by algebraic manipulation to reach tan form, then solving a basic tan equation. The steps are standard and well-practiced: expand sin(60°-x), collect terms, divide by cos x. Part (ii) is routine calculator work. Slightly easier than average due to the mechanical nature and clear path to solution. |
| 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 trig formulae to express equation in terms of \(\sin x\) and \(\cos x\) | M1 | |
| Use \(\cos 60° = \frac{1}{2}\) and \(\sin 60° = \frac{\sqrt{3}}{2}\), or equivalent | M1 | |
| Obtain equation in \(\sin x\) and \(\cos x\) in any correct form | A1 | |
| Obtain \(\tan x = \frac{\sqrt{3}}{5}\), or \(0.3464...\), or equivalent | A1 | [4 marks] |
| (ii) Obtain answer \(x = 19.1°\) | B1 | |
| Obtain answer \(x = 199.1°\) and no others in the range [ignore answers outside the given range.] | B1√ | [2 marks] |
**(i)** Use trig formulae to express equation in terms of $\sin x$ and $\cos x$ | M1 |
Use $\cos 60° = \frac{1}{2}$ and $\sin 60° = \frac{\sqrt{3}}{2}$, or equivalent | M1 |
Obtain equation in $\sin x$ and $\cos x$ in any correct form | A1 |
Obtain $\tan x = \frac{\sqrt{3}}{5}$, or $0.3464...$, or equivalent | A1 | [4 marks]
**(ii)** Obtain answer $x = 19.1°$ | B1 |
Obtain answer $x = 199.1°$ and no others in the range [ignore answers outside the given range.] | B1√ | [2 marks]4 (i) Show that the equation $\sin \left( 60 ^ { \circ } - x \right) = 2 \sin x$ can be written in the form $\tan x = k$, where $k$ is a constant.\\
(ii) Hence solve the equation $\sin \left( 60 ^ { \circ } - x \right) = 2 \sin x$, for $0 ^ { \circ } < x < 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P2 2009 Q4 [6]}}