| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Solve equation with tan(θ ± α) |
| Difficulty | Standard +0.3 This is a standard application of the tan addition formula requiring algebraic manipulation to reach a given form, followed by solving a quadratic in tan θ. The 'show that' structure guides students through the harder algebraic work, and the techniques are routine for P3 level with no novel insight required—slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(\tan(A \pm B)\) formula correctly at least once to obtain an equation in \(\tan\,\theta\) | M1 | |
| Obtain a correct horizontal equation in any form | A1 | |
| Use correct exact values of \(\tan 30°\) and \(\tan 60°\) throughout | M1 | |
| Obtain the given equation correctly | A1 | [4 marks] |
| (ii) Make reasonable attempt to solve the given quadratic in \(\tan\,\theta\) | M1 | |
| Obtain answer \(\theta = 24.7°\) | A1 | |
| Obtain answer \(\theta = 95.3°\) and no others in the given range | A1 | [Ignore answers outside the given range.] [Treat answers in radians as MR and deduct one mark from the marks for the angles.] |
**(i)** Use $\tan(A \pm B)$ formula correctly at least once to obtain an equation in $\tan\,\theta$ | M1 |
Obtain a correct horizontal equation in any form | A1 |
Use correct exact values of $\tan 30°$ and $\tan 60°$ throughout | M1 |
Obtain the given equation correctly | A1 | [4 marks]
**(ii)** Make reasonable attempt to solve the given quadratic in $\tan\,\theta$ | M1 |
Obtain answer $\theta = 24.7°$ | A1 |
Obtain answer $\theta = 95.3°$ and no others in the given range | A1 | [Ignore answers outside the given range.] [Treat answers in radians as MR and deduct one mark from the marks for the angles.] | [3 marks]
---4 (i) Show that the equation $\tan \left( 30 ^ { \circ } + \theta \right) = 2 \tan \left( 60 ^ { \circ } - \theta \right)$ can be written in the form
$$\tan ^ { 2 } \theta + ( 6 \sqrt { } 3 ) \tan \theta - 5 = 0$$
(ii) Hence, or otherwise, solve the equation
$$\tan \left( 30 ^ { \circ } + \theta \right) = 2 \tan \left( 60 ^ { \circ } - \theta \right) ,$$
for $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2008 Q4 [7]}}