| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| 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 structured two-part question where part (i) guides students through algebraic manipulation using the tan addition formula, and part (ii) applies this to solve a quadratic in tan θ. The addition formula application is standard, and the algebraic steps are clearly signposted. While it requires multiple techniques (addition formulae, algebraic manipulation, solving quadratic equations), the scaffolding makes it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Use \(\tan(A \pm B)\) formula correctly at least once and obtain an equation in \(\tan\theta\) | M1 | |
| Obtain a correct horizontal equation in any form | A1 | |
| Use \(\tan 60° = \sqrt{3}\) throughout | M1 | |
| Obtain the given equation correctly | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Set \(k = 3\sqrt{3}\) and obtain \(\tan^2\theta = \frac{1}{11}\) | B1 | |
| Obtain answer \(16.8°\) | B1√ | |
| Obtain answer \(163.2°\) | B1√ | [3] |
| [Ignore answers outside the given interval. Treat answers in radians (0.293 and 2.85) as a misread.] |
**(i)**
Use $\tan(A \pm B)$ formula correctly at least once and obtain an equation in $\tan\theta$ | M1 |
Obtain a correct horizontal equation in any form | A1 |
Use $\tan 60° = \sqrt{3}$ throughout | M1 |
Obtain the given equation correctly | A1 | [4]
**(ii)**
Set $k = 3\sqrt{3}$ and obtain $\tan^2\theta = \frac{1}{11}$ | B1 |
Obtain answer $16.8°$ | B1√ |
Obtain answer $163.2°$ | B1√ | [3]
[Ignore answers outside the given interval. Treat answers in radians (0.293 and 2.85) as a misread.] |4 (i) Show that the equation
$$\tan \left( 60 ^ { \circ } + \theta \right) + \tan \left( 60 ^ { \circ } - \theta \right) = k$$
can be written in the form
$$( 2 \sqrt { } 3 ) \left( 1 + \tan ^ { 2 } \theta \right) = k \left( 1 - 3 \tan ^ { 2 } \theta \right)$$
(ii) Hence solve the equation
$$\tan \left( 60 ^ { \circ } + \theta \right) + \tan \left( 60 ^ { \circ } - \theta \right) = 3 \sqrt { } 3$$
giving all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2011 Q4 [7]}}