| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Two angles with tan relationships |
| Difficulty | Hard +2.3 This AEA question requires recognizing a geometric series, applying the sum formula, manipulating the resulting equation using double angle formulae (tan 2θ = 2tan θ/(1-tan²θ)), and solving a quartic equation in tan θ. The multi-step algebraic manipulation and the need to connect the solution to angle bounds makes this significantly harder than standard A-level trigonometry questions. |
| Spec | 1.04j Sum to infinity: convergent geometric series |r|<11.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.06g Equations with exponentials: solve a^x = b |
3.The angle $\theta , 0 < \theta < \frac { \pi } { 2 }$ ,satisfies
$$\tan \theta \tan 2 \theta = \sum _ { r = 0 } ^ { \infty } 2 \cos ^ { r } 2 \theta$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\tan \theta = 3 ^ { p }$ ,where $p$ is a rational number to be found.
\item Hence show that $\frac { \pi } { 6 } < \theta < \frac { \pi } { 4 }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2012 Q3 [10]}}