| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2004 |
| Session | June |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Topic | Sine and Cosine Rules |
| Type | Proving angle or length value |
| Difficulty | Hard +2.3 This is a sophisticated multi-part AEA question requiring circle theorems, coordinate geometry or Pythagoras, algebraic manipulation with the AP condition, sine and cosine rules, and proving cotangent relationships. Part (a) uses the diameter property to establish b²+a²=c² combined with the AP condition—moderately challenging algebra. Parts (c)-(e) require proving the sine rule from scratch and manipulating the cosine rule with the AP constraint to establish non-trivial trigonometric identities. The extended proof structure, need for strategic algebraic insight, and synthesis across multiple parts places this well above average difficulty but not at the extreme end of AEA questions. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.04h Arithmetic sequences: nth term and sum formulae1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 |
Triangle $ABC$, with $BC = a$, $AC = b$ and $AB = c$ is inscribed in a circle. Given that $AB$ is a diameter of the circle and that $a^2$, $b^2$ and $c^2$ are three consecutive terms of an arithmetic progression (arithmetic series),
\begin{enumerate}[label=(\alph*)]
\item express $b$ and $c$ in terms of $a$, [4]
\item verify that $\cot A$, $\cot B$ and $\cot C$ are consecutive terms of an arithmetic progression. [3]
\end{enumerate}
In an acute-angled triangle $PQR$ the sides $QR$, $PR$ and $PQ$ have lengths $p$, $q$ and $r$ respectively.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Prove that
$$\frac{p}{\sin P} = \frac{q}{\sin Q} = \frac{r}{\sin R}.$$ [3]
\end{enumerate}
Given now that triangle $PQR$ is such that $p^2$, $q^2$ and $r^2$ are three consecutive terms of an arithmetic progression,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item use the cosine rule to prove that
$$\frac{2\cos Q}{q} = \frac{\cos P}{p} + \frac{\cos R}{r}.$$ [6]
\item Using the results given in parts $(c)$ and $(d)$, prove that $\cot P$, $\cot Q$ and $\cot R$ are consecutive terms in an arithmetic progression. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2004 Q7 [19]}}