| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sine and Cosine Rules |
| Type | Given area find angle/side |
| Difficulty | Standard +0.3 This is a straightforward application of standard formulas (area = ½ab sin θ, then cosine rule) with routine algebraic manipulation. Part (a) uses the Pythagorean identity to find cos θ from sin θ, and part (b) applies the cosine rule directly. The multi-step nature and 7 marks suggest slightly above-average difficulty for AS level, but all techniques are standard with no novel insight required. |
| Spec | 1.05b 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 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sin\theta = \frac{4}{5}\), \(\cos\theta = \pm\frac{3}{5}\) | M1, A1, M1, A1 | Use the area formula and attempts to find the value of \(\sin\theta\); Correct value of \(\sin\theta\); Uses their value of \(\sin\theta\) to find two values of \(\cos\theta\) with the correct formula; Correct value of \(\cos\theta\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(QR = 2\sqrt{185}\), Perimeter: \(30 + 2\sqrt{185}\) | M1, A1, A1 | Uses a suitable method of finding the longest side by choosing a negative value of \(\cos\theta\) and proceeds to find \(QR\) using a cosine rule; The value of \(QR\) is correct; Perimeter = \(30 + 2\sqrt{185}\) |
### Part a:
$\sin\theta = \frac{4}{5}$, $\cos\theta = \pm\frac{3}{5}$ | M1, A1, M1, A1 | Use the area formula and attempts to find the value of $\sin\theta$; Correct value of $\sin\theta$; Uses their value of $\sin\theta$ to find two values of $\cos\theta$ with the correct formula; Correct value of $\cos\theta$
### Part b:
$QR = 2\sqrt{185}$, Perimeter: $30 + 2\sqrt{185}$ | M1, A1, A1 | Uses a suitable method of finding the longest side by choosing a negative value of $\cos\theta$ and proceeds to find $QR$ using a cosine rule; The value of $QR$ is correct; Perimeter = $30 + 2\sqrt{185}$In a triangle $PQR$, $PQ = 20$ cm, $PR = 10$ cm and angle $QPR = \theta$, where $\theta$ is measured in degrees. The area of triangle $PQR$ is 80 cm$^2$.
\begin{enumerate}[label=(\alph*)]
\item Show that the two possible values of $\cos \theta = \pm \frac{3}{5}$ [4]
\end{enumerate}
Given that $QR$ is the longest side of the triangle,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the exact perimeter of the triangle $PQR$, giving your answer as a simplified surd. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 1 Q7 [7]}}