| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve tan·sin or tan·trig product |
| Difficulty | Standard +0.3 This is a multi-part question with routine components: (a) requires basic knowledge of cosine graph properties, (b) involves standard transformations (horizontal stretch and translation), and (c) requires solving a trig equation by rearranging to tan θ form and using a calculator in a specified range. While part (c) requires some algebraic manipulation and understanding of the tangent function, all techniques are standard AS-level material with no novel problem-solving required. The question is slightly easier than average due to the scaffolded structure and straightforward application of known methods. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05f Trigonometric function graphs: symmetries and periodicities1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((-180°, -3)\) | B1 | Deduces \(P(-180°, -3)\) or \(c = -180°, d = -3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((-720°, -3)\) | B1ft | Follow through on their \((c,d) \to (4c,d)\) where \(d\) is negative |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((-144°, -3)\) | B1ft | Follow through on their \((c,d) \to (c+36°, d)\) where \(d\) is negative |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts to use \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), \(\sin^2\theta + \cos^2\theta = 1\) and solves quadratic in \(\sin\theta\) | M1 | Overall problem solving mark, condoning slips |
| \(3\cos\theta = 8\tan\theta \Rightarrow 3\cos^2\theta = 8\sin\theta\) | B1 | Uses correct identity and multiplies across |
| \(3\sin^2\theta + 8\sin\theta - 3 = 0\) and \((3\sin\theta - 1)(\sin\theta + 3) = 0\) | M1 | Uses \(\sin^2\theta + \cos^2\theta = 1\) to form 3TQ in \(\sin\theta\), attempts to solve |
| \(\sin\theta = \frac{1}{3}\) | A1 | Accept sight of \(\frac{1}{3}\); ignore reference to other root |
| awrt \(520.5°\) only | A1 | Full method with all identities correct; no other values |
# Question 9:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(-180°, -3)$ | B1 | Deduces $P(-180°, -3)$ or $c = -180°, d = -3$ |
## Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(-720°, -3)$ | B1ft | Follow through on their $(c,d) \to (4c,d)$ where $d$ is negative |
## Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(-144°, -3)$ | B1ft | Follow through on their $(c,d) \to (c+36°, d)$ where $d$ is negative |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to use $\tan\theta = \frac{\sin\theta}{\cos\theta}$, $\sin^2\theta + \cos^2\theta = 1$ and solves quadratic in $\sin\theta$ | M1 | Overall problem solving mark, condoning slips |
| $3\cos\theta = 8\tan\theta \Rightarrow 3\cos^2\theta = 8\sin\theta$ | B1 | Uses correct identity and multiplies across |
| $3\sin^2\theta + 8\sin\theta - 3 = 0$ and $(3\sin\theta - 1)(\sin\theta + 3) = 0$ | M1 | Uses $\sin^2\theta + \cos^2\theta = 1$ to form 3TQ in $\sin\theta$, attempts to solve |
| $\sin\theta = \frac{1}{3}$ | A1 | Accept sight of $\frac{1}{3}$; ignore reference to other root |
| awrt $520.5°$ only | A1 | Full method with all identities correct; no other values |
---9.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{bcbd842f-b2e2-4587-ab4c-15a57a449e5d-20_810_1214_255_427}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows part of the curve with equation $y = 3 \cos x ^ { \circ }$.\\
The point $P ( c , d )$ is a minimum point on the curve with $c$ being the smallest negative value of $x$ at which a minimum occurs.
\begin{enumerate}[label=(\alph*)]
\item State the value of $c$ and the value of $d$.
\item State the coordinates of the point to which $P$ is mapped by the transformation which transforms the curve with equation $y = 3 \cos x ^ { \circ }$ to the curve with equation
\begin{enumerate}[label=(\roman*)]
\item $y = 3 \cos \left( \frac { x ^ { \circ } } { 4 } \right)$
\item $y = 3 \cos ( x - 36 ) ^ { \circ }$
\end{enumerate}\item Solve, for $450 ^ { \circ } \leqslant \theta < 720 ^ { \circ }$,
$$3 \cos \theta = 8 \tan \theta$$
giving your solution to one decimal place.\\
In part (c) you must show all stages of your working.\\
Solutions relying entirely on calculator technology are not acceptable.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 1 2020 Q9 [8]}}