2 Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < { } _ { 2 } ^ { 1 } \pi\).-
Show mark scheme
Show mark scheme source
Question 2:
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(4\cos\theta - \sin\theta = R\cos(\theta + \alpha) = R\cos\theta\cos\alpha - R\sin\theta\sin\alpha\) \(\Rightarrow R\cos\alpha = 4,\ R\sin\alpha = 1\) M1
correct pairs
\(\Rightarrow R^2 = 1^2 + 4^2 = 17,\ R = \sqrt{17} = 4.123\) B1
\(R = \sqrt{17} = 4.123\)
\(\tan\alpha = \frac{1}{4} \Rightarrow \alpha = 0.245\) M1
\(\tan\alpha = \frac{1}{4}\) o.e.
A1 \(\alpha = 0.245\)
\(\sqrt{17}\cos(\theta + 0.245) = 3\) \(\Rightarrow \cos(\theta + 0.245) = \frac{3}{\sqrt{17}}\) \(\Rightarrow \theta + 0.245 = 0.756,\ 5.527\) M1
\(\theta + 0.245 = \arccos\frac{3}{\sqrt{17}}\), ft their \(R\), \(\alpha\) for method
\(\Rightarrow \theta = 0.511,\ 5.282\) A1A1 [7]
penalise extra solutions in the range \((-1)\)
Copy
## Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $4\cos\theta - \sin\theta = R\cos(\theta + \alpha) = R\cos\theta\cos\alpha - R\sin\theta\sin\alpha$ | | |
| $\Rightarrow R\cos\alpha = 4,\ R\sin\alpha = 1$ | M1 | correct pairs |
| $\Rightarrow R^2 = 1^2 + 4^2 = 17,\ R = \sqrt{17} = 4.123$ | B1 | $R = \sqrt{17} = 4.123$ |
| $\tan\alpha = \frac{1}{4} \Rightarrow \alpha = 0.245$ | M1 | $\tan\alpha = \frac{1}{4}$ o.e. |
| | A1 | $\alpha = 0.245$ |
| $\sqrt{17}\cos(\theta + 0.245) = 3$ | | |
| $\Rightarrow \cos(\theta + 0.245) = \frac{3}{\sqrt{17}}$ | | |
| $\Rightarrow \theta + 0.245 = 0.756,\ 5.527$ | M1 | $\theta + 0.245 = \arccos\frac{3}{\sqrt{17}}$, ft their $R$, $\alpha$ for method |
| $\Rightarrow \theta = 0.511,\ 5.282$ | A1A1 [7] | penalise extra solutions in the range $(-1)$ |
---
Show LaTeX source
Copy
2 Express $4 \cos \theta - \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 < \alpha < { } _ { 2 } ^ { 1 } \pi$.-\\
Hence solve the equation $4 \cos \theta - \sin \theta = 3$, for $0 \leqslant \theta \leqslant 2 \pi$.
\hfill \mbox{\textit{OCR C4 Q2 [7]}}