5 Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
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Question 5:
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\(4\cos\theta - \sin\theta = R\cos(\theta + \alpha) = R(\cos\theta\cos\alpha - R\sin\theta\sin\alpha)\) M1
\(R\cos\): correct pairs
\(R\cos\alpha = 4\), \(R\sin\alpha = 1\) B1
\(R = \sqrt{17} = 4.123\)
\(R^2 = 1^2 + 4^2 = 17\), \(R = \sqrt{17} = 4.123\) M1
\(\tan\alpha = \frac{1}{4}\) o.e.
\(\tan\alpha = \frac{1}{4}\) A1
\(\alpha = 0.245\)
\(\Rightarrow \alpha = 0.245\) \(\sqrt{17}\cos(\theta + 0.245) = 3\) \(\cos(\theta + 0.245) = \frac{3}{\sqrt{17}}\) M1
\(\theta + 0.245 = \arccos\frac{3}{\sqrt{17}}\), ft their \(R\), \(\alpha\) for method
\(\theta + 0.245 = 0.756, 5.527\) \(\theta = 0.511, 5.282\) A1 A1
(penalise extra solutions in the range \((-1)\))
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## Question 5:
$4\cos\theta - \sin\theta = R\cos(\theta + \alpha) = R(\cos\theta\cos\alpha - R\sin\theta\sin\alpha)$ | M1 | $R\cos$: correct pairs |
$R\cos\alpha = 4$, $R\sin\alpha = 1$ | B1 | $R = \sqrt{17} = 4.123$ |
$R^2 = 1^2 + 4^2 = 17$, $R = \sqrt{17} = 4.123$ | M1 | $\tan\alpha = \frac{1}{4}$ o.e. |
$\tan\alpha = \frac{1}{4}$ | A1 | $\alpha = 0.245$ |
$\Rightarrow \alpha = 0.245$ | | |
$\sqrt{17}\cos(\theta + 0.245) = 3$ | | |
$\cos(\theta + 0.245) = \frac{3}{\sqrt{17}}$ | M1 | $\theta + 0.245 = \arccos\frac{3}{\sqrt{17}}$, ft their $R$, $\alpha$ for method |
$\theta + 0.245 = 0.756, 5.527$ | | |
$\theta = 0.511, 5.282$ | A1 A1 | (penalise extra solutions in the range $(-1)$) |
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5 Express $4 \cos \theta - \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$.\\
Hence solve the equation $4 \cos \theta - \sin \theta = 3$, for $0 \leqslant \theta \leqslant 2 \pi$.
\hfill \mbox{\textit{OCR MEI C4 Q5 [7]}}