2 Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { - \pi } { 2 }\).
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Question 2:
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\(3\cos\theta + 4\sin\theta = R\cos(\theta - \alpha) = R(\cos\theta\cos\alpha + \sin\theta\sin\alpha)\) M1
\(R\cos\alpha = 3\), \(R\sin\alpha = 4\) B1
\(R^2 = 3^2 + 4^2 = 25 \Rightarrow R = 5\) M1A1
\(R = 5\) cwo
\(\tan\alpha = \frac{4}{3} \Rightarrow \alpha = 0.9273\) \(5\cos(\theta - 0.9273) = 2\) M1
\(\cos(\theta - 0.9273) = \frac{2}{5}\) \(\theta - 0.9273 = 1.1593, -1.1593\) A1 A1
and no others in the range
\(\theta = 2.087, -0.232\)
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## Question 2:
$3\cos\theta + 4\sin\theta = R\cos(\theta - \alpha) = R(\cos\theta\cos\alpha + \sin\theta\sin\alpha)$ | M1 | |
$R\cos\alpha = 3$, $R\sin\alpha = 4$ | B1 | |
$R^2 = 3^2 + 4^2 = 25 \Rightarrow R = 5$ | M1A1 | $R = 5$ cwo |
$\tan\alpha = \frac{4}{3} \Rightarrow \alpha = 0.9273$ | | |
$5\cos(\theta - 0.9273) = 2$ | M1 | |
$\cos(\theta - 0.9273) = \frac{2}{5}$ | | |
$\theta - 0.9273 = 1.1593, -1.1593$ | A1 A1 | and no others in the range |
$\theta = 2.087, -0.232$ | | |
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2 Express $3 \cos \theta + 4 \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { - \pi } { 2 }$.\\
Hence solve the equation $3 \cos \theta + 4 \sin \theta = 2$ for $\quad - \pi \leqslant \theta \leqslant \pi$.
\hfill \mbox{\textit{OCR MEI C4 Q2 [7]}}