6 Express \(\sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
Hence solve the equation \(\sin \theta - 3 \cos \theta = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
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Question 6:
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(\sin\theta - 3\cos\theta = R\sin(\theta - \alpha) = R(\sin\theta\cos\alpha - \cos\theta\sin\alpha)\) \(\Rightarrow R\cos\alpha = 1,\ R\sin\alpha = 3\) M1
equating correct pairs
\(\Rightarrow R^2 = 1^2 + 3^2 = 10 \Rightarrow R = \sqrt{10}\) B1
o.e. ft
\(\tan\alpha = 3 \Rightarrow \alpha = 71.57°\) M1
www cao (\(71.6°\) or better)
A1 \(\sqrt{10}\sin(\theta - 71.57°) = 1\) \(\Rightarrow \theta - 71.57° = \sin^{-1}(\frac{1}{\sqrt{10}})\) M1
o.e. ft \(R\), \(\alpha\)
\(\theta - 71.57° = 18.43°,\ 161.57°\) B1
www
\(\Rightarrow \theta = 90°,\ 233.1°\) A1 [7]
and no others in range (MR-1 for radians)
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## Question 6:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sin\theta - 3\cos\theta = R\sin(\theta - \alpha) = R(\sin\theta\cos\alpha - \cos\theta\sin\alpha)$ | | |
| $\Rightarrow R\cos\alpha = 1,\ R\sin\alpha = 3$ | M1 | equating correct pairs |
| $\Rightarrow R^2 = 1^2 + 3^2 = 10 \Rightarrow R = \sqrt{10}$ | B1 | o.e. ft |
| $\tan\alpha = 3 \Rightarrow \alpha = 71.57°$ | M1 | www cao ($71.6°$ or better) |
| | A1 | |
| $\sqrt{10}\sin(\theta - 71.57°) = 1$ | | |
| $\Rightarrow \theta - 71.57° = \sin^{-1}(\frac{1}{\sqrt{10}})$ | M1 | o.e. ft $R$, $\alpha$ |
| $\theta - 71.57° = 18.43°,\ 161.57°$ | B1 | www |
| $\Rightarrow \theta = 90°,\ 233.1°$ | A1 [7] | and no others in range (MR-1 for radians) |
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6 Express $\sin \theta - 3 \cos \theta$ in the form $R \sin ( \theta - \alpha )$, where $R$ and $\alpha$ are constants to be determined, and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
Hence solve the equation $\sin \theta - 3 \cos \theta = 1$ for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
\hfill \mbox{\textit{OCR C4 Q6 [7]}}