Standard +0.3 This is a standard C3 harmonic form question with routine steps: expand R cos(x+α), equate coefficients to find R and α, then solve a simple equation of form R cos(x+α)=4. The method is well-practiced and requires no novel insight, making it slightly easier than average for A-level.
3. (a) Express \(5 \cos x - 3 \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
(b) Hence, or otherwise, solve the equation
$$5 \cos x - 3 \sin x = 4$$
for \(0 \leqslant x < 2 \pi\), giving your answers to 2 decimal places.
*Note: If there are any EXTRA solutions inside the range \(0 \leq x < 2\pi\), then withhold the final accuracy mark if the candidate would otherwise score all 5 marks. Also ignore EXTRA solutions outside the range \(0 \leq x < 2\pi\).*
## Question 3:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $5\cos x - 3\sin x = R\cos x\cos\alpha - R\sin x\sin\alpha$ | | |
| Equate $\cos x$: $5 = R\cos\alpha$ | | |
| Equate $\sin x$: $3 = R\sin\alpha$ | | |
| $R = \sqrt{5^2+3^2} = \sqrt{34}\ \{= 5.83095...\}$ | M1; A1 | $R^2=5^2+3^2$; $\sqrt{34}$ or awrt 5.8 |
| $\tan\alpha = \frac{3}{5} \Rightarrow \alpha = 0.5404195003...^c$ | M1 | $\tan\alpha = \pm\frac{3}{5}$ or $\tan\alpha = \pm\frac{5}{3}$ or $\sin\alpha = \pm\frac{3}{\text{their }R}$ or $\cos\alpha = \pm\frac{5}{\text{their }R}$ |
| | A1 | $\alpha=$ awrt 0.54 or $\alpha=$ awrt $0.17\pi$ or $\alpha = \frac{\pi}{\text{awrt }5.8}$ |
| Hence, $5\cos x - 3\sin x = \sqrt{34}\cos(x+0.5404)$ | | |
**Total: (4)**
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| $5\cos x - 3\sin x = 4$ | | |
| $\sqrt{34}\cos(x+0.5404) = 4$ | | |
| $\cos(x+0.5404) = \frac{4}{\sqrt{34}}\ \{=0.68599...\}$ | M1 | $\cos(x \pm \text{their }\alpha) = \frac{4}{\text{their }R}$ |
| $(x+0.5404) = 0.814826916...^c$ | M1 | For applying $\cos^{-1}\left(\frac{4}{\text{their }R}\right)$ |
| $x = 0.2744...^c$ | A1 | awrt $0.27^c$ |
| $(x+0.5404) = 2\pi - 0.814826916...^c\ \{=5.468358...^c\}$ | ddM1 | $2\pi - \text{their }0.8148$ |
| $x = 4.9279...^c$ | A1 | awrt $4.93^c$ |
| Hence $x = \{0.27, 4.93\}$ | | |
**Total: (5)**
**[9]**
*Note: If there are any EXTRA solutions inside the range $0 \leq x < 2\pi$, then withhold the final accuracy mark if the candidate would otherwise score all 5 marks. Also ignore EXTRA solutions outside the range $0 \leq x < 2\pi$.*
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3. (a) Express $5 \cos x - 3 \sin x$ in the form $R \cos ( x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$.\\
(b) Hence, or otherwise, solve the equation
$$5 \cos x - 3 \sin x = 4$$
for $0 \leqslant x < 2 \pi$, giving your answers to 2 decimal places.\\
\hfill \mbox{\textit{Edexcel C3 2010 Q3 [9]}}