Moderate -0.8 This is a straightforward C2 trigonometry question requiring a phase-shifted sine sketch, reading intercepts from the transformation, and solving a basic trig equation using calculator and symmetry. All techniques are routine with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part structure.
9. (a) Sketch, for \(0 \leqslant x \leqslant 2 \pi\), the graph of \(y = \sin \left( x + \frac { \pi } { 6 } \right)\).
(b) Write down the exact coordinates of the points where the graph meets the coordinate axes.
(c) Solve, for \(0 \leqslant x \leqslant 2 \pi\), the equation
$$\sin \left( x + \frac { \pi } { 6 } \right) = 0.65$$
giving your answers in radians to 2 decimal places.
Starting on positive \(y\)-axis, going up to max., then min. below \(x\)-axis, no further turning points in range, finishing above \(x\)-axis at \(x = 2\pi\) or \(360°\); some indication of scale on \(y\)-axis (e.g. 1, \(-1\) or 0.5)
## Question 9:
### Part (a):
| Sine wave with at least 2 turning points | M1 | |
| Starting on positive $y$-axis, going up to max., then min. below $x$-axis, no further turning points in range, finishing above $x$-axis at $x = 2\pi$ or $360°$; some indication of scale on $y$-axis (e.g. 1, $-1$ or 0.5) | A1 | Ignore parts outside $0$ to $2\pi$ |
### Part (b):
| $\left(0, \frac{1}{2}\right)$, $\left(\frac{5\pi}{6}, 0\right)$, $\left(\frac{11\pi}{6}, 0\right)$ | B1, B1, B1 | Ignore any extra solutions; not $150°$, $330°$ |
### Part (c):
| awrt $0.71$ radians ($0.70758...$) or awrt $40.5°$ ($40.5416...$) $(\alpha)$ | B1 | |
| $(\pi - \alpha)$ ($2.43...$) or $(180° - \alpha)$ if $\alpha$ in degrees | M1 | NOT $\pi - \left(\alpha - \frac{\pi}{6}\right)$ |
| Subtract $\frac{\pi}{6}$ from $\alpha$ (or from $(\pi - \alpha)$) or subtract $30°$ if $\alpha$ in degrees | M1 | |
| $0.18$ (or $0.06\pi$), $1.91$ (or $0.61\pi$) | A1, A1 | Allow awrt; 1st A mark dependent on 2nd M mark |
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9. (a) Sketch, for $0 \leqslant x \leqslant 2 \pi$, the graph of $y = \sin \left( x + \frac { \pi } { 6 } \right)$.\\
(b) Write down the exact coordinates of the points where the graph meets the coordinate axes.\\
(c) Solve, for $0 \leqslant x \leqslant 2 \pi$, the equation
$$\sin \left( x + \frac { \pi } { 6 } \right) = 0.65$$
giving your answers in radians to 2 decimal places.
\hfill \mbox{\textit{Edexcel C2 2007 Q9 [10]}}