C2 June 2007 Q9

EdexcelOld spec10 marksRadians Trigonometry

(a) Sketch, for \(0 \leqslant x \leqslant 2\pi\), the graph of \(y = \sin\left(x + \dfrac{\pi}{6}\right)\). (2)

(b) Write down the exact coordinates of the points where the graph meets the coordinate axes. (3)

(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. (5)

Graph of y = sin(x + π/6) for 0 to 2π: starts at 0.5 on the y-axis, rises to a maximum of 1, falls through a minimum of −1 and ends at 0.5 at x = 2π — Mark scheme (a)

Scheme	Marks
Sine wave (anywhere) with at least 2 turning points.	M1
Starting on positive \(y\)-axis, going up to a max., then min. below \(x\)-axis, no further turning points in range, finishing above \(x\)-axis at \(x = 2\pi\) or \(360^\circ\). There must be some indication of scale on the \(y\)-axis… (e.g. 1, \(-1\) or 0.5) Ignore parts of graph outside 0 to \(2\pi\).	A1
	(2)

Mark scheme (b)

Scheme	Marks
\(\left(0, \dfrac{1}{2}\right),\ \left(\dfrac{5\pi}{6}, 0\right),\ \left(\dfrac{11\pi}{6}, 0\right)\) (Ignore any extra solutions) (Not \(150^\circ, 330^\circ\)) \(\left(\pi - \dfrac{\pi}{6}\right)\) and \(\left(2\pi - \dfrac{\pi}{6}\right)\) are insufficient, but if both are seen allow B1 B0.	B1, B1, B1
	(3)

Notes

The zeros are not required, i.e. allow 0.5, etc. (and also allow coordinates the wrong way round).
These marks are also awarded if the exact intercept values are seen in part (a), but if values in (b) and (a) are contradictory, (b) takes precedence.

Mark scheme (c)

Scheme	Marks
awrt 0.71 radians (0.70758…), or awrt \(40.5^\circ\) (40.5416…) \((\alpha)\)	B1
\((\pi - \alpha)\) (2.43…) or \((180 - \alpha)\) if \(\alpha\) is in degrees. \(\left[\underline{\text{NOT}}\ \pi - \left(\alpha - \dfrac{\pi}{6}\right)\right]\)	M1
Subtract \(\dfrac{\pi}{6}\) from \(\alpha\) (or from \((\pi - \alpha)\))… or subtract 30 if \(\alpha\) is in degrees	M1
0.18 (or \(0.06\pi\)), 1.91 (or \(0.61\pi\)) Allow awrt (The 1^st A mark is dependent on just the 2^nd M mark)	A1, A1
	(5)
	10

Notes

B1: If the required value of \(\alpha\) is not seen, this mark can be given by implication if a final answer rounding to 0.18 or 0.19 (or a final answer rounding to 1.91 or 1.90) is achieved. (Also see premature approx. note*).

Special case: \(\sin\left(x + \dfrac{\pi}{6}\right) = 0.65 \Rightarrow \sin x + \sin\dfrac{\pi}{6} = 0.65 \Rightarrow \sin x = 0.15\)
\(x = \arcsin 0.15 = 0.15056...\) and \(x = \pi - 0.15056 = 2.99\) (B0 M1 M0 A0 A0)
(This special case mark is also available for degrees… \(180 - 8.62\ldots\))

Extra solutions outside 0 to \(2\pi\): Ignore.
Extra solutions between 0 and \(2\pi\): Loses the final A mark.

*Premature approximation in part (c):
e.g. \(\alpha = 41^\circ,\ 180 - 41 = 139,\ 41 - 30 = 11\) and \(139 - 30 = 109\)
Changing to radians: 0.19 and 1.90
This would score B1 (required value of \(\alpha\) not seen, but there is a final answer 0.19 (or 1.90)), M1 M1 A0 A0.