Sketch the graph of $y = \left| {\cos \left( {\frac{x}{4}} \right)} \right|$ for $0 \leqslant x \leqslant 8\pi$.

Solve $\left| {\cos \left( {\frac{x}{4}} \right)} \right| = \frac{1}{2}$ for $0 \leqslant x \leqslant 8\pi$.

    A1A1

Note:     Award A1 for correct shape and A1 for correct domain and range.

[2 marks]

$\left| {\cos \left( {\frac{x}{4}} \right)} \right| = \frac{1}{2}$

$x = \frac{{4\pi }}{3}$     A1

attempting to find any other solutions     M1

Note:     Award (M1) if at least one of the other solutions is correct (in radians or degrees) or clear use of symmetry is seen.

$x = 8\pi  - \frac{{4\pi }}{3} = \frac{{20 \pi }}{3}$

$x = 4\pi  - \frac{{4\pi }}{3} = \frac{{8\pi }}{3}$

$x = 4\pi  + \frac{{4\pi }}{3} = \frac{{16\pi }}{3}$     A1

Note:     Award A1 for all other three solutions correct and no extra solutions.

Note:     If working in degrees, then max A0M1A0.

[3 marks]