Solve the equation \(3{\cos ^2}x - 8\cos x + 4 = 0\), where \(0 \leqslant x \leqslant 180^\circ \), expressing your answer(s) to the nearest degree.

Find the exact values of \(\sec x\) satisfying the equation \(3{\sec ^4}x - 8{\sec ^2}x + 4 = 0\).

attempting to solve for \(\cos x\) or for u where \(u = \cos x\) or for x graphically.     (M1)

EITHER

\(\cos x = \frac{2}{3}{\text{ (and 2)}}\)     (A1)

OR

\(x = 48.1897 \ldots ^\circ \)     (A1)

THEN

\(x = 48^\circ \)     A1

Note: Award (M1)(A1)A0 for \(x = 48^\circ ,{\text{ }}132^\circ \).

Note: Award (M1)(A1)A0 for 0.841 radians.

[3 marks]

attempting to solve for \(\sec x\) or for \(v\) where \(v = \sec x\).     (M1)

\(\sec x = \pm \sqrt 2 {\text{ }}\left( {{\text{and }} \pm \sqrt {\frac{2}{3}} } \right)\)     (A1)

\(\sec x = \pm \sqrt 2 \)     A1

[3 marks]

Part (a) was generally well done. Some candidates did not follow instructions and express their final answer correct to the nearest degree. A large number of candidates successfully employed a graphical approach.

Part (b) was not well done. Common errors included attempting to solve for x rather than for \(\sec x\), either omitting or not considering \(\sec x =  - \sqrt 2 \), not rejecting \(\sec x =  \pm \sqrt {\frac{2}{3}} \) and not working with exact values.