Standard +0.3 This is a standard trigonometric equation requiring the double angle formula (cos 2θ = 2cos²θ - 1) to form a quadratic in cos θ, then solving and finding angles in the given range. It's slightly above average difficulty due to the multi-step process and need for inverse trig, but follows a well-practiced technique with no novel insight required.
By forming and solving a suitable quadratic equation, find the solutions of the equation
$$3 \cos 2\theta - 5 \cos \theta + 2 = 0$$
in the interval \(0° < \theta < 360°\), giving your answers to the nearest \(0.1°\).
[5 marks]
Both correct and no extra values in the interval but ignore any values outside of the interval including \(0^0\) and \(360^0\).
To earn the m1 mark, candidate's factors must give their \(6\cos^2\theta\) and their \(-1\) i.e. the first and last terms of their quadratic.
If the quadratic formula is used it must be used correctly for their quadratic.
If they get the correct quadratic and NMS, both correct answers for \(\cos\theta\) implies the m1 mark, or
If they get the correct quadratic and NMS, one correct answer for \(\theta\) implies the m1 mark.
If they get a wrong quadratic, they must show the working to (possibly) score the m1 mark.
Interval is specified to be \(0^0 < \theta < 360^0\) ; hence the reason for 'ignoring' solutions \(0^0\) and \(360^0\) that might come from\(\cos\theta = 1\).
Degree signs are not required; 99.6 and 260.4 are sufficient for the A marks.
Allow SC1 if both rounded correctly to greater accuracy 99.5940... and 260.4059... if A0 A0.
$\cos2\theta = 2\cos^2\theta - 1$ used | B1 | PI: Correct expression in terms of $\cos\theta$ used.
$3(2\cos^2\theta - 1) - 5\cos\theta + 2 \,(= 0)$ | M1 | Attempt to use identity for $\cos2\theta$ of the form $a\cos^2\theta + b$ to obtain a quadratic in $\cos\theta$.
$6\cos^2\theta - 5\cos\theta - 1 = 0$
$(\cos\theta - 1)(\cos\theta + 1) = 0$ | m1 | Attempt to factorise their quadratic or correct use of quadratic formula.
$(\cos\theta = 1)$ $\cos\theta = -\frac{1}{6}$ | |
$\theta = 99.6^0, 260.4^0$ | A1 | Either correct – CAO
| A1 | Total: 5 | Both correct and no extra values in the interval but ignore any values outside of the interval including $0^0$ and $360^0$.
To earn the m1 mark, candidate's factors must give their $6\cos^2\theta$ and their $-1$ i.e. the first and last terms of their quadratic.
If the quadratic formula is used it must be used correctly for their quadratic.
If they get the correct quadratic and NMS, both correct answers for $\cos\theta$ implies the m1 mark, or
If they get the correct quadratic and NMS, one correct answer for $\theta$ implies the m1 mark.
If they get a wrong quadratic, they must show the working to (possibly) score the m1 mark.
Interval is specified to be $0^0 < \theta < 360^0$ ; hence the reason for 'ignoring' solutions $0^0$ and $360^0$ that might come from$\cos\theta = 1$.
Degree signs are not required; 99.6 and 260.4 are sufficient for the A marks.
Allow SC1 if both rounded correctly to greater accuracy 99.5940... and 260.4059... if A0 A0.
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By forming and solving a suitable quadratic equation, find the solutions of the equation
$$3 \cos 2\theta - 5 \cos \theta + 2 = 0$$
in the interval $0° < \theta < 360°$, giving your answers to the nearest $0.1°$.
[5 marks]
\hfill \mbox{\textit{AQA C4 2016 Q2 [5]}}