Standard +0.3 This is a straightforward C2-level trigonometric equation requiring standard techniques: cross-multiply, use sin²x + cos²x = 1 to convert to a quadratic in cos x, then solve the quadratic and find angles. The algebraic manipulation is routine and the question explicitly guides students through the process in two parts. Slightly easier than average due to the scaffolding provided in part (a).
12. (a) Show that
$$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$
may be expressed in the form
$$a \cos ^ { 2 } x + b \cos x + c = 0$$
where \(a , b\) and \(c\) are constants to be found.
(b) Hence solve, for \(0 \leqslant x < 2 \pi\), the equation
$$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$
giving your answers in radians to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
\includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-37_81_65_2640_1886}
12. (a) Show that
$$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$
may be expressed in the form
$$a \cos ^ { 2 } x + b \cos x + c = 0$$
where $a , b$ and $c$ are constants to be found.\\
(b) Hence solve, for $0 \leqslant x < 2 \pi$, the equation
$$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$
giving your answers in radians to 2 decimal places.\\
(Solutions based entirely on graphical or numerical methods are not acceptable.)\\
\includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-37_81_65_2640_1886}\\
\hfill \mbox{\textit{Edexcel C12 2019 Q12 [8]}}