| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | Double angle equations requiring identity expansion and factorisation |
| Difficulty | Moderate -0.3 This is a straightforward double angle equation requiring the standard technique of converting cos 2θ to 1-2sin²θ, leading to a quadratic in sin θ. The algebraic manipulation is routine and finding solutions in the given interval is standard practice. Slightly easier than average due to the clean quadratic factorization and simple angle values. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2(1 - 2\sin^2\theta) = 1 - 2\sin\theta\) | M1 | Substitutes \(1-2\sin^2\theta\) or \(2\cos^2\theta - 1\) or \(\cos^2\theta - \sin^2\theta\) for \(\cos 2\theta\) |
| \(4\sin^2\theta - 2\sin\theta - 1 = 0\) | M1(*) | Forms a "quadratic in sine" \(= 0\) |
| \(\sin\theta = \frac{2 \pm \sqrt{4 - 4(4)(-1)}}{8}\) | M1 | Applies the quadratic formula |
| PVs: \(\alpha_1 = 54°\) or \(\alpha_2 = -18°\) | A1 | Any one correct answer |
| \(\theta = \{54,\ 126,\ 198,\ 342\}\) | dM1(*) | 180 \(-\) their pv |
| A1 | All four solutions correct [6] |
## Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2(1 - 2\sin^2\theta) = 1 - 2\sin\theta$ | M1 | Substitutes $1-2\sin^2\theta$ or $2\cos^2\theta - 1$ or $\cos^2\theta - \sin^2\theta$ for $\cos 2\theta$ |
| $4\sin^2\theta - 2\sin\theta - 1 = 0$ | M1(*) | Forms a "quadratic in sine" $= 0$ |
| $\sin\theta = \frac{2 \pm \sqrt{4 - 4(4)(-1)}}{8}$ | M1 | Applies the quadratic formula |
| PVs: $\alpha_1 = 54°$ or $\alpha_2 = -18°$ | A1 | Any one correct answer |
| $\theta = \{54,\ 126,\ 198,\ 342\}$ | dM1(*) | 180 $-$ their pv |
| | A1 | All four solutions correct **[6]** |\begin{enumerate}
\item Find all the solutions of
\end{enumerate}
$$2 \cos 2 \theta = 1 - 2 \sin \theta$$
in the interval $0 \leqslant \theta < 360 ^ { \circ }$.\\
\hfill \mbox{\textit{Edexcel C3 2011 Q3 [6]}}