| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Expand compound angle then solve |
| Difficulty | Standard +0.3 Part (a) is a straightforward application of the Pythagorean identity requiring algebraic manipulation with surds. Part (b) involves solving a standard trigonometric equation with a compound angle and multiple solutions within an interval. Both parts use routine C2 techniques (trigonometric identities, solving trig equations) with no novel insight required, making this slightly easier than average but still requiring careful execution across multiple steps. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\sin^2 \theta = (2-\sqrt{2})^2 = 4 - 4\sqrt{2} + 2 = 6 - 4\sqrt{2}\) | M1 | |
| \(\cos^2 \theta = 1 - (6 - 4\sqrt{2}) = -5 + 4\sqrt{2}\) | M1 A1 | |
| (b) \(2x - \frac{\pi}{6} = \frac{\pi}{3}, 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}\) | B1 M1 | |
| \(2x = \frac{\pi}{3}\), \(\frac{11\pi}{6}\) | M1 A1 | |
| \(x = \frac{\pi}{4}\), \(\frac{11\pi}{12}\) | M1 A2 | (10 marks) |
**(a)** $\sin^2 \theta = (2-\sqrt{2})^2 = 4 - 4\sqrt{2} + 2 = 6 - 4\sqrt{2}$ | M1 |
$\cos^2 \theta = 1 - (6 - 4\sqrt{2}) = -5 + 4\sqrt{2}$ | M1 A1 |
**(b)** $2x - \frac{\pi}{6} = \frac{\pi}{3}, 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}$ | B1 M1 |
$2x = \frac{\pi}{3}$, $\frac{11\pi}{6}$ | M1 A1 |
$x = \frac{\pi}{4}$, $\frac{11\pi}{12}$ | M1 A2 | (10 marks)\begin{enumerate}[label=(\alph*)]
\item Given that $\sin \theta = 2 - \sqrt{2}$, find the value of $\cos^2 \theta$ in the form $a + b\sqrt{2}$ where $a$ and $b$ are integers. [3]
\item Find, in terms of $\pi$, all values of $x$ in the interval $0 \leq x < \pi$ for which
$$\cos(2x - \frac{\pi}{6}) = \frac{1}{2}.$$ [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q8 [10]}}