| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Given sin/cos/tan, find other expressions |
| Difficulty | Standard +0.3 This is a straightforward application of standard double angle and addition formulae. Part (a) uses cos 2x = 2cos²x - 1 with simple algebraic manipulation. Part (b) requires expanding compound angles and solving for tan y, but follows a routine procedure. Slightly above average difficulty due to the algebraic manipulation with surds, but still a standard C3 exercise. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae |
| Answer | Marks |
|---|---|
| \(\cos^2 x = (\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}\) | M1 |
| \(\cos 2x = 2\cos^2 x - 1 = 2(4 - 2\sqrt{3}) - 1 = 7 - 4\sqrt{3}\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(2(\cos y \cos 30° - \sin y \sin 30°) = \sqrt{3}(\sin y \cos 30° - \cos y \sin 30°)\) | M1 A1 | |
| \(\sqrt{3}\cos y - \sin y = \frac{3}{2}\sin y - \frac{1}{2}\sqrt{3}\cos y\) | B1 | |
| \(\frac{3}{2}\sqrt{3}\cos y = \frac{5}{2}\sin y\) | ||
| \(\tan y = \frac{3}{5}\sqrt{3} = \frac{3}{5}\sqrt{3}\) | M1 A1 | (8 marks) |
**(a)**
$\cos^2 x = (\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}$ | M1 |
$\cos 2x = 2\cos^2 x - 1 = 2(4 - 2\sqrt{3}) - 1 = 7 - 4\sqrt{3}$ | M1 A1 |
**(b)**
$2(\cos y \cos 30° - \sin y \sin 30°) = \sqrt{3}(\sin y \cos 30° - \cos y \sin 30°)$ | M1 A1 |
$\sqrt{3}\cos y - \sin y = \frac{3}{2}\sin y - \frac{1}{2}\sqrt{3}\cos y$ | B1 |
$\frac{3}{2}\sqrt{3}\cos y = \frac{5}{2}\sin y$ | |
$\tan y = \frac{3}{5}\sqrt{3} = \frac{3}{5}\sqrt{3}$ | M1 A1 | (8 marks)\begin{enumerate}
\item (a) Given that $\cos x = \sqrt { 3 } - 1$, find the value of $\cos 2 x$ in the form $a + b \sqrt { 3 }$, where $a$ and $b$ are integers.\\
(b) Given that
\end{enumerate}
$$2 \cos ( y + 30 ) ^ { \circ } = \sqrt { 3 } \sin ( y - 30 ) ^ { \circ }$$
find the value of $\tan y$ in the form $k \sqrt { 3 }$ where $k$ is a rational constant.\\
\hfill \mbox{\textit{Edexcel C3 Q1 [8]}}