| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Product to sum using compound angles |
| Difficulty | Standard +0.8 Part (a) requires deriving a product-to-sum formula from compound angle identities, which demands algebraic manipulation and substitution insight (setting A+B=P, A-B=Q). Part (b) applies this to solve a trigonometric equation involving multiple angles, requiring strategic rearrangement and solving multiple cases. This is more challenging than routine compound angle questions but standard for C3 proof-then-apply problems. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks |
|---|---|
| \(\cos(A+B) \equiv \cos A \cos B - \sin A \sin B\) | |
| \(\cos(A-B) \equiv \cos A \cos B + \sin A \sin B\) | |
| subtracting, \(\cos(A+B) - \cos(A-B) \equiv -2\sin A \sin B\) | M1 A1 |
| \(P = A+B, Q = A-B\) | |
| adding, \(P+Q = 2A \Rightarrow A = \frac{P+Q}{2}\) | M1 |
| subtracting, \(P-Q = 2B \Rightarrow B = \frac{P-Q}{2}\) | |
| \(\therefore \cos P - \cos Q \equiv -2\sin\frac{P+Q}{2}\sin\frac{P-Q}{2}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((\cos 5x - \cos x) + \sin 3x = 0\) | ||
| \(-2\sin 3x \sin 2x + \sin 3x = 0\) | M1 | |
| \(\sin 3x(1-2\sin 2x) = 0\) | M1 | |
| \(\sin 3x = 0\) or \(\sin 2x = \frac{1}{2}\) | A1 | |
| \(3x = 0, 180, 360\) or \(2x = 30, 150\) | B1 | |
| \(x = 0, 15, 60, 75, 120\) | M1 A2 | (11) |
**(a)**
$\cos(A+B) \equiv \cos A \cos B - \sin A \sin B$ | |
$\cos(A-B) \equiv \cos A \cos B + \sin A \sin B$ | |
subtracting, $\cos(A+B) - \cos(A-B) \equiv -2\sin A \sin B$ | M1 A1 |
$P = A+B, Q = A-B$ | |
adding, $P+Q = 2A \Rightarrow A = \frac{P+Q}{2}$ | M1 |
subtracting, $P-Q = 2B \Rightarrow B = \frac{P-Q}{2}$ | |
$\therefore \cos P - \cos Q \equiv -2\sin\frac{P+Q}{2}\sin\frac{P-Q}{2}$ | A1 |
**(b)**
$(\cos 5x - \cos x) + \sin 3x = 0$ | |
$-2\sin 3x \sin 2x + \sin 3x = 0$ | M1 |
$\sin 3x(1-2\sin 2x) = 0$ | M1 |
$\sin 3x = 0$ or $\sin 2x = \frac{1}{2}$ | A1 |
$3x = 0, 180, 360$ or $2x = 30, 150$ | B1 |
$x = 0, 15, 60, 75, 120$ | M1 A2 | (11)
---\begin{enumerate}
\item (a) Use the identities for $\cos ( A + B )$ and $\cos ( A - B )$ to prove that
\end{enumerate}
$$\cos P - \cos Q \equiv - 2 \sin \frac { P + Q } { 2 } \sin \frac { P - Q } { 2 }$$
(b) Hence find all solutions in the interval $0 \leq x < 180$ to the equation
$$\cos 5 x ^ { \circ } + \sin 3 x ^ { \circ } - \cos x ^ { \circ } = 0$$
\hfill \mbox{\textit{Edexcel C3 Q6 [11]}}