Standard +0.3 Part (a) is trivial recall of a standard double angle formula. Part (b) requires converting cosec to sin, multiplying through, applying the double angle formula sin(2x) = 2sin(x)cos(x), and solving a quadratic in cos(x). This is a standard C3 technique with straightforward execution, making it slightly easier than average overall.
8. (a) Write down \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
(b) Find, for \(0 < x < \pi\), all the solutions of the equation
$$\operatorname { cosec } x - 8 \cos x = 0$$
giving your answers to 2 decimal places.
\(\cosec x - 8\cos x = 0\), using \(\cosec x = \frac{1}{\sin x}\) to give \(\frac{1}{\sin x} - 8\cos x = 0\)
M1
Using \(\cosec x = \frac{1}{\sin x}\)
\(\frac{1}{\sin x} = 8\cos x \Rightarrow 1 = 8\sin x \cos x \Rightarrow 1 = 4(2\sin x \cos x) \Rightarrow 1 = 4\sin 2x\)
\(\sin 2x = \frac{1}{4}\) reached
M1
\(\sin 2x = k\), where \(-1 < k < 1\) and \(k \neq 0\)
\(\sin 2x = \frac{1}{4}\)
A1
\(\sin 2x = \frac{1}{4}\)
Radians: \(2x = \{0.25268..., 2.88891...\}\)
Degrees: \(2x = \{14.4775..., 165.5225...\}\)
Radians: \(x = \{0.12634..., 1.44445...\}\)
A1
Either awrt \(7.24\) or \(82.76\) or \(0.13\) or \(1.44\) or \(1.45\) or awrt \(0.04\pi\) or awrt \(0.46\pi\)
Degrees: \(x = \{7.23875..., 82.76124...\}\)
Both \(\underline{0.13}\) and \(\underline{1.44}\)
A1 cao
Solutions must be given in \(x\) only. If there are EXTRA solutions inside \(0 < x < \pi\) withhold final accuracy mark. Ignore EXTRA solutions outside \(0 < x < \pi\)
(5 marks)
[Total: 6 marks]
## Question 8:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\sin 2x = 2\sin x \cos x$ | B1 aef | $2\sin x \cos x$ |
**(1 mark)**
---
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\cosec x - 8\cos x = 0$, using $\cosec x = \frac{1}{\sin x}$ to give $\frac{1}{\sin x} - 8\cos x = 0$ | M1 | Using $\cosec x = \frac{1}{\sin x}$ |
| $\frac{1}{\sin x} = 8\cos x \Rightarrow 1 = 8\sin x \cos x \Rightarrow 1 = 4(2\sin x \cos x) \Rightarrow 1 = 4\sin 2x$ | | |
| $\sin 2x = \frac{1}{4}$ reached | M1 | $\sin 2x = k$, where $-1 < k < 1$ and $k \neq 0$ |
| $\sin 2x = \frac{1}{4}$ | A1 | $\sin 2x = \frac{1}{4}$ |
| Radians: $2x = \{0.25268..., 2.88891...\}$ | | |
| Degrees: $2x = \{14.4775..., 165.5225...\}$ | | |
| Radians: $x = \{0.12634..., 1.44445...\}$ | A1 | Either awrt $7.24$ or $82.76$ or $0.13$ or $1.44$ or $1.45$ or awrt $0.04\pi$ or awrt $0.46\pi$ |
| Degrees: $x = \{7.23875..., 82.76124...\}$ | | |
| Both $\underline{0.13}$ and $\underline{1.44}$ | A1 cao | Solutions must be given in $x$ only. If there are EXTRA solutions inside $0 < x < \pi$ withhold final accuracy mark. Ignore EXTRA solutions outside $0 < x < \pi$ |
**(5 marks)**
**[Total: 6 marks]**
8. (a) Write down $\sin 2 x$ in terms of $\sin x$ and $\cos x$.\\
(b) Find, for $0 < x < \pi$, all the solutions of the equation
$$\operatorname { cosec } x - 8 \cos x = 0$$
giving your answers to 2 decimal places.\\
\hfill \mbox{\textit{Edexcel C3 2009 Q8 [6]}}