C3 June 2009 Q8
8.
(a) Write down \(\sin 2x\) in terms of \(\sin x\) and \(\cos x\). (1)
(b) Find, for \(0 \lt x \lt \pi\), all the solutions of the equation\[\operatorname{cosec} x - 8\cos x = 0\]giving your answers to 2 decimal places. (5)
| Scheme | Marks |
|---|---|
| \(\sin 2x = \underline{2\sin x\cos x}\) | B1 aef |
| (1) |
Notes
B1 aef: \(\underline{2\sin x\cos x}\)
| Scheme | Marks |
|---|---|
| \(\operatorname{cosec} x - 8\cos x = 0, \quad 0 \lt x \lt \pi\) | |
| \(\dfrac{1}{\sin x} - 8\cos x = 0\) | M1 |
| \(\dfrac{1}{\sin x} = 8\cos x\) \(1 = 8\sin x\cos x\) \(1 = 4(2\sin x\cos x)\) \(1 = 4\sin 2x\) | |
| \(\underline{\sin 2x = \tfrac{1}{4}}\) | M1 A1 |
| Radians \(\quad 2x = \{0.25268\ldots, 2.88891\ldots\}\) Degrees \(\quad 2x = \{14.4775\ldots, 165.5225\ldots\}\) | |
| Radians \(\quad x = \{0.12634\ldots, 1.44445\ldots\}\) Degrees \(\quad x = \{7.23875\ldots, 82.76124\ldots\}\) | A1 A1 cao |
| (5) | |
| (6 marks) |
Notes
M1: Using \(\operatorname{cosec} x = \dfrac{1}{\sin x}\)
M1: \(\sin 2x = k\), where \(-1 \lt k \lt 1\) and \(k \ne 0\)
A1: \(\underline{\sin 2x = \tfrac{1}{4}}\)
A1: Either arwt 7.24 or 82.76 or 0.13 or 1.44 or 1.45 or awrt \(0.04\pi\) or awrt \(0.46\pi\).
A1 cao: Both 0.13 and 1.44
Solutions for the final two A marks must be given in \(x\) only. If there are any EXTRA solutions inside the range \(0 \lt x \lt \pi\) then withhold the final accuracy mark. Also ignore EXTRA solutions outside the range \(0 \lt x \lt \pi\).