C3 June 2007 Q7
7.
(a) Prove that\[\frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} = 2\,\mathrm{cosec}\,2\theta, \qquad \theta \neq 90n^\circ.\] (4)
(b) On the axes, sketch the graph of \(y = 2\,\mathrm{cosec}\,2\theta\) for \(0^\circ < \theta < 360^\circ\). (2)
(c) Solve, for \(0^\circ < \theta < 360^\circ\), the equation\[\frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} = 3,\]giving your answers to 1 decimal place. (6)
| Scheme | Marks |
|---|---|
| \(\dfrac{\sin\theta}{\cos\theta} + \dfrac{\cos\theta}{\sin\theta} = \dfrac{\sin^2\theta + \cos^2\theta}{\cos\theta\sin\theta}\) M1 Use of common denominator to obtain single fraction | M1 |
| \(= \dfrac{1}{\cos\theta\sin\theta}\) M1 Use of appropriate trig identity (in this case \(\sin^2\theta + \cos^2\theta = 1\)) | M1 |
| \(= \dfrac{1}{\frac{1}{2}\sin 2\theta}\) Use of \(\sin 2\theta = 2\sin\theta\cos\theta\) | M1 |
| \(= 2\,\mathrm{cosec}\,2\theta\) (✱) | A1 cso |
| (4) |
Alternative
| \(\dfrac{\sin\theta}{\cos\theta} + \dfrac{\cos\theta}{\sin\theta} = \tan\theta + \dfrac{1}{\tan\theta} = \dfrac{\tan^2\theta + 1}{\tan\theta}\) | M1 |
| \(= \dfrac{\sec^2\theta}{\tan\theta}\) | M1 |
| \(= \dfrac{1}{\cos\theta\sin\theta} = \dfrac{1}{\frac{1}{2}\sin 2\theta}\) | M1 |
| \(= 2\,\mathrm{cosec}\,2\theta\) (✱) (cso) | A1 |
Notes
If show two expressions are equal, need conclusion such as QED, tick, true.
| Scheme | Marks |
|---|---|
| Shape (May be translated but need to see 4 “sections”) | B1 |
| T.P.s at \(y = \pm 2\), asymptotic at correct \(x\)-values (dotted lines not required) | B1 dep. |
| (2) |
| Scheme | Marks |
|---|---|
| \(2\,\mathrm{cosec}\,2\theta = 3\) \(\sin 2\theta = \dfrac{2}{3}\) Allow \(\dfrac{2}{\sin 2\theta} = 3\) [M1 for equation in \(\sin 2\theta\)] | M1, A1 |
| \((2\theta) = [\ 41.810\ldots^\circ,\ 138.189\ldots^\circ;\ \ 401.810\ldots^\circ,\ 498.189\ldots^\circ\ ]\) 1st M1 for \(\alpha,\ 180 - \alpha\); 2nd M1 adding \(360^\circ\) to at least one of values | M1; M1 |
| \(\theta = 20.9^\circ,\ 69.1^\circ,\ 200.9^\circ,\ 249.1^\circ\) (1 d.p.) awrt | A1,A1 |
| (6) | |
| (12 marks) |
Notes
1st A1 for any two correct, 2nd A1 for other two
Extra solutions in range lose final A1 only
Alternative
| \(\tan\theta + \dfrac{1}{\tan\theta} = 3\) and form quadratic, \(\tan^2\theta - 3\tan\theta + 1 = 0\) (M1 for attempt to multiply through by \(\tan\theta\), A1 for correct equation above) | M1, A1 |
| Solving quadratic \(\left[\tan\theta = \dfrac{3 \pm \sqrt{5}}{2} = 2.618\ldots \text{ or } = 0.3819\ldots\right]\) | M1 |
| \(\theta = 69.1^\circ,\ 249.1^\circ\) \(\theta = 20.9^\circ,\ 200.9^\circ\) (1 d.p.) (M1 is for one use of \(180^\circ + \alpha^\circ\), A1A1 as for main scheme) | M1, A1, A1 |