| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Prove identity then solve equation |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: algebraic manipulation to prove an identity using the double angle formula, sketching a reciprocal trig graph, and solving a trig equation. All parts follow predictable patterns taught in C3 with no novel insight required, making it slightly easier than average. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} = \frac{\sin^2\theta + \cos^2\theta}{\cos\theta\sin\theta}\) | M1 | |
| M1 Use of common denominator to obtain single fraction | ||
| \(= \frac{1}{\cos\theta\sin\theta}\) | M1 | |
| M1 Use of appropriate trig identity (in this case \(\sin^2\theta + \cos^2\theta = 1\)) | ||
| \(= \frac{1}{\frac{1}{2}\sin 2\theta}\) | M1 | |
| Use of \(\sin 2\theta = 2\sin\theta\cos\theta\) | ||
| \(= 2\operatorname{cosec}2\theta\) (✱) | A1 cso | (4 marks) |
| Alt.(a) \(\frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} = \tan\theta + \frac{1}{\tan\theta} = \frac{\tan^2\theta + 1}{\tan\theta}\) | M1 | |
| \(= \frac{\sec^2\theta}{\tan\theta}\) | M1 | |
| \(= \frac{1}{\cos\theta\sin\theta} = \frac{1}{\frac{1}{2}\sin 2\theta}\) | M1 | |
| \(= 2\operatorname{cosec}2\theta\) (✱) (cso) | A1 | |
| If show two expressions are equal, need conclusion such as QED, tick, true. | ||
| (b) | B1 | |
| Shape (May be translated but need to see 4"sections") | ||
| T.P.s at \(y = \pm2\), asymptotic at correct x-values (dotted lines not required) | B1 dep. | (2 marks) |
| (c) \(2\operatorname{cosec}2\theta = 3\) | ||
| \(\sin 2\theta = \frac{2}{3}\) | Allow \(\frac{2}{\sin 2\theta} = 3\) | M1, A1 |
| \((2\theta) = [41.810...°, 138.189...°;\) 401.810...°, 498.189...°]$ | ||
| 1st M1 for \(\alpha\), 180° – \(\alpha\); 2nd M1 adding 360° to at least one of values | M1; M1 | |
| \(\theta = 20.9°, 69.1°, 200.9°, 249.1°\) (1 d.p.) | awrt | A1, A1 |
| Answer | Marks |
|---|---|
| Alt.(c) \(\tan\theta + \frac{1}{\tan\theta} = 3\) and form quadratic, \(\tan^2\theta - 3\tan\theta + 1 = 0\) | M1, A1 |
| (M1 for attempt to multiply through by \(\tan\theta\), A1 for correct equation above) | |
| Solving quadratic [\(\tan\theta = \frac{3 \pm \sqrt{5}}{2} = 2.618...\) or \(0.3819...\)] | M1 |
| \(\theta = 69.1°, 249.1°\) | |
| \(\theta = 20.9°, 200.9°\) (1 d.p.) | M1, A1, A1 |
| (M1 is for one use of 180° + \(\alpha\)°, A1A1 as for main scheme) | |
| (12 marks) |
**(a)** $\frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} = \frac{\sin^2\theta + \cos^2\theta}{\cos\theta\sin\theta}$ | M1 | |
M1 Use of common denominator to obtain single fraction | | |
$= \frac{1}{\cos\theta\sin\theta}$ | M1 | |
M1 Use of appropriate trig identity (in this case $\sin^2\theta + \cos^2\theta = 1$) | | |
$= \frac{1}{\frac{1}{2}\sin 2\theta}$ | M1 | |
Use of $\sin 2\theta = 2\sin\theta\cos\theta$ | | |
$= 2\operatorname{cosec}2\theta$ (**✱**) | A1 cso | (4 marks) |
**Alt.(a)** $\frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} = \tan\theta + \frac{1}{\tan\theta} = \frac{\tan^2\theta + 1}{\tan\theta}$ | M1 | |
$= \frac{\sec^2\theta}{\tan\theta}$ | M1 | |
$= \frac{1}{\cos\theta\sin\theta} = \frac{1}{\frac{1}{2}\sin 2\theta}$ | M1 | |
$= 2\operatorname{cosec}2\theta$ (**✱**) (cso) | A1 | |
If show two expressions are equal, need conclusion such as QED, tick, true. | | |
**(b)** | B1 | |
Shape (May be translated but need to see 4"sections") | | |
T.P.s at $y = \pm2$, asymptotic at correct x-values (dotted lines not required) | B1 dep. | (2 marks) |
**(c)** $2\operatorname{cosec}2\theta = 3$ | | |
$\sin 2\theta = \frac{2}{3}$ | Allow $\frac{2}{\sin 2\theta} = 3$ | M1, A1 | |
$(2\theta) = [41.810...°, 138.189...°;$ 401.810...°, 498.189...°]$ | | |
1st M1 for $\alpha$, 180° – $\alpha$; 2nd M1 adding 360° to at least one of values | M1; M1 | |
$\theta = 20.9°, 69.1°, 200.9°, 249.1°$ (1 d.p.) | awrt | A1, A1 | (6 marks) |
**Note:** 1st A1 for any two correct, 2nd A1 for other two. Extra solutions in range lose final A1 only. **SC: Final 4 marks of $\theta = 20.9°$, after M0M0 is B1; record as M0M0A1A0**
**Alt.(c)** $\tan\theta + \frac{1}{\tan\theta} = 3$ and form quadratic, $\tan^2\theta - 3\tan\theta + 1 = 0$ | M1, A1 | |
(M1 for attempt to multiply through by $\tan\theta$, A1 for correct equation above) | | |
Solving quadratic [$\tan\theta = \frac{3 \pm \sqrt{5}}{2} = 2.618...$ or $0.3819...$] | M1 | |
$\theta = 69.1°, 249.1°$ | | |
$\theta = 20.9°, 200.9°$ (1 d.p.) | M1, A1, A1 | |
(M1 is for one use of 180° + $\alpha$°, A1A1 as for main scheme) | | |
| (12 marks) |
---\begin{enumerate}
\item (a) Prove that
\end{enumerate}
$$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta } = 2 \operatorname { cosec } 2 \theta , \quad \theta \neq 90 n ^ { \circ }$$
(b) On the axes on page 20, sketch the graph of $y = 2 \operatorname { cosec } 2 \theta$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.\\
(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.\\
\includegraphics[max width=\textwidth, alt={}, center]{f3c3c777-7808-4d82-a1f4-2dee6674be1e-11_899_1253_315_347}\\
\hfill \mbox{\textit{Edexcel C3 2007 Q7 [12]}}