| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sine and Cosine Rules |
| Type | Exact trigonometric values |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing standard techniques: part (a) uses Pythagorean identity to find exact trig values from given ratios (routine C2 content), and part (b) applies the sine rule directly with a special angle. All steps are mechanical with no problem-solving or insight required, making it easier than average. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\cos\alpha = \frac{5}{\sqrt{29}}\) | M1 | Attempt \(\cos\alpha\). Could draw triangle and use Pythagoras to find hypotenuse, or use trig identities. Must get as far as attempting \(\cos\alpha\). Must be working in exact values. Must be using correct ratios for \(\tan\alpha\) and \(\cos\alpha\) |
| A1 | Obtain \(\frac{5}{\sqrt{29}}\). Allow any exact equiv, including rationalised surd or \(\sqrt{\frac{25}{29}}\). isw if decimal equiv subsequently given. Answer only gets full credit. SR B1 for exact answer following decimal working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\cos\beta = -\frac{\sqrt{40}}{7}\) | M1 | Attempt \(\cos\beta\). Could draw triangle and use Pythagoras to find adjacent, or use trig identities. Must get as far as attempting \(\cos\beta\). Must be working in exact values. Must be using correct ratios for \(\sin\alpha\) and \(\cos\alpha\) |
| A1 | Obtain \(\frac{\sqrt{40}}{7}\). Allow any exact equiv, including \(\sqrt{\frac{40}{49}}\). Allow \(\pm\frac{\sqrt{40}}{7}\) (from using \(\cos^2 x = 1 - \sin^2 x\)). isw if decimal equiv subsequently given. Answer only gets M1A1 | |
| A1 FT | Obtain \(-\frac{\sqrt{40}}{7}\), or \(-\)ve of their exact numerical value for \(\cos\beta\). A1 FT can only be awarded following M1. isw if decimal equiv subsequently given. Answer only gets full credit. SR B1 for \(\frac{\sqrt{40}}{7}\) or equiv, following decimal working. SR B2 for \(-\frac{\sqrt{40}}{7}\) or equiv, following decimal working. SR B1 for decimal answer in range \([-0.904, -0.903]\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{\sin \gamma}{6} = \frac{\sin 60}{8}\) | M1* | Attempt use of correct sine rule. Must be correct sine rule, either way up (just need to substitute values in – no rearrangement needed). |
| Use \(\sin 60° = \frac{\sqrt{3}}{2}\) | M1d* | Could be implied e.g. \(\frac{6}{\sin \gamma} = \frac{16}{3}\sqrt{3}\) |
| \(\sin \gamma = \frac{3\sqrt{3}}{8}\) | A1 | Must be seen simplified to this, or \(0.375\sqrt{3}\) or \(\frac{9}{8\sqrt{3}}\), but isw if decimal equiv subsequently given. isw any attempt to find the angle. A0 if only ever seen as \(\sin^{-1}\frac{3\sqrt{3}}{8}\) |
| [3] |
## Question 7:
### Part (a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\cos\alpha = \frac{5}{\sqrt{29}}$ | M1 | Attempt $\cos\alpha$. Could draw triangle and use Pythagoras to find hypotenuse, or use trig identities. Must get as far as attempting $\cos\alpha$. Must be working in exact values. Must be using correct ratios for $\tan\alpha$ and $\cos\alpha$ |
| | A1 | Obtain $\frac{5}{\sqrt{29}}$. Allow any exact equiv, including rationalised surd or $\sqrt{\frac{25}{29}}$. isw if decimal equiv subsequently given. Answer only gets full credit. **SR B1** for exact answer following decimal working |
### Part (a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\cos\beta = -\frac{\sqrt{40}}{7}$ | M1 | Attempt $\cos\beta$. Could draw triangle and use Pythagoras to find adjacent, or use trig identities. Must get as far as attempting $\cos\beta$. Must be working in exact values. Must be using correct ratios for $\sin\alpha$ and $\cos\alpha$ |
| | A1 | Obtain $\frac{\sqrt{40}}{7}$. Allow any exact equiv, including $\sqrt{\frac{40}{49}}$. Allow $\pm\frac{\sqrt{40}}{7}$ (from using $\cos^2 x = 1 - \sin^2 x$). isw if decimal equiv subsequently given. Answer only gets M1A1 |
| | A1 FT | Obtain $-\frac{\sqrt{40}}{7}$, or $-$ve of their exact numerical value for $\cos\beta$. A1 FT can only be awarded following M1. isw if decimal equiv subsequently given. Answer only gets full credit. **SR B1** for $\frac{\sqrt{40}}{7}$ or equiv, following decimal working. **SR B2** for $-\frac{\sqrt{40}}{7}$ or equiv, following decimal working. **SR B1** for decimal answer in range $[-0.904, -0.903]$ |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{\sin \gamma}{6} = \frac{\sin 60}{8}$ | M1* | Attempt use of correct sine rule. Must be correct sine rule, either way up (just need to substitute values in – no rearrangement needed). |
| Use $\sin 60° = \frac{\sqrt{3}}{2}$ | M1d* | Could be implied e.g. $\frac{6}{\sin \gamma} = \frac{16}{3}\sqrt{3}$ |
| $\sin \gamma = \frac{3\sqrt{3}}{8}$ | A1 | Must be seen simplified to this, or $0.375\sqrt{3}$ or $\frac{9}{8\sqrt{3}}$, but isw if decimal equiv subsequently given. isw any attempt to find the angle. A0 if only ever seen as $\sin^{-1}\frac{3\sqrt{3}}{8}$ |
| **[3]** | | |
---7
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Given that $\alpha$ is the acute angle such that $\tan \alpha = \frac { 2 } { 5 }$, find the exact value of $\cos \alpha$.
\item Given that $\beta$ is the obtuse angle such that $\sin \beta = \frac { 3 } { 7 }$, find the exact value of $\cos \beta$.
\end{enumerate}\item \\
\includegraphics[max width=\textwidth, alt={}, center]{f25e2580-ba0b-42ce-bf86-63f2c2075223-3_316_662_955_700}
The diagram shows a triangle $A B C$ with $A C = 6 \mathrm {~cm} , B C = 8 \mathrm {~cm}$, angle $B A C = 60 ^ { \circ }$ and angle $A B C = \gamma$. Find the exact value of $\sin \gamma$, simplifying your answer.
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2012 Q7 [8]}}