| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Given sin/cos/tan, find other expressions |
| Difficulty | Moderate -0.3 Part (i) requires constructing a right triangle from tan A = 2 to find cosec A, which is straightforward but tests understanding of trig ratios. Part (ii) applies the tan addition formula, requiring algebraic manipulation but following a standard procedure. Both parts are routine applications of C3 content with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Either Attempt to find exact value of \(\sin A\) | M1 | using right-angled triangle or identity or … |
| Obtain \(\frac{1}{2}\sqrt{5}\) or \(\sqrt{\frac{1}{4}}\) or exact equiv | A1 | final \(\pm\frac{1}{2}\sqrt{5}\) is A0; correct answer only earns M1A1 |
| Or Attempt use of identity \(1 + \cot^2 A = \cosec^2 A\) | M1 | using \(\cot A = \frac{1}{2}\); allow sign error in attempt at identity |
| Obtain \(\frac{1}{2}\sqrt{5}\) or \(\sqrt{\frac{1}{4}}\) or exact equiv | A1 | final \(\pm\frac{1}{2}\sqrt{5}\) is A0; correct answer only earns M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply \(\frac{2 + \tan B}{1 - 2\tan B} = 3\) | B1 | |
| Attempt solution of equation of form \(\frac{\text{linear in } t}{\text{linear in } t} = 3\) | M1 | by sound process at least as far as \(k \tan B = c\) |
| Obtain \(\tan B = \frac{1}{7}\) | A1 | answer must be exact; ignore subsequent attempt to find angle \(B\) |
## (i)
**Either** Attempt to find exact value of $\sin A$ | M1 | using right-angled triangle or identity or …
Obtain $\frac{1}{2}\sqrt{5}$ or $\sqrt{\frac{1}{4}}$ or exact equiv | A1 | final $\pm\frac{1}{2}\sqrt{5}$ is A0; correct answer only earns M1A1
**Or** Attempt use of identity $1 + \cot^2 A = \cosec^2 A$ | M1 | using $\cot A = \frac{1}{2}$; allow sign error in attempt at identity
Obtain $\frac{1}{2}\sqrt{5}$ or $\sqrt{\frac{1}{4}}$ or exact equiv | A1 | final $\pm\frac{1}{2}\sqrt{5}$ is A0; correct answer only earns M1A1
**Total: [2]**
## (ii)
State or imply $\frac{2 + \tan B}{1 - 2\tan B} = 3$ | B1 |
Attempt solution of equation of form $\frac{\text{linear in } t}{\text{linear in } t} = 3$ | M1 | by sound process at least as far as $k \tan B = c$
Obtain $\tan B = \frac{1}{7}$ | A1 | answer must be exact; ignore subsequent attempt to find angle $B$
**Total: [3]**
---The acute angle $A$ is such that $\tan A = 2$.
\begin{enumerate}[label=(\roman*)]
\item Find the exact value of $\cosec A$. [2]
\item The angle $B$ is such that $\tan (A + B) = 3$. Using an appropriate identity, find the exact value of $\tan B$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2013 Q2 [5]}}