| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Find exact trigonometric values |
| Difficulty | Challenging +1.2 This question requires systematic application of compound angle formulas and trigonometric identities across three connected parts. While part (i) uses standard addition formulas, part (ii) requires working backwards with the double angle formula (recognizing 10° = 2×5°), and part (iii) involves converting a trigonometric equation to tan form. The multi-step nature, need to manipulate expressions in terms of p throughout, and the non-routine approach in part (ii) make this moderately harder than average, though still within standard C3 technique application. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Identify \(\tan 55°\) as \(\tan(45°+10°)\) | B1 | or equiv |
| Use correct angle sum formula for \(\tan(A+B)\) | M1 | or equiv |
| Obtain \(\frac{1+p}{1-p}\) | A1 | 3 with \(\tan 45°\) replaced by 1 |
| (ii) Either: Attempt use of identity for \(\tan 2A\) | *M1 | linking 10° and 5° |
| Obtain \(p=\frac{2t}{1-t^2}\) | A1 | |
| Attempt solution for \(t\) of quadratic equation | M1 | dep *M |
| Obtain \(\frac{-1+\sqrt{1+p^2}}{p}\) | A1 | 4 or equiv; and no second expression |
| Or (1): Attempt expansion of \(\tan(60°-55°)\) | *M1 | |
| Obtain \(\frac{\sqrt{3}-\frac{1+p}{1-p}}{1+\sqrt{3}\frac{1+p}{1-p}}\) | A1∇ | follow their answer from (i) |
| Attempt simplification to remove denominators | M1 | dep *M |
| Obtain \(\frac{\sqrt{3}(1-p)-(1+p)}{1-p+\sqrt{3}(1+p)}\) | A1 | (4) or equiv |
| Or (2): State or imply \(\tan 15°=2-\sqrt{3}\) | B1 | |
| Attempt expansion of \(\tan(15°-10°)\) | M1 | with exact attempt for \(\tan 15°\) |
| Obtain \(\frac{2-\sqrt{3}-p}{1+p(2-\sqrt{3})}\) | A2 | (4) |
| Or (3): State or imply \(\tan 5°=\frac{\sqrt{3}-1}{\sqrt{3}+1}\) | B1 | or exact equiv |
| Attempt expansion of \(\tan(15°-10°)\) | M1 | with exact attempt for \(\tan 5°\) |
| Obtain \(\frac{\sqrt{3}-1-p\sqrt{3}-p}{\sqrt{3}+1+p\sqrt{3}-p}\) | A2 | (4) or equiv |
| Or (4): Attempt expansion of \(\tan(10°-5°)\) | *M1 | |
| Obtain \(t=\frac{p-t}{1+pt}\) | A1 | |
| Attempt solution for \(t\) of quadratic equation | M1 | dep *M |
| Obtain \(\frac{-2+\sqrt{4+4p^2}}{2p}\) | A1 | (4) or equiv; and no second expression |
| (iii) Attempt expansion of both sides | M1 | |
| Obtain \(3\sin\theta\cos 10°+3\cos\theta\sin 10°=7\cos\theta\cos 10°+7\sin\theta\sin 10°\) | A1 | or equiv |
| Attempt division throughout by \(\cos\theta\cos 10°\) | M1 | or by \(\cos\theta\) (or \(\cos 10°\)) only |
| Obtain \(3t+3p=7+7pt\) | A1 | or equiv |
| Obtain \(\frac{3p-7}{7p-3}\) | A1 | 5 or equiv |
(i) Identify $\tan 55°$ as $\tan(45°+10°)$ | B1 | or equiv
Use correct angle sum formula for $\tan(A+B)$ | M1 | or equiv
Obtain $\frac{1+p}{1-p}$ | A1 | 3 with $\tan 45°$ replaced by 1
(ii) Either: Attempt use of identity for $\tan 2A$ | *M1 | linking 10° and 5°
Obtain $p=\frac{2t}{1-t^2}$ | A1
Attempt solution for $t$ of quadratic equation | M1 | dep *M
Obtain $\frac{-1+\sqrt{1+p^2}}{p}$ | A1 | 4 or equiv; and no second expression
Or (1): Attempt expansion of $\tan(60°-55°)$ | *M1
Obtain $\frac{\sqrt{3}-\frac{1+p}{1-p}}{1+\sqrt{3}\frac{1+p}{1-p}}$ | A1∇ | follow their answer from (i)
Attempt simplification to remove denominators | M1 | dep *M
Obtain $\frac{\sqrt{3}(1-p)-(1+p)}{1-p+\sqrt{3}(1+p)}$ | A1 | (4) or equiv
Or (2): State or imply $\tan 15°=2-\sqrt{3}$ | B1
Attempt expansion of $\tan(15°-10°)$ | M1 | with exact attempt for $\tan 15°$
Obtain $\frac{2-\sqrt{3}-p}{1+p(2-\sqrt{3})}$ | A2 | (4)
Or (3): State or imply $\tan 5°=\frac{\sqrt{3}-1}{\sqrt{3}+1}$ | B1 | or exact equiv
Attempt expansion of $\tan(15°-10°)$ | M1 | with exact attempt for $\tan 5°$
Obtain $\frac{\sqrt{3}-1-p\sqrt{3}-p}{\sqrt{3}+1+p\sqrt{3}-p}$ | A2 | (4) or equiv
Or (4): Attempt expansion of $\tan(10°-5°)$ | *M1
Obtain $t=\frac{p-t}{1+pt}$ | A1
Attempt solution for $t$ of quadratic equation | M1 | dep *M
Obtain $\frac{-2+\sqrt{4+4p^2}}{2p}$ | A1 | (4) or equiv; and no second expression
(iii) Attempt expansion of both sides | M1
Obtain $3\sin\theta\cos 10°+3\cos\theta\sin 10°=7\cos\theta\cos 10°+7\sin\theta\sin 10°$ | A1 | or equiv
Attempt division throughout by $\cos\theta\cos 10°$ | M1 | or by $\cos\theta$ (or $\cos 10°$) only
Obtain $3t+3p=7+7pt$ | A1 | or equiv
Obtain $\frac{3p-7}{7p-3}$ | A1 | 5 or equiv
**Total: 12**The value of $\tan 10°$ is denoted by $p$. Find, in terms of $p$, the value of
\begin{enumerate}[label=(\roman*)]
\item $\tan 55°$, [3]
\item $\tan 5°$, [4]
\item $\tan \theta$, where $\theta$ satisfies the equation $3 \sin(\theta + 10°) = 7 \cos(\theta - 10°)$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2010 Q9 [12]}}