| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| 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 straightforward manipulation of trigonometric identities to show tan θ = 6. Parts (ii)(a) and (ii)(b) are direct applications of standard formulae (tan subtraction and double angle) with no conceptual challenges. This is slightly easier than average as it's a routine multi-step question testing standard techniques with clear signposting. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(\sec\theta = \frac{1}{\cos\theta}\) | B1 | |
| Attempt to express in terms of \(\tan\theta\) only | M1 | |
| Obtain \(\tan^2\theta = 36\) and hence \(\tan\theta = 6\) | A1 | AG; necessary detail needed (but no need to justify exclusion of \(\tan\theta = -6\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute 6 in attempt at formula | M1 | Of form \(\frac{\tan\theta \pm \tan 45°}{1 \mp \tan\theta\tan 45°}\) with different signs in numerator and denominator; any apparent use of angle 80.5… means M0 |
| Obtain \(\frac{5}{7}\) | A1 | Or exact equiv; answer only: 0/2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute 6 in attempt at formula | M1 | Of form \(\frac{\tan\theta + \tan\theta}{1 \pm \tan\theta\tan\theta}\); any apparent use of angle 80.5… means M0 |
| Obtain \(-\frac{12}{35}\) | A1 | Or exact equiv; allow \(\frac{12}{-35}\); answer only: 0/2 |
# Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $\sec\theta = \frac{1}{\cos\theta}$ | B1 | |
| Attempt to express in terms of $\tan\theta$ only | M1 | |
| Obtain $\tan^2\theta = 36$ and hence $\tan\theta = 6$ | A1 | AG; necessary detail needed (but no need to justify exclusion of $\tan\theta = -6$) |
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# Question 3(ii)(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute 6 in attempt at formula | M1 | Of form $\frac{\tan\theta \pm \tan 45°}{1 \mp \tan\theta\tan 45°}$ with different signs in numerator and denominator; any apparent use of angle 80.5… means M0 |
| Obtain $\frac{5}{7}$ | A1 | Or exact equiv; answer only: 0/2 |
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# Question 3(ii)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute 6 in attempt at formula | M1 | Of form $\frac{\tan\theta + \tan\theta}{1 \pm \tan\theta\tan\theta}$; any apparent use of angle 80.5… means M0 |
| Obtain $-\frac{12}{35}$ | A1 | Or exact equiv; allow $\frac{12}{-35}$; answer only: 0/2 |
---3 It is given that $\theta$ is the acute angle such that $\sec \theta \sin \theta = 36 \cot \theta$.\\
(i) Show that $\tan \theta = 6$.\\
(ii) Hence, using an appropriate formula in each case, find the exact value of
\begin{enumerate}[label=(\alph*)]
\item $\tan \left( \theta - 45 ^ { \circ } \right)$,
\item $\quad \tan 2 \theta$.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2012 Q3 [7]}}