| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Given one function find others |
| Difficulty | Moderate -0.8 This is a straightforward application of Pythagorean identity to find cos θ, then using basic definitions (cot θ = cos θ/sin θ) and the double angle formula (cos 2θ = 1 - 2sin²θ). All values work out to exact fractions with no problem-solving required, making it easier than average but not trivial since it requires multiple standard steps. |
| Spec | 1.05g Exact trigonometric values: for standard angles1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt complete method for finding \(\cot\theta\) | M1 | rt-angled triangle, identities, calculator... |
| Obtain \(\frac{5}{12}\) | A1 | 2 or exact equiv |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt relevant identity for \(\cos 2\theta\) | M1 | \(\pm 2\cos^2\theta \pm 1\) or \(\pm 1 \pm 2\sin^2\theta\) or \(\pm(\cos^2\theta - \sin^2\theta)\) |
| State correct identity with correct value(s) substituted | A1 | |
| Obtain \(-\frac{119}{169}\) | A1 | 3 correct answer only earns 3/3 |
# Question 2:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt complete method for finding $\cot\theta$ | M1 | rt-angled triangle, identities, calculator... |
| Obtain $\frac{5}{12}$ | A1 | **2** or exact equiv |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt relevant identity for $\cos 2\theta$ | M1 | $\pm 2\cos^2\theta \pm 1$ or $\pm 1 \pm 2\sin^2\theta$ or $\pm(\cos^2\theta - \sin^2\theta)$ |
| State correct identity with correct value(s) substituted | A1 | |
| Obtain $-\frac{119}{169}$ | A1 | **3** correct answer only earns 3/3 |
---2 It is given that $\theta$ is the acute angle such that $\sin \theta = \frac { 12 } { 13 }$. Find the exact value of\\
(i) $\cot \theta$,\\
(ii) $\cos 2 \theta$.
\hfill \mbox{\textit{OCR C3 2007 Q2 [5]}}