| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Given one function find others |
| Difficulty | Standard +0.3 This question requires knowledge of reciprocal trig functions and the tan addition formula, but follows a standard pattern: use cot θ = 4 to find tan θ = 1/4, apply tan(A+B) formula for part (i), and use Pythagorean identity to find cosec θ for part (ii). The calculations are straightforward with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05l Double angle formulae: and compound angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply \(\tan\theta=\frac{1}{4}\) | B1 | Note that both parts are to be answered without calculator so sufficient detail is needed |
| State or imply use of \(\frac{\tan\theta+1}{1-\tan\theta}\) | B1 | |
| Obtain \(\frac{5}{3}\) or \(1\frac{2}{3}\) or \(\frac{20}{12}\) or exact equiv | B1 | But not unsimplified equiv (such as \(\frac{5}{4}/\frac{3}{4}\)) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt use of correct relevant identity or of right-angled triangle | M1 | Such as \(\cosec^2\theta=1+\cot^2\theta\), or \(\cosec\theta=\frac{1}{\sin\theta}\) with attempt at \(\sin\theta\), or use of Pythagoras' theorem in right-angled triangle |
| Obtain \(\sqrt{17}\) | A1 | Final answer \(\pm\sqrt{17}\) earns A0 |
| [2] |
# Question 2(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $\tan\theta=\frac{1}{4}$ | B1 | Note that both parts are to be answered without calculator so sufficient detail is needed |
| State or imply use of $\frac{\tan\theta+1}{1-\tan\theta}$ | B1 | |
| Obtain $\frac{5}{3}$ or $1\frac{2}{3}$ or $\frac{20}{12}$ or exact equiv | B1 | But not unsimplified equiv (such as $\frac{5}{4}/\frac{3}{4}$) |
| **[3]** | | |
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# Question 2(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt use of correct relevant identity or of right-angled triangle | M1 | Such as $\cosec^2\theta=1+\cot^2\theta$, or $\cosec\theta=\frac{1}{\sin\theta}$ with attempt at $\sin\theta$, or use of Pythagoras' theorem in right-angled triangle |
| Obtain $\sqrt{17}$ | A1 | Final answer $\pm\sqrt{17}$ earns A0 |
| **[2]** | | |
---2 It is given that $\theta$ is the acute angle such that $\cot \theta = 4$. Without using a calculator, find the exact value of\\
(i) $\tan \left( \theta + 45 ^ { \circ } \right)$,\\
(ii) $\operatorname { cosec } \theta$.
\hfill \mbox{\textit{OCR C3 2015 Q2 [5]}}