| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Double angle with reciprocal functions |
| Difficulty | Standard +0.3 This question requires proving a standard reciprocal/double angle identity using known formulas (cosec 2θ = 1/sin 2θ, cot 2θ = cos 2θ/sin 2θ, combining to get cot θ), then solving a straightforward equation by substitution. While it involves multiple steps and reciprocal functions, the proof follows a predictable algebraic path and the equation solving is routine once the identity is established. Slightly easier than average due to the mechanical nature of the manipulation. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use \(\cot A = 1/\tan A\) or \(\cos A/\sin A\) and/or \(\text{cosec}\, A = 1/\sin A\) on at least two terms | M1 | |
| Use a correct double angle formula or the \(\sin(A - B)\) formula at least once | M1 | |
| Obtain given result | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Solve \(\cot\theta = 2\) for \(\theta\) and obtain answer \(26.6°\) | B1 | |
| Obtain answer \(206.6°\) and no others in the given range | B1\(\sqrt{}\) | Ignore answers outside given range. Treat answers in radians as misread |
## Question 3:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use $\cot A = 1/\tan A$ or $\cos A/\sin A$ and/or $\text{cosec}\, A = 1/\sin A$ on at least two terms | M1 | |
| Use a correct double angle formula or the $\sin(A - B)$ formula at least once | M1 | |
| Obtain given result | A1 | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Solve $\cot\theta = 2$ for $\theta$ and obtain answer $26.6°$ | B1 | |
| Obtain answer $206.6°$ and no others in the given range | B1$\sqrt{}$ | Ignore answers outside given range. Treat answers in radians as misread |
---3 (i) Prove the identity $\operatorname { cosec } 2 \theta + \cot 2 \theta \equiv \cot \theta$.\\
(ii) Hence solve the equation $\operatorname { cosec } 2 \theta + \cot 2 \theta = 2$, for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2009 Q3 [5]}}