| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | March |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Solve with multiple compound angles |
| Difficulty | Standard +0.3 This question requires systematic application of compound angle formulae to expand sin(θ+45°) and cos(θ+60°), then collecting terms to form an equation in sin θ and cos θ that yields tan θ. While it involves multiple steps and algebraic manipulation, it follows a standard procedure without requiring novel insight. Part (ii) is routine once tan θ is found. Slightly easier than average due to the methodical nature of the approach. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.05g Exact trigonometric values: for standard angles1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use trig formulae and obtain an equation in \(\sin\theta\) and \(\cos\theta\) | M1 | |
| Obtain a correct equation in any form | A1 | |
| Substitute exact trig ratios and obtain an expression for \(\tan\theta\) | M1 | |
| Obtain answer \(\tan\theta = \frac{2\sqrt{2}-1}{1-\sqrt{6}}\), or equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State answer, e.g. \(\theta = 128.4°\) | B1 | |
| State second answer, e.g. \(\theta = 308.4°\) | B1ft |
## Question 3(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use trig formulae and obtain an equation in $\sin\theta$ and $\cos\theta$ | M1 | |
| Obtain a correct equation in any form | A1 | |
| Substitute exact trig ratios and obtain an expression for $\tan\theta$ | M1 | |
| Obtain answer $\tan\theta = \frac{2\sqrt{2}-1}{1-\sqrt{6}}$, or equivalent | A1 | |
## Question 3(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State answer, e.g. $\theta = 128.4°$ | B1 | |
| State second answer, e.g. $\theta = 308.4°$ | B1ft | |3 (i) Given that $\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta$, find the exact value of $\tan \theta$ in a form involving surds. You need not simplify your answer.\\
(ii) Hence solve the equation $\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.\\
\hfill \mbox{\textit{CAIE P3 2019 Q3 [6]}}