June 2024 Paper 1 Q9
9
(a) Show that, for small values of \(\theta\) measured in radians\[\cos 4\theta + 2\sin 3\theta - \tan 2\theta \approx A + B\theta + C\theta^2\]where \(A\), \(B\) and \(C\) are constants to be found. [3 marks]
(b) Use your answer to part (a) to find an approximation for\[\cos 0.28 + 2\sin 0.21 - \tan 0.14\]Give your answer to three decimal places. [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| Substitutes at least one small angle identity correctly into \(\cos 4\theta + 2\sin 3\theta - \tan 2\theta\) | M1 | 1.1a |
| Obtains a correct expression in terms of \(\theta\) ACF | A1 | 1.1b |
| Completes argument to obtain \(1 + 4\theta - 8\theta^2\) | R1 | 2.1 |
| (3) |
Typical solution
\[\cos 4\theta + 2\sin 3\theta - \tan 2\theta \approx 1 - \frac{(4\theta)^2}{2} + 2(3\theta) - (2\theta)\]\[= 1 + 4\theta - 8\theta^2\]| Scheme | Marks | AO |
|---|---|---|
| Substitutes \(\theta = 0.07\) into their \(1 + 4\theta - 8\theta^2\) | M1 | 3.1a |
| Obtains AWRT 1.241 CSO | A1 | 1.1b |
| (2) | ||
| (5 marks) |