It is given that $y = \ln \left[ \sin \left( x + \frac{\pi}{4} \right) \right]$, where \(-\frac{\p

Question

It is given that $y = \ln \left[ \sin \left( x + \frac{\pi}{4} \right) \right]$, where $-\frac{\pi}{4} < x < \frac{3\pi}{4}$. Show that $\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)^2 + 1 = 0$. Hence find the first four non-zero terms of the Maclaurin expansion of $y$, leaving your answer in exact form.[6] Verify the result obtained in part (a) using standard series from the List of Formulae (MF27).[5]

Accepted Answer

(a) $y = \ln\left( \frac{1}{\sqrt{2}} \right) + x - x^2 + \frac{2}{3}x^3 + \dots$

2025 HCI P2 Q2

2025 HCI P2 Q2