Standard +0.8 This is a quotient rule differentiation problem requiring trigonometric manipulation to reach a specific target form. While the quotient rule application is standard, simplifying the result to match the given form (involving sin 2θ) requires recognizing double angle identities and algebraic manipulation of trigonometric expressions. The 5-mark allocation and 'show that' format indicate substantial working is needed beyond routine differentiation, placing it moderately above average difficulty.
Given that
$$y = \frac{3\sin \theta}{2\sin \theta + 2\cos \theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$
show that
$$\frac{dy}{d\theta} = \frac{A}{1 + \sin 2\theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$
where \(A\) is a rational constant to be found.
[5]
Given that
$$y = \frac{3\sin \theta}{2\sin \theta + 2\cos \theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$
show that
$$\frac{dy}{d\theta} = \frac{A}{1 + \sin 2\theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$
where $A$ is a rational constant to be found.
[5]
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q4 [5]}}