In the triangle \(ABC\), \(AB=2\), \(BC=\sqrt{3}\), \(AC=x\) and angle \(ABC=\frac{\pi}{6}+\theta\) radians (see diagram).
Show that \(x^2=7-6\cos\theta+2\sqrt{3}\sin\theta\).[2]
Show that \(x\frac{d^2x}{d\theta^2}+\left(\frac{dx}{d\theta}\right)^2=3\cos\theta-\sqrt{3}\sin\theta\).[2]
Use the result from part (b) to find the Maclaurin expansion for \(x\), up to and including the term in \(\theta^3\). Give the coefficients as exact values in their simplest form.[3]
Using the result from part (c), deduce the approximate value of \(x\) when angle \(ABC\) is \(35^\circ\), giving your answer correct to 6 decimal places.[2]