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\includegraphics[max width=\textwidth, alt={}, center]{798da17d-0af5-4aa6-b731-564642dc28d5-3_766_611_251_703} The diagram shows the curve with parametric equations $$x = a \sin \theta , \quad y = a \theta \cos \theta$$ where \(a\) is a positive constant and \(- \pi \leqslant \theta \leqslant \pi\). The curve meets the positive \(y\)-axis at \(A\) and the positive \(x\)-axis at \(B\).

1. Write down the value of \(\theta\) corresponding to the origin, and state the coordinates of \(A\) and \(B\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \theta \tan \theta\), and hence find the equation of the tangent to the curve at the origin.