(a) By the factor theorem, \[p(-3)=-162+9a+12-3=0\quad\Longrightarrow\quad a=17.\]

(b) Division by \(x+3\) gives quotient \(6x^2-x-1=(3x+1)(2x-1)\). Thus \[p(x)=(x+3)(3x+1)(2x-1).\]

(c) The roots give \(\operatorname{cosec}\theta=-3,-1/3,1/2\). The last two are impossible for a real angle since \(|\operatorname{cosec}\theta|\ge1\). Thus \(\sin\theta=-1/3\). If \(\phi=\sin^{-1}(1/3)\approx19.4712^\circ\), then \[\theta=180^\circ+\phi\approx\boxed{199.5^\circ},\qquad\theta=360^\circ-\phi\approx\boxed{340.5^\circ}.\]