Answer
$$\left\{ {{{38.354}^ \circ },{{104.775}^ \circ },{{218.354}^ \circ },{{284.775}^ \circ }} \right\}$$
Work Step by Step
$$\eqalign{
& 3{\cot ^2}\theta - 3\cot \theta - 1 = 0 \cr
& {\text{Let }}x = \cot \theta \cr
& 3{x^2} - 3x - 1 = 0 \cr
& {\text{Solve by using the quadratic formula}} \cr
& x = \frac{{3 \pm \sqrt {{{\left( { - 3} \right)}^2} - 4\left( 3 \right)\left( { - 1} \right)} }}{{2\left( 3 \right)}} \cr
& x = \frac{{3 \pm \sqrt {21} }}{6} \cr
& {x_1} = \frac{{3 + \sqrt {21} }}{6}{\text{ or }}{x_2} = \frac{{3 - \sqrt {21} }}{6} \cr
& {\text{,then}} \cr
& \cot \theta = \frac{{3 + \sqrt {21} }}{6}{\text{ or}}\cot \theta = \frac{{3 - \sqrt {21} }}{6} \cr
& {\text{Definition of inverse cotangent}} \cr
& \theta = {\cot ^{ - 1}}\left( {\frac{{3 + \sqrt {21} }}{6}} \right),\,\,\,\,\,\,\,\theta = {\cot ^{ - 1}}\left( {\frac{{3 + \sqrt {21} }}{6}} \right) + 180 \cr
& \theta \approx {38.354^ \circ },\,\,\,\,\,\,\,\theta \approx {218.354^ \circ } \cr
& \cr
& \theta = {\cot ^{ - 1}}\left( {\frac{{3 - \sqrt {21} }}{6}} \right) + 180,\,\,\,\,\,\,\,\theta = {\cot ^{ - 1}}\left( {\frac{{3 - \sqrt {21} }}{6}} \right) + 360 \cr
& \theta \approx {104.775^ \circ }\,\,\,\,\,\,\,or\,\,\,\,\,\,\theta \approx {284.775^ \circ } \cr
& {\text{The solution set is}} \cr
& \left\{ {{{38.354}^ \circ },{{104.775}^ \circ },{{218.354}^ \circ },{{284.775}^ \circ }} \right\} \cr} $$
