Answer
$$\left\{ {{{24.3}^ \circ },{{114.295}^ \circ }\,\,} \right\}$$
Work Step by Step
$$\eqalign{
& 2\sin \theta = 1 - 2\cos \theta \cr
& {\text{Square each side}} \cr
& 4{\sin ^2}\theta = 1 - 4\cos \theta + 4{\cos ^2}\theta \cr
& 4\left( {1 - {{\cos }^2}\theta } \right) = 1 - 4\cos \theta + 4{\cos ^2}\theta \cr
& 4 - 4{\cos ^2}\theta = 1 - 4\cos \theta + 4{\cos ^2}\theta \cr
& 8{\cos ^2}\theta - 4\cos \theta - 3 = 0 \cr
& {\text{Let }}x = \cos \theta \cr
& {\text{8}}{x^2} - 4x - 3 = 0 \cr
& {\text{Solve by using the quadratic formula}} \cr
& x = \frac{{ - \left( { - 4} \right) \pm \sqrt {{{\left( { - 4} \right)}^2} - 4\left( 8 \right)\left( { - 3} \right)} }}{{2\left( 8 \right)}} \cr
& x = \frac{{4 \pm \sqrt {112} }}{{16}} \cr
& x = \frac{{1 \pm \sqrt 7 }}{4} \cr
& {\text{,then}} \cr
& \cos \theta = \frac{{1 + \sqrt 7 }}{4}{\text{ or }}\cos \theta = \frac{{1 - \sqrt 7 }}{4} \cr
& {\text{Solve each equation}} \cr
& \theta = {\cos ^{ - 1}}\left( {\frac{{1 + \sqrt 7 }}{4}} \right)\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\theta = {\cos ^{ - 1}}\left( {\frac{{1 - \sqrt 7 }}{4}} \right)\, \cr
& \theta \approx {24.3^ \circ }\,\,\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,\,\,\,\,\theta \approx {114.295^ \circ } \cr
& {\text{The solution set is}} \cr
& \theta \approx \left\{ {{{24.3}^ \circ },{{114.295}^ \circ }\,\,} \right\} \cr} $$
