Answer
$$\left\{ {{{78.04}^ \circ },\,\,\,\,{{281.96}^ \circ }} \right\}$$
Work Step by Step
$$\eqalign{
& 4{\cos ^2}\theta + 4\cos \theta = 1 \cr
& {\text{Subtract 1}} \cr
& 4{\cos ^2}\theta + 4\cos \theta - 1 = 0 \cr
& {\text{Let }}x = \cos \theta \cr
& 4{x^2} + 4x - 1 = 0 \cr
& {\text{Solve by using the quadratic formula}} \cr
& x = \frac{{ - 4 \pm \sqrt {{{\left( 4 \right)}^2} - 4\left( 4 \right)\left( { - 1} \right)} }}{{2\left( 4 \right)}} \cr
& x = \frac{{ - 4 \pm \sqrt {32} }}{8} \cr
& x = \frac{{ - 4 \pm 4\sqrt 2 }}{8} \cr
& {x_1} = \frac{{ - 1 + \sqrt 2 }}{2}{\text{ or }}{x_2} = \frac{{ - 1 - \sqrt 2 }}{2} \cr
& {\text{,then}} \cr
& \cos \theta = \frac{{ - 1 + \sqrt 2 }}{2}{\text{ or }}\cos \theta = \frac{{ - 1 - \sqrt 2 }}{2} \cr
& {\text{Solve each equation}} \cr
& \theta = {\cos ^{ - 1}}\left( {\frac{{ - 1 + \sqrt 2 }}{2}} \right)\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\theta = {\cos ^{ - 1}}\left( {\frac{{ - 1 - \sqrt 2 }}{2}} \right)\, \cr
& \theta \approx {78.04^ \circ }\,\,\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,\,\,\,\,\theta \approx 281.95 \cr
& {\text{The solution set is}} \cr
& \left\{ {{{78.04}^ \circ },\,\,\,\,{{281.96}^ \circ }} \right\} \cr} $$
