Answer
$\{106.3^{o}, 149.6^{o},286.3^{o},329.6^{o}\}$
Work Step by Step
Substitute t=$\tan\theta$
$t^{2}+4t+2=0$... quadratic formula...
$t=\displaystyle \frac{-4\pm\sqrt{16-8}}{2}=\frac{-4\pm 2\sqrt{2}}{2}$
$=\displaystyle \frac{2(-2\pm\sqrt{2})}{2}=-2\pm\sqrt{2}$
$\tan\theta=-2\pm\sqrt{2}$
(tan is negative in quadrants II and IV)
Calculator:
$\tan^{-1}(-2+\sqrt{2})\approx$106.324949937$\approx 106.3^{o}$
and, $106.3^{o}+180^{o}=286.3^{o}$
Calculator:
$\tan^{-1}(-2-\sqrt{2})\approx$149.6388065957$\approx 149.6^{o}$
and, $149.6^{o}+180^{o}=329.6^{o}$
Solution set = $\{106.3^{o}, 149.6^{o},286.3^{o},329.6^{o}\}$
