# Find all values of Θ in the interval [0°, 360°)? tan²Θ - 2 tan Θ -1 = 0

Jun 4, 2018

$\arctan \left(1 + \sqrt{2}\right) , \arctan \left(1 + \sqrt{2}\right) + \pi , \pi + \arctan \left(1 - \sqrt{2}\right) , 2 \pi + \arctan \left(1 + \sqrt{2}\right)$

#### Explanation:

Substituting
$t = \tan \left(\theta\right)$
so we get

${t}^{2} - 2 t - 1 = 0$
${t}_{1 , 2} = 1 \pm \sqrt{2}$

Jun 4, 2018

67^@50; 157^@51; 247^@50: 337^@51

#### Explanation:

Solve this quadratic equation for tan t:
${\tan}^{2} t - 2 \tan t - 1 = 0$
$D = {d}^{2} = {b}^{2} - 4 a c = 4 + 4 = 8$ --> $d = \pm 2 \sqrt{2}$
The 2 real roots are:
$\tan t = - \frac{b}{2 a} \pm \frac{d}{2 a} = \frac{2}{2} \pm 2 \frac{\sqrt{2}}{2} = 1 \pm \sqrt{2}$
a. $\tan t = 1 + \sqrt{2} = 2.414$
Calculator and unit circle give 2 solutions for t:
$t = {67}^{\circ} 50$ and $t = {67}^{\circ} 50 + 180 = {247}^{\circ} 50$
b. $\tan t = 1 - \sqrt{2} = - 0.414$
Calculator and unit circle give:
$t = - {22}^{\circ} 49$ and $t = - 22.49 + 180 = {157}^{\circ} 51$
Note: t = - 22.49 is co-terminal to t = 337.51
The answers for (0, 360) are:
67^@50; 157^@51; 247^@50; 337^@51