Question

Find the principal solution of the angle in the equation: 3 tan² θ = 1.

Answer 1

this is the principal solution

Answer 2

To find the principal solution of the angle in the equation
$3 {tan}^{2} \theta = 1 \\ \\ {tan}^{2}\theta = \frac{1}{3} \\ \\ tan \: \theta = + - \frac{1}{ \sqrt{3} }$
there can be two cases

$tan \: \theta = \frac{1}{ \sqrt{3} } \\ \\ \theta = {tan}^{ - 1} ( \frac{1}{ \sqrt{3} } ) \\ \\ \theta = {tan}^{ - 1} (tan \frac{\pi}{6} ) \\ \\ \theta = \frac{\pi}{6}$
case 2:
$tan \: \theta = \frac{ - 1}{ \sqrt{3} } \\ \\ \theta = {tan}^{ - 1} ( \frac{ - 1}{ \sqrt{3} } ) \\ \\ \theta = {tan}^{ - 1} (tan \frac{ - \pi}{6} ) \\ \\ \theta = \frac{ - \pi}{6}$
Because principal value branch of tan is [-π/2,π/2]

so value can be $\theta$=π/6,-π/6