Question

\tan\frac{5\pi}{7} x+\sqrt{3}=0 tan 7 5π ​ x+ 3 ​ =0

Answer 1

I'm guessing the equation is supposed to read

$\tan\left(\dfrac{5\pi}7 x\right) + \sqrt3 = 0$

Move the constant term to the other side:

$\tan\left(\dfrac{5\pi}7 x\right) = -\sqrt3$

Take the inverse tangent of both sides:

$\tan^{-1}\left(\tan\left(\dfrac{5\pi}7 x\right)\right) = \tan^{-1}\left(-\sqrt3\right) + n\pi$

(where n is an integer)

$\dfrac{5\pi}7 x = - \tan^{-1}\left(\sqrt3\right) + n\pi$

Since tan(π/3) = √3, we get

$\dfrac{5\pi}7 x = -\dfrac\pi3 + n\pi$

and solving for x gives

$\boxed{x = -\dfrac7{15} + \dfrac{7n}5}$