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(\frac{5\pi}7 x\right) + \sqrt3 = 0`

Move the constant term to the other side:

`\tan\left(\frac{5\pi}7 x\right) = -\sqrt3`

Take the inverse tangent of both sides:

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

(where n is an integer)

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

Since tan(π/3) = √3, we get

`\frac{5\pi}7 x = -\frac\pi3 + n\pi`

and solving for x gives

`\boxed{x = -\frac7{15} + \frac{7n}5}`