In polar coordinates, we have
[tex]x=r\cos\theta\implies\cos\theta=\dfrac xr[/tex]
[tex]y=r\sin\theta\implies\sin\theta=\dfrac yr[/tex]
From this we can write
[tex]\tan\theta=\dfrac{\sin\theta}{\cos\theta}=\dfrac{\frac yr}{\frac xr}=\dfrac yx[/tex]
If [tex]\theta=\dfrac{4\pi}3[/tex], then [tex]\tan\theta=\sqrt[/tex], so we get
[tex]\dfrac yx=\sqrt3\implies y=\sqrt3\,x[/tex]
