sin(x)=-⅓, where π<x<3π/2 (3rd quadrant).
(sin(x+π)=sin(x)cos(π)+cos(x)sin(π)=-sin(x), where 0<x<π/2, 1st quadrant, which is diametrically opposite to 3rd quadrant.)
In 3rd quadrant, tan(x)>0.
In a right triangle, if the hypotenuse has length 3 and one leg has length 1, the other leg, by Pythagoras, has length √(3²-1²)=√8 or 2√2
If we move x into 1st quadrant (where sine and tangent are positive) by subtracting or adding π, sin(x)=⅓=opp/hyp, then tan(x)=opp/adj=1/√8=√8/8=2√2/8=√2/4, which is the same when x is in the 3rd quadrant.
tan(x)=tan(x+π)=√2/4=0.3536 approx.