construct a right angled triangle in quadrants one, label the angle at the origin as x
Reflect the triangle in the origin, it will fall into the third quadrant.
notice both the run and rise of the first have now become negative, while the hypotenuse is always considered positive.
so in 1st: sinx = opposite/hypotenuse
in third quadrant:
sinx = -opposite/hyptenuse = -sinx
test with some value of x
e.g. let x = 23.89°
sin 23.89° = .404982013..
sin(23.89+180)
= sin 203.89° = -.404982013