Why is sin(20) = sin(160) = -sin(200) = -sin(340)?

The sine is positive in second quadrant   so sin 20 = sin (180-20) = sin 160.

The sine is negative in the third quadrant  so sin (180+20)  = -sin 20
so sin 20 = - sin 200

The sine is negative in fourth quadrant so sin 340 = -sin 20 and sin 20 = - sin 340.