Question

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

Answer 1

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.

Answer 2

Sin(20°)
20° is in the first quadrant. Every trigonometry function is positive in quadrant 1.
Sin(20)  = 0.3420 [I just used a calculator to get this amount]

Sin(160°)
160 is in the second quadrant. The sine function positive in quad 2.
To find the reference number of any angle in quad 2, Subtract the base angle from 180° 

Sin(180 - 20) = sin(160) = 0.3420 [obtained from the calculator.

Sin(200)
200° is in the third quadrant. The sine function is negative in the third quadrant. The general formula is 180 + 20  degrees in this case.
Sin(200) = - 0.3420 

To change this to  a value = to sin(20) you require a minus sign in front
- sin(200) = -(-0.3420) = 0.3420

Sin(340)
340 degrees is in the 4th quadrant. The sine is minus in quadrant 4.
The general formula is 360 - Ф. In this case Ф = 20°

Sin(340) = - 0.3420 [I just put sin(340) into the calculator. 
- sin(340) = sin(20) = 0.3420