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

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

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.