Think about a line that goes from the center of the unit circle to it's perimeter and makes some angle with the x axis.

The sine of is the y coordinate of the intersection of that line and the unit circle because it's radius is 1.

Now what does do. It reflects this line about the y axis. The x coordinate of the intersection with the unit circle is multiplied by -1, and the y coordinate remains the same. Thus the sine of remains the same.