From Förberedande kurs i matematik 1

Because the sine value for an angle is equal to the angle's \displaystyle y -coordinate on a unit circle, two angles have the same sine value only if they have the same \displaystyle y-coordinate. Therefore, if we draw in the angle \displaystyle {\pi }/{7}\; on a unit circle, we see that the only angle between \displaystyle {\pi }/{2}\; and \displaystyle \pi which has the same sine value lies in the second quadrant, where the line \displaystyle {y=\sin \pi }/{7}\; cuts the unit circle.

FIGURE1 FIGURE2 the line \displaystyle {y=\sin \pi }/{7}\; the line \displaystyle {y=\sin \pi }/{7}\;

Because of symmetry, we have that this angle is the reflection of the angle \displaystyle {\pi }/{7}\; in the \displaystyle y-axis, i.e.

\displaystyle v=\pi -{\pi }/{7}\;={6\pi }/{7}\;.