From Förberedande kurs i matematik 1

The angle \displaystyle \pi -v\text{ } makes the same angle with the negative \displaystyle x -axis as the angle \displaystyle v makes with the positive \displaystyle x -axis and this means that \displaystyle \pi -v\text{ } is the reflection of \displaystyle v in the y-axis.

Under such reflection, the angle's \displaystyle y -coordinate does not change (but the \displaystyle x -coordinate changes sign) and therefore \displaystyle \text{sin}\left( \pi -v \right)=\text{sin }v\text{ }=a.