From Förberedande kurs i matematik 1

We can add and subtract multiples of \displaystyle \text{2}\pi to or from the argument of the sine function without changing its value. The angle \displaystyle \text{2}\pi corresponds to a whole turn in a unit circle and the sine function returns to the same value every time the angle changes by a complete revolution.

For example, if we can subtract sufficiently many \displaystyle \text{2}\pi s from \displaystyle \text{9}\pi , we will obtain a more manageable argument which lies between \displaystyle 0 and \displaystyle \text{2}\pi ,

\displaystyle \text{sin 9}\pi =\text{sin}\left( 9\pi -2\pi -2\pi -2\pi -2\pi \right)=\sin \pi

The line which makes an angle \displaystyle \pi with the positive part of the \displaystyle x -axis is the negative part of the \displaystyle x -axis and it cuts the unit circle at the point \displaystyle \left( -1 \right.,\left. 0 \right), which is why we can see from the \displaystyle y -coordinate that \displaystyle \text{sin 9}\pi =\text{sin }\pi =0.