Round and Round and Round

This problem follows on from Going Round in Circles.
Watch the film below.





Imagine the dot starts at the point $(1,0)$, turns through $60$ degrees anticlockwise and then stops.

I was wondering, if the point hadn't stopped, and instead carried on until it had turned through $30$ $000$ degrees, might it have finished the same distance above the horizontal axis?

I took out my calculator and typed $30$ $000$ $\div$ $360$
The answer on the screen was $83.333333$.

How can I use this to help me solve my problem?

There are ideas for follow-up problems in the Notes .