How do I find the position of the minute hand in a broken clock after some time k?

Question

I have a clock which is broken.Initially, the end point of the minute hand is at coordinate (0,a).After 1 second, the its y coordinate becomes b.I have to find its y coordinate after some given time k.The catch is that I have to find the answer in terms of fraction rather than decimals in order to prevent loosing accuracy. Now, I was thinking of using cos-1(b/a) to find the angle of the minute hand in 1 second and then solving it for the time k. However, doing so would result in a decimal answer whereas I needed fraction for the above mentioned reason.