what is the maximum value of sin(314.159t)+sin(251.327t)? At what value of t the function attains its maximum?

from the graph, the first time is when t=0.3144

You can stick the t into the formula to find the maximum value :)

Oh, the second graph is the differential of the first.  The turning points are when f'(t)=0

Here's the graph.......https://www.desmos.com/calculator/juj2ievcfv

This graph has a period of about 5.73 degrees....it attains a max of 1.97 at .314 degrees and every ±5.73 degrees on either side of this.