maximum displacement

[Guest]

A simply supported beam is subjected to two vibrations along its length, emanating from two machines at opposite ends of the beam. The displacement caused by the vibrations can be modelled by the following equations.
x1=5sin(100π+π/2)
x2=6.5sin(100π-π/3)
when both machines are swithced on, how long does it take for each machine to produce its maximum displacement?

[HallsofIvy]

If both machines are switched on the two machines will produce a wave which will, at some time, reach a maximum. That is completely different from asking when each machine will produce a maximum! Also, there is no variable, "t" in your functions were they supposed to be
x1=5sin(100t+π/2)
x2=6.5sin(100t-π/3)?

If so, then the first is maximum when 100t+ π/2= nπ/2, for integer n, so 100t= (n-1)π/2 so t= (n-1)π/200. The second is maximum when 100t- π/3= nπ/2 so 100t= nπ/2- π/3= (3n- 2)π/6 so t= (3n- 2)π/600.

But I suspect your real question is about the maximum of the sum of these two functions, not the maximum of each. You will need to use sum and difference trig formulas.