What is the period of 4sin(6(x-pi/3))+5?

Period, T => Time so that the cycle is the same. An example would be sin(0) = sin(2*pi), in fact adding any multiple of 2*pi is the same for sin(x)

The period and cycle is to do with the trig function so you can ignore the 4 and 5 from 4sin(6(x-pi/3))+5

So consider: sin( 6(x-pi/3) )

Consider now: sin( 6(x-pi/3) ) = sin( 6([x+T]-pi/3) ) where x is the same variable and T is the time period.

6(x-pi/3) = 6x -2pi
6([x+T]-pi/3) = 6x + 6T - 2pi

we know that adding 2*pi doesn't change it so this implies:
6T = 2pi
=> T = pi/3

Try for any x, it should work!

4 sin [ 6x - 2π ] + 5

Period = 2π / 6 = π / 3

34.