We can look at this graph y=cosx+5 as a translated graph. This is the same as the standard graph y=cosx being shifted 5 units upward.
The function y=cosx has the minimum y value of -1, meaning the smallest number the graph can have on the y axis is -1. Adding 5 to -1, we have 4.
Therefore, the minimum y value of y=cosx+5 is y=4.