Lagrange Remainder Max

When working with lagrange remainder theorem the M, or the maximum value of the n+1 derivative over a given interval, is always 1 for sine and cosine.  If the c value is 1.5pi and the desired x value at which we wish to find the error is 1.3pi, why do we use 1 instead of a more specific value?  In the specific problem I was working I wound up using a 4th or 5th derivative where I was dealing with cosine which meant 1.3pi would have a maximum value far less than one (since cos(1.5pi) is zero I knew 1.3pi would yield a max). 

It seems calculating M is rather arbitrary in this manner, could someone clarify.