How do you evaluate ##arccos(cos(5pi/4))##?

##(3pi)/4##
##arccosx## can be thought of as an angle that measures between ##0## and ##pi## radians whose cosine is x.
(It can also be thought of as simply a number between ##0## and ##pi## whose cosine is ##x##.)
The restriction to angles between ##0## and ##pi## makes ##arccos## a function.
##arccos(cos((5pi)/4))## is an angle between ##0## and ##pi## whose cosine is the same as the cosine of ##(5pi)/4##.
The angle we want is ##(3pi)/4##
We know that ##cos((5pi)/4) = -sqrt2/2##
and the Quadrant II angle with cosine equal to ##-sqrt2/2##
is ##(3pi)/4##