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##