Any angle of the form ##(11pi)/4 + 2n pi## with ##n in ZZ## is coterminal with ##(11pi)/4##
The coterminal angle of ##(11pi)/4## in ##[0, 2pi)## is ##(3pi)/4##
Coterminal angles are angles which are equal modulo ##2 pi##
That is: ##alpha## and ##beta## are coterminal angles if ##alpha – beta = 2n pi## for some integer ##n##.
For example, ##(11pi)/4## and ##(3pi)/4## are coterminal, since:
##(11pi)/4 – (3pi)/4 = (8pi)/4 = 2pi = 2n pi## with ##n = 1##
Every angle has a unique coterminal angle in the range ##[0, 2 pi)##
If ##theta >= 0## then ##theta – 2 floor(theta/(2pi)) pi in [0, 2 pi)##
If ##theta < 0## then ##theta + 2 ceil((-theta)/(2pi)) pi in [0, 2 pi)##
Coterminality is an example of an equivalence relation
If we use the symbol ##~## to mean “is coterminal with” then we find:
Reflexive: For all ##alpha##: ##alpha ~ alpha##
Commutative: For all ##alpha, beta##: ##alpha ~ beta <=> beta ~ alpha##
Transitive: For all ##alpha##, ##beta##, ##gamma##: if ##alpha ~ beta## and ##beta ~ gamma## then ##alpha ~ gamma##
