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How do you find the coterminal angle for #(11pi) / 4#?

1 Answer
Jul 15, 2015

Answer:

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#

Explanation:

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#