Question

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

Answer 1

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`