There are several q-analogs of the cosine function.
The two natural definitions of the -cosine defined by Koekoek and Swarttouw (1998) are given by
(1)
| |||
(2)
| |||
(3)
|
where and are q-exponential functions. The -cosine and -sine functions satisfy the relations
(4)
| |||
(5)
|
Another definition of the -cosine considered by Gosper (2001) is given by
(6)
| |||
(7)
| |||
(8)
| |||
(9)
|
where is a Jacobi theta function and is defined via
(10)
|
This is an even function of unit amplitude, period , and double and triple angle formulas and addition formulas which are analogous to ordinary sine and cosine. For example,
(11)
| |||
(12)
|
where is the q-sine, and is q-pi (Gosper 2001). The -cosine also satisfies
(13)
|