q-Sine

There are several q-analogs of the sine function.

The two natural definitions of the -sine 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 -sine considered by Gosper (2001) is given by

			(6)
			(7)
			(8)

where is a Jacobi theta function and is defined via

(9)

This is an odd function of unit amplitude and period with double and triple angle formulas and addition formulas which are analogous to ordinary sine and cosine. For example,

(10)

where is the q-cosine and is q-pi (Gosper 2001).