The -analogs of trigonometric functions are built out of -exponentials in the same way that the classical trigonometric functions are built out of the classical exponential function. The existence of two flavors of -exponents makes for two kinds of -trigonometric functions: and for and and for . These triples are plotted for various on the interval . You can see how the -analogs approach their classical counterparts as tends to 1.
The -sine and -cosine functions can also be defined by the following series:
The -sine and -cosine functions have -analogs of the defining algebraic identity satisfied by their classical counterparts: . Additionally they satisfy .
V. Kac and P. Cheung, Quantum Calculus, New York: Springer, 2001.