q-Trigonometric Functions
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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.
Contributed by: Oleksandr Pavlyk (March 2011)
Open content licensed under CC BY-NC-SA
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Details
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.
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