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4 Elementary FunctionsTrigonometric Functions

§4.14 Definitions and Periodicity

4.14.1 sinz =eiz-e-iz2i,
4.14.2 cosz =eiz+e-iz2,
4.14.3 cosz±isinz =e±iz,
4.14.4 tanz =sinzcosz,
4.14.5 cscz =1sinz,
4.14.6 secz =1cosz,
4.14.7 cotz =coszsinz=1tanz.

The functions sinz and cosz are entire. In the zeros of sinz are z=kπ, k; the zeros of cosz are z=(k+12)π, k. The functions tanz, cscz, secz, and cotz are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7).

For k

4.14.8 sin(z+2kπ) =sinz,
4.14.9 cos(z+2kπ) =cosz,
4.14.10 tan(z+kπ) =tanz.