Delta sine
Let $\mathbb{T}$ be a time scale and let $t_0 \in \mathbb{T}$ and let $p \colon \mathbb{T} \rightarrow \mathbb{R}$ be a regressive function. We define the trigonometric function $\sin_p \colon \mathbb{T} \rightarrow \mathbb{R}$ by $$\sin_p(t,t_0)=\dfrac{e_{ip}(t,t_0)-e_{-ip}(t,t_0)}{2i}$$
Properties
Theorem
The following formula holds: $$\sin_p^{\Delta}(t,t_0)=p(t)\cos_p(t,t_0),$$ where $\sin_p$ denotes the $\Delta$-$\sin_p$ function and $\cos_p$ denotes the $\Delta$-$\cos_p$ function.
Proof
Compute $$\begin{array}{ll} \sin^{\Delta}_p(t,t_0) &= \dfrac{1}{2i} \dfrac{\Delta}{\Delta t} \left( e_{ip}(t,t_0) - e_{-ip}(t,t_0) \right) \\ &= \dfrac{ip}{2i} ( e_{ip}(t,t_0) + e_{-ip}(t,t_0) ) \\ &= \dfrac{1}{2} (e_{ip}(t,t_0)+e_{-ip}(t,t_0)) \\ &= p\cos_p(t,t_0), \end{array}$$ as was to be shown. █
References
Examples
| $\mathbb{T}$ | |
| $\mathbb{R}$ | $\sin_p(t,s)= $ |
| $\mathbb{Z}$ | $\sin_p(t,s) = \dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) - \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2i}$ |
| $h\mathbb{Z}$ | $\sin_p(t,s) = $ |
| $\mathbb{Z}^2$ | $\sin_p(t,s) = $ |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $\sin_p(t,s) = $ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | $\sin_p(t,s) =$ |
| $\mathbb{H}$ | $\sin_p(t,s) = $ |
