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Trigonometric and Hyperbolic Sine and Cosine Reverse Theory
We use the reverse theory standard math function definition for the functions $H$ and $G$. In addition, we use the following definitions for $s$ and $c$ and the integer $\ell$
 Coefficients $s$ $c$ $\ell$ Trigonometric Case $\sin [ X(t) ]$ $\cos [ X(t) ]$ 1 Hyperbolic Case $\sinh [ X(t) ]$ $\cosh [ X(t) ]$ -1
We use the value $$z^{(j)} = ( s^{(j)} , c^{(j)} )$$ in the definition for $G$ and $H$. The forward mode formulas for the sine and cosine functions are $$\begin{array}{rcl} s^{(j)} & = & \frac{1 + \ell}{2} \sin ( x^{(0)} ) + \frac{1 - \ell}{2} \sinh ( x^{(0)} ) \\ c^{(j)} & = & \frac{1 + \ell}{2} \cos ( x^{(0)} ) + \frac{1 - \ell}{2} \cosh ( x^{(0)} ) \end{array}$$ for the case $j = 0$, and for $j > 0$, $$\begin{array}{rcl} s^{(j)} & = & \frac{1}{j} \sum_{k=1}^{j} k x^{(k)} c^{(j-k)} \\ c^{(j)} & = & \ell \frac{1}{j} \sum_{k=1}^{j} k x^{(k)} s^{(j-k)} \end{array}$$ If $j = 0$, we have the relation $$\begin{array}{rcl} \D{H}{ x^{(j)} } & = & \D{G}{ x^{(j)} } + \D{G}{ s^{(j)} } c^{(0)} + \ell \D{G}{ c^{(j)} } s^{(0)} \end{array}$$ If $j > 0$, then for $k = 1, \ldots , j-1$ $$\begin{array}{rcl} \D{H}{ x^{(k)} } & = & \D{G}{ x^{(k)} } + \D{G}{ s^{(j)} } \frac{1}{j} k c^{(j-k)} + \ell \D{G}{ c^{(j)} } \frac{1}{j} k s^{(j-k)} \\ \D{H}{ s^{(j-k)} } & = & \D{G}{ s^{(j-k)} } + \ell \D{G}{ c^{(j)} } k x^{(k)} \\ \D{H}{ c^{(j-k)} } & = & \D{G}{ c^{(j-k)} } + \D{G}{ s^{(j)} } k x^{(k)} \end{array}$$
Input File: omh/appendix/theory/sin_cos_reverse.omh