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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/sin_cos_reverse.omh