Trigonometric and Hyperbolic Sine and Cosine Reverse Theory
Keywords: trigonometric hyperbolic sine cosine reverse theory sin sinh cos cosh

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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 ℓ

Coefficients |  | s |  | c |  | ℓ
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 s ( j ) = 1 + ℓ 2 sin ( x ( 0 ) ) + 1 - ℓ 2 sinh ( x ( 0 ) ) c ( j ) = 1 + ℓ 2 cos ( x ( 0 ) ) + 1 - ℓ 2 cosh ( x ( 0 ) ) for the case j = 0 , and for j > 0 , s ( j ) = 1 j ∑ k = 1 j k x ( k ) c ( j - k ) c ( j ) = ℓ 1 j ∑ k = 1 j k x ( k ) s ( j - k ) If j = 0 , we have the relation ∂ H ∂ x ( j ) = ∂ G ∂ x ( j ) + ∂ G ∂ s ( j ) c ( 0 ) + ℓ ∂ G ∂ c ( j ) s ( 0 ) If j > 0 , then for k = 1 , … , j -1 ∂ H ∂ x ( k ) = ∂ G ∂ x ( k ) + ∂ G ∂ s ( j ) 1 j k c ( j - k ) + ℓ ∂ G ∂ c ( j ) 1 j k s ( j - k ) ∂ H ∂ s ( j - k ) = ∂ G ∂ s ( j - k ) + ℓ ∂ G ∂ c ( j ) k x ( k ) ∂ H ∂ c ( j - k ) = ∂ G ∂ c ( j - k ) + ∂ G ∂ s ( j ) k x ( k )

Input File: omh/theory/sin_cos_reverse.omh