A parameterization of trigonometric functions.
Version of Thursday 24 December 2015.
Dave Barber's e-mail and other pages.

This report presents a simple parameterization that subsumes the circular and hyperbolic functions of trigonometry. The sole parameter is a complex number, usually named p or q, whose magnitude is unity. (Although the formulas will give well-defined results for any nonzero parameter, such is not the intention here.) Meanwhile, the conventional argument to the function is often named z or w.

These functions should not be confused with the Jacobi elliptic functions, which generalize trigonometry in a much different direction.

The functions are named sinp, cosp, tanp, cotp, secp, and cscp. The development starts with two formulas:

The prefix tilde ~ is used to notate complex conjugation.

 table one
  general case p = i p = 1
definitions sinp (p, z) =
(exp (+p · z) − exp (−p · z)) ÷ (2 · p)
sin (z) =
(exp (+i · z) − exp (−i · z)) ÷ (2 · i)
sinh (z) =
(exp (+z) − exp (−z)) ÷ 2
cosp (p, z) =
(exp (+p · z) + exp (−p · z)) ÷ 2
cos (z) =
(exp (+i · z) + exp (−i · z)) ÷ 2
cosh (z) =
(exp (+z) + exp (−z)) ÷ 2
tanp (p, z) =
sinp (p, z) ÷ cosp (p, z)
tan (z) =
sin (z) ÷ cos (z)
tanh (z) =
sinh (z) ÷ cosh (z)
periods
k is a
real integer
sinp (p, z) =
sinp (p, z + 2 · k · π · i · ~p)
sin (z) =
sin (z + 2 · k · π)
sinh (z) =
sinh (z + 2 · k · π · i)
cosp (p, z) =
cosp (p, z + 2 · k · π · i · ~p)
cos (z) =
cos (z + 2 · k · π)
cosh (z) =
cosh (z + 2 · k · π · i)
tanp (p, z) =
tanp (p, z + k · π · i · ~p)
tan (z) =
tan (z + k · π)
tanh (z) =
tanh (z + k · π · i)
derivatives d sinp (p, z) / dz = cosp (p, z) d sin (z) / dz = cos (z) d sinh (z) / dz = cosh (z)
d cosp (p, z) / dz = p2 · sinp (p, z) d cos (z) / dz = − sin (z) d cosh (z) / dz = sinh (z)
d tanp (p, z) / dz = secp2 (p, z) d tan (z) / dz = sec2 (z) d tanh (z) / dz = sech2 (z)

The other three functions are defined as:


Many well-known trigonometric identities have a parameterized adaptation, as in table two. Plenty of other identities can be generated by means of elementary algebra.

table two
general case p = i p = 1
sinp (p, −z) = −sinp (p, z) sin (−z) = −sin (z) sinh (−z) = −sinh (z)
cosp (p, −z) = cosp (p, z) cos (−z) = cos (z) cosh (−z) = cosh (z)
2 · p2 · sinp (p, z) · sinp (p, w) =
cosp (p, z + w) − cosp (p, zw)
2 · sin (z) · sin (w) =
cos (zw) − cos (z + w)
2 · sinh (z) · sinh (w) =
cosh (z + w) − cosh (zw)
2 · cosp (p, z) · cosp (p, w) =
cosp (p, z + w) + cosp (p, zw)
2 · cos (z) · cos (w) =
cos (zw) + cos (z + w)
2 · cosh (z) · cosh (w) =
cosh (z + w) + cosh (zw)
2 · sinp (p, z) · cosp (p, w) =
sinp (p, z + w) + sinp (p, zw)
2 · sin (z) · cos (w) =
sin (zw) + sin (z + w)
2 · sinh (z) · cosh (w) =
sinh (z + w) + sinh (zw)
2 · p2 · sinp2 (p, z) = cosp (p, 2 · z) − 1 2 · sin2 (z) = 1 − cos (2 · z) 2 · sinh2 (z) = cosh (2 · z) − 1
2 · cosp2 (p, z) = cosp (p, 2 · z) + 1 2 · cos2 (z) = 1 + cos (2 · z) 2 · cosh2 (z) = cosh (2 · z) + 1
2 · sinp (p, z) · cosp (p, z) = sinp (p, 2 · z) 2 · sin (z) · cos (z) = sin (2 · z) 2 · sinh (z) · cosh (z) = sinh (2 · z)
cosp2 (p, z) − p2 · sinp2 (p, z) = 1 cos2 (z) + sin2 (z) = 1 cosh2 (z) − sinh2 (z) = 1
secp2 (p, z) + p2 · tanp2 (p, z) = 1 sec2 (z) − tan2 (z) = 1 sech2 (z) + tanh2 (z) = 1
cotp2 (p, z) − cscp2 (p, z) = p2 cot2 (z) − csc2 (z) = −1 coth2 (z) − csch2 (z) = 1
sinp (p, z + w) =
sinp (p, z) · cosp (p, w) + sinp (p, w) · cosp (p, z)
sin (z + w) =
sin (z) · cos (w) + sin (w) · cos (z)
sinh (z + w) =
sinh (z) · cosh (w) + sinh (w) · cosh (z)
cosp (p, z + w) =
cosp (p, z) · cosp (p, w) + p2 · sinp (p, w) · sinp (p, z)
cos (z + w) =
cos (z) · cos (w) − sin (w) · sin (z)
cosh (z + w) =
cosh (z) · cosh (w) + sinh (w) · sinh (z)
sinp (p, z) + sinp (p, w) =
2 · sinp (p, (z + w) ÷ 2) · cosp (p, (zw) ÷ 2)
sin (z) + sin (w) =
2 · sin ((z + w) ÷ 2) · cos ((zw) ÷ 2)
sinh (z) + sinh (w) =
2 · sinh ((z + w) ÷ 2) · cosh ((zw) ÷ 2)
sinp (p, z) − sinp (p, w) =
2 · cosp (p, (z + w) ÷ 2) · sinp (p, (zw) ÷ 2)
sin (z) − sin (w) =
2 · cos ((z + w) ÷ 2) · sin ((zw) ÷ 2)
sinh (z) − sinh (w) =
2 · cosh ((z + w) ÷ 2) · sinh ((zw) ÷ 2)
cosp (p, z) + cosp (p, w) =
2 · cosp (p, (z + w) ÷ 2) · cosp (p, (zw) ÷ 2)
cos (z) + cos (w) =
2 · cos ((z + w) ÷ 2) · cos ((zw) ÷ 2)
cosh (z) + cosh (w) =
2 · cosh ((z + w) ÷ 2) · cosh ((zw) ÷ 2)
cosp (p, z) − cosp (p, w) =
2 · p2 · sinp (p, (z + w) ÷ 2) · sinp (p, (zw) ÷ 2)
cos (z) − cos (w) =
− 2 · sin ((z + w) ÷ 2) · sin ((zw) ÷ 2)
cosh (z) − cosh (w) =
2 · sinh ((z + w) ÷ 2) · sinh ((zw) ÷ 2)


Because d4 sin (z) / dz4 equals sin (z) itself, this function is said to be cyclodifferentiable of order four. Similarly, sinh (z) is of order two. However, sinp (p, z) and cosp (p, z) are not always cyclodifferentiable. If the parameter is written in polar form p = (r, θ), then consideration of table three reveals that two necessary conditions for cyclodifferentiability are that r equal 1, and that π ÷ θ be a rational number. With suitable choice of θ, the order of cyclodifferentiability can be any desired even number. For instance, if θ = π ÷ k (k nonzero integer), then the order will be 2 · k. As another example, the order will be ten if θ = 0.2 · π, 0.6 · π, 1.4 · π, or 1.8 · π.

 table three
  p = i p = 1 general case
d 0 / dz 0 sin (z) sinh (z) sinp (p, z)
d 1 / dz 1 cos (z) cosh (z) cosp (p, z)
d 2 / dz 2 − sin (z) sinh (z) p2 · sinp (p, z) where p2 = (r2, 2 · θ)
d 3 / dz 3 − cos (z) cosh (z) p2 · cosp (p, z)
d 4 / dz 4 sin (z) sinh (z) p4 · sinp (p, z) where p4 = (r4, 4 · θ)


The parameter can be changed, as in table four.

 table four
general case p · sinp (p, q · z) = q sinp (q, p · z) cosp (p, q · z) = cosp (q, p · z)
special cases i · sin (z) = sinh (i · z) cos (z) = cosh (i · z)
i · sinh (z) = sin (i · z) cosh (z) = cos (i · z)
sinp (−p, z) = sinp (p, z) cosp (−p, z) = cosp (p, z)
sinp (i · p, z) = − i · sinp (p, i · z) cosp (i · p, z) = cosp (p, i · z)
sin (z) = − i · p · sinp (p, i · z ÷ p) cos (z) = cosp (p, i · z ÷ p)

The two identities in the bottom row of table four may be of aid in converting the "round numbers" of trigonometry such as sin (π ÷ 6) = 0.5, sin (π ÷ 4) = √0.5, and sin (π ÷ 3) = √0.75 to parameters other than i. The bottom row also yields a formula to convert from sinp to cosp. Recall that sin (z + π ÷ 2) = cos (z). Therefore:

i · p · sinp (p, i · (z + π ÷ 2) ÷ p) = cosp (p, i · z ÷ p)

or, with a change of variable, more nearly symmetrically:

p · sinp (p, i · (w + π ÷ 4) ÷ p) =
i · cosp (p, i · (w − π ÷ 4) ÷ p)

Setting p = 1 gives the hyperbolic case:

sinh (i · (z + π ÷ 2)) = i · cosh (i · z)


Table five contains several Maclaurin series:

table five
general case p = i p = 1
sinp (p, z) = p0 · z1 ÷ 1!
+ p2 · z3 ÷ 3!
+ p4 · z5 ÷ 5!
+ p6 · z7 ÷ 7!
+ p8 · z9 ÷ 9!
et cetera
sin (z) = z1 ÷ 1!
z3 ÷ 3!
+ z5 ÷ 5!
z7 ÷ 7!
+ z9 ÷ 9!
et cetera
sinh (z) = z1 ÷ 1!
+ z3 ÷ 3!
+ z5 ÷ 5!
+ z7 ÷ 7!
+ z9 ÷ 9!
et cetera
cosp (p, z) = p0 · z0 ÷ 0!
+ p2 · z2 ÷ 2!
+ p4 · z4 ÷ 4!
+ p6 · z6 ÷ 6!
+ p8 · z8 ÷ 8!
et cetera
cos (z) = z0 ÷ 0!
z2 ÷ 2!
+ z4 ÷ 4!
z6 ÷ 8!
+ z8 ÷ 9!
et cetera
cosh (z) = z0 ÷ 0!
+ z2 ÷ 2!
+ z4 ÷ 4!
+ z8 ÷ 6!
+ z9 ÷ 8!
et cetera