Arc SOHCAHTOA method

[Albert Chan]

For arc hyperbolic half-angle formulas, we can use cosh(2x) = 2 cosh(x)^2 - 1, and SOHCAHTOA rules.

acoshq(x) = 2*acoshq((sqrt(x)+1)/2) = 2*asinhq((sqrt(x)-1)/2)

asinhq(x) = acoshq(1+x) = 2*asinhq((sqrt(1+x)-1)/2) = 2*asinhq(x/2/(sqrt(1+x)+1))

atanhq(x)
= acoshq(1/(1-x))
= 2*acoshq((1/sqrt(1-x)+1)/2)
= 2*acoshq((1+sqrt(1-x)) / (2*sqrt(1-x)))      // CAH, A = 1+sqrt(1-x), H = 2*sqrt(1-x)
= 2*atanhq((1-sqrt(1-x)) / (1+sqrt(1-x)))       // TOA, O = A-H = 1-sqrt(1-x)
= 2*atanhq(x/(1+sqrt(1-x))^2)

Code:

function asinhq(x)
    if x > 4e-4 then return 2 * asinhq(0.5*x/(sqrt(1+x)+1)) end
    return sqrt(x) * (1-x/6*(1-x*9/20/(1+x*25/42)))
end

function acoshq(x) return asinhq(x-1) end
function atanhq(x) return asinhq(x/(1-x)) end