Trigonometric Functions and Existence of Circle Ratio

Let m, k be natural numbers. One can check that RHK(m,k) is real. Let m, k be natural numbers. Then RHK(m,k) is a real number. In this article we present several logical schemes. The scheme ExComplex CASE deals with a binary functor F yielding an element of C, and states that: For every k there exists s1 such that for every n holds if n ≤ k, then s1(n) = F (k,n) and if n > k, then s1(n) = 0C for all values of the parameter. The scheme ExReal CASE deals with a binary functor F yielding a real number, and states that: For every k there exists r1 such that for every n holds if n ≤ k, then r1(n) = F (k,n) and if n > k, then r1(n) = 0