円と三角関数

概要

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The main purpose of this paper is to describe fundamental relations of generalized trigo-nometric functions. Let n be a natural number, and O-xy be the orthogonal coordi-nate in the Euclidean plane R^2. We consider an equation |x|^n + |y|^n= 1 which is the unit circle if n=2. We call its graph as C(n). Let A(O, 1) and P(x, y) be any point on C(n). Here, we define a measure to ∠AOP by a twice of area of a graphical figure AOP encircled by the curved segment AP and radi-uses OA, OP. If ∠AOP=u then functions c(n, u) and s(n, u) are defined as follows : x=c(n, u), y=s(n, u) , which correspond to cosine and sine functions. Here, let t(n, u)= s(n, u)/c(n, u), which correspond to tangent, and the following hold : Let π_n be the area of a closed domain D(n) encircled by C(n). Then π_n=4〓ydx which is π if n= 2. Here we have s(n, 2 π_n+u)=s(n, u), c(n, 2 π_n+u)=c(n, u), t(n, π_n+u)=t(n, u) which show the cyclicity of c, s and t.
敬愛大学・千葉敬愛短期大学の論文