Functional Equations Characterizing the Sine and Cosine Functions Over a Convex Polygon∗

Home , Functional equation

Applied Mathematics E-Notes, 19(2019), 277-287 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ ∼

Functional Equations Characterizing The Sine And Cosine Functions Over A Convex Polygon∗

Kanet Ponpetch†, Vichian Laohakosol‡, Sukrawan Mavecha§

Received 22 May 2018

Abstract Two functional equations exhibiting functions with a constant sum over points lying in a hyperplane are solved. These functional equations are employed to characterize the sine and cosine functions.

1 Introduction

In [2], Benz solved the functional equation

f(x)f(y)f(z) = f(x) + f(y) + f(z)(x, y, z (0, π/2)) ∈ with x + y + z = π obtaining a general solution f : (0, π/2) (0, ) of the form → ∞ π f(x) = tan kx + (1 k) (x (0, π/2)) , − 3 ∈ with an arbitrary constant k [ 1/2, 1]. This conﬁrmed a question posed by Davison [1]. Such a result can be regarded∈ − as a functional equation characterizing the trigonometric tangent function over a triangle. In [3] and [4], Hengrawit et al extended this result by solving a generalized functional equation over a convex polygon, which can also be regarded as characterizing the tangent function. Analyzing the work in [3] and [4], in a recent paper [5], the following two functional equations, with a constant pa- rameter sum over a hyperplane, which can be used to characterize the sine and cosine functions, respectively, are solved:

n 1 − 2M+1 n b 2 c M f (xik ) ( 1) g (xj) = 0, (1) −  g (xik )   M=0 1 i1<

277 278 Functional Equations Characterizing

n n 2M n b 2 c M f (xik ) n g (xj) + ( 1) g (xj) = ( 1) , (2) −  g (xik )   − j=1 M=1 1 i1<

2 Preliminary Results

We start with a theorem and a lemma taken from [5] which are needed.

THEOREM 1 ([5, Theorem 1.2]). Let n be an integer 3, and let I1 := (a, b),I2 := ≥ (c, d) be two non-empty open intervals. Then the function φ : I1 I2 satisﬁes the constant sum functional equation →

φ(xi) = U1, i=1 X where U1 is a real constant, subject to the hyperplane condition

xi = U2, i=1 X where U2 is a real constant, if and only if, U U φ(x) = k x 2 + 1 − n n for some ﬁxed k lying in the range

nc U1 nd U1 nc U1 nd U1 max − , − < k < min − , − . nb U2 na U2 na U2 nb U2 − − − −

LEMMA 1 ([5, Lemma 4.2]). Let n be an integer 2. If x1, . . . , xn (0, π), then ≥ ∈

sin(x1 + + xn) n 1 ··· − 2M+1 n b 2 c M sin xik = ( 1) cos xj , −  cos xik   M=0 1 i1<

cos(x1 + + xn) ··· n n 2M n b 2 c M sin xik = cos xj + ( 1) cos xj . −  cos xik   j=1 M=1 1 i1<

We now prove our main theorem.

THEOREM 2. Let n be an integer 3 and let I1 := (a, b),I2 := (c, d) be two non-empty open intervals. ≥

A) Let (N1,...,Nn,R1,...,Rn) be a function of 2n variables and let t I1,T R. F ∈ ∈ Suppose that S, C : I1 I2 are two bijections satisfying →

(S(x1),...,S(xn),C(x1),...,C(xn)) = S (x1 + + xn) . (3) F ···

Suppose also that the functions U, V : I1 I2 satisfy → 1 1 S− U = C− V (4) ◦ ◦ (U(α1),...,U(αn),V (α1),...,V (αn)) = S(t) (5) F subject to the condition

α1 + + αn = T (α1, . . . , αn I1). ··· ∈ Then T t T t U(x) = S k x + ,V (x) = C k x + (x I1), − n n − n n ∈ for some ﬁxed k R lying in the range ∈ nc t nd t nc t nd t max − , − < k < min − , − . nb T na T na T nb T − − − −

B) Let (Y1,...,Yn 1,Z1,...,Zn 1) be a function of 2(n 1) variables and let H − − − W, w R. Suppose that S, C : I1 I2 are two bijections satisfying ∈ →

(S(x1),...,S(xn 1),C(x1),...,C(xn 1)) = C (x1 + + xn 1) . (6) H − − ··· −

Suppose also that the functions U, V : I1 I2 satisfy → 1 1 S− U = C− V (7) ◦ ◦ 1 (U(α1),...,U(αn 1),V (α1),...,V (αn 1)) = C w S− U (αn) , (8) H − − − ◦ 280 Functional Equations Characterizing

for some α1, . . . , αn I1 such that ∈ 1 w S− U (αn) I1 and α1 + + αn = W. − ◦ ∈ ··· Then W w W w U(x) = S ` x + ,V (x) = C ` x + (x I1), − n n − n n ∈ for some ﬁxed ` R lying in the range ∈ nc w nd w nc w nd w max − , − < ` < min − , − . nb W na W na W nb W − − − −

PROOF. A) By (4), there exists φ : I1 I1 such that →

U(x) = S (φ (x)) and V (x) = C (φ (x)) (x I1). ∈ Thus, (5) becomes

(S(φ(α1)),...,S(φ(αn)),C(φ(α1)),...,C(φ(αn))) = S(t). F By (3), we have S (φ(α1) + + φ(αn)) = S(t). ··· Then φ(α1) + + φ(αn) = t subject to α1 + + αn = T. ··· ··· By Theorem 1 [5, Theorem 1.2], we have

T t φ(x) = k x + (x I1), − n n ∈ where k R lying in the range ∈ nc t nd t nc t nd t max − , − < k < min − , − . nb T na T na T nb T − − − − B) By (7), there exists φ : I1 I1 such that →

U(x) = S (φ (x)) and V (x) = C (φ (x)) (x I1). ∈ Thus, (8) becomes

(S(φ(α1)),...,S(φ(αn 1)),C(φ(α1)),...,C(φ(αn 1))) = C (w φ (αn)) . H − − − By (6), we have C (φ(α1) + + φ(αn 1)) = C (w φ (αn)) . ··· − − Then φ(α1) + + φ(αn) = w ··· Ponpetch et al. 281 subject to α1 + + αn = W. ··· By Theorem 1 [5, Theorem 1.2], we have W w φ(x) = ` x + (x I1), − n n ∈ where ` R lying in the range ∈ nc w nd w nc w nd w max − , − < ` < min − , − . nb W na W na W nb W − − − − The proof is complete.

EXAMPLE 1. Let n 3,I1 := (0, π/2),I2 := (0, 1), u I1. The trigonometric ≥ ∈ sine and cosine functions sin, cos : I1 I2 are two bijections satisfying → (sin x1,..., sin xn, cos x1,..., cos xn) Fn 1 − 2M+1 n b 2 c M sin xik ( 1) cos xj ≡ −  cos xik   M=0 1 i1<

1 1 (n 2)π sin− U = cos− V and α1 + + αn = − (α1, . . . , αn I1). ◦ ◦ ··· 2 ∈ Then (n 2)π u (n 2)π u U(x) = sin m x − + ,V (x) = cos m x − + (x I1), − 2n n − 2n n ∈ for some ﬁxed m R lying in the range ∈ u 2(n u) 2u (n u) max − , − − < m < min , − . π (n 2)π (n 2)π π − −

EXAMPLE 2. Let n 3,I1 := (0, π/2),I2 := (0, 1), v I1. The trigonometric ≥ ∈ sine and cosine functions sin, cos : I1 I2 are two bijections satisfying → (cos x1,..., cos xn, sin x1,..., sin xn) G n n 2M n b 2 c M sin xik cos xj + ( 1) cos xj ≡ −  cos xik   j=1 M=1 1 i1<

Suppose the functions U, V : I1 I2 satisfy →

(V (α1),...,V (αn),U(α1),...,U(αn)) = cos(v) G subject to the two conditions

1 1 sin− U = cos− V, ◦ ◦ (n 2)π α1 + + αn = − (α1, . . . , αn I1). ··· 2 ∈ Then (n 2)π v (n 2)π v U(x) = sin c x − + ,V (x) = cos c x − + (x I1), − 2n n − 2n n ∈ for some ﬁxed c R lying in the range ∈ v 2(n v) 2v (n v) max − , − − < c < min , − . π (n 2)π (n 2)π π − −

EXAMPLE 3. Let n 3,I1 := (0, π/2),I2 := (0, 1), s R. The trigonometric ≥ ∈ sine and cosine functions sin, cos : I1 I2 are two bijections satisfying →

(sin x1,..., sin xn 1, cos x1,..., cos xn 1) − − H n 1 n 1 −2 2M n 1 − b c − M sin xik cos xj + ( 1) cos xj ≡ −  cos xik   j=1 M=1 1 i1<

Suppose the functions U, V : I1 I2 satisfy → 1 1 sin− U = cos− V, ◦ ◦ 1 (U(α1),...,U(αn 1),V (α1),...,V (αn 1)) = cos s sin− U (αn) , H − − − ◦ for some α1, . . . , αn I1 such that ∈

1 (n 2)π s sin− U (αn) I1 and α1 + + αn = − . − ◦ ∈ ··· 2 Then (n 2)π s (n 2)π s U(x) = sin y x − + ,V (x) = cos y x − + (x I1), − 2n n − 2n n ∈ for some ﬁxed y R lying in the range ∈ s 2(n s) 2s (n s) max − , − − < y < min , − . π (n 2)π (n 2)π π − − Ponpetch et al. 283

EXAMPLE 4. Let n 3,I1 := (0, π/2),I2 := (0, 1), r R. The trigonometric ≥ ∈ sine and cosine functions sin, cos : I1 I2 are two bijections satisfying → (cos x1,..., cos xn 1, sin x1,..., sin xn 1) − − En 2 −2 2M+1 n 1 b c − M sin xik ( 1) cos xj ≡ −  cos xik   M=0 1 i1<

In this section, we investigate the results of Theorem 2 when n = 1, 2. We illustrate by examples that the (implicit) uniqueness of solution is lost in the case n = 2, while the existence of solution is lost in the case n = 1.

PROPOSITION 1. Let (N1,N2,R1,R2) be a function of 4 variables, let I1 := F (a, b),I2 := (c, d) be two non-empty open intervals and let t I1,T R. Suppose that ∈ ∈ S, C : I1 I2 are two bijections satisfying → (S(x1),S(x2),C(x1),C(x2)) = S (x1 + x2) . (9) F Suppose the functions U, V : I1 I2 satisfy → (U(α1),U(α2),V (α1),V (α2)) = S(t) (10) F subject to the two conditions

1 1 S− U = C− V (11) ◦ ◦ α1 + α2 = T (α1, α2 I1). (12) ∈ 284 Functional Equations Characterizing

Then one pair of solutions to (10) is given by

T t T t U(x) = S A x + ,V (x) = C A x + (x I1), − 2 2 − 2 2 ∈ where A : J (c t/2, d t/2) is an odd function on J := (a T/2, b T/2). Moreover, another→ − pair of− solutions to (10) is given by − −

T t T t U(x) = S k x + ,V (x) = C k x + (x I1), − 2 2 − 2 2 ∈ for some ﬁxed k R lying in the range ∈ 2c t 2d t 2c t 2d t max − , − < k < min − , − . 2b T 2a T 2a T 2b T − − − −

PROOF. By (11), there exists φ : I1 I1 such that →

U(x) = S (φ (x)) ,V (x) = C (φ (x)) (x I1). ∈ Thus, (10) becomes

(S(φ(α1)),S(φ(α2)),C(φ(α1)),C(φ(α2))) = S(t). F By (9), we have S (φ(α1) + φ(α2)) = S(t). Then, φ(α1) + φ(α2) = t subject to α1 + α2 = T. (13) From the condition (12), we have 2a < T < 2b. Let J := (a T/2, b T/2) and deﬁne − − ψ : J I2 by → T ψ(y) = φ y + (y J). 2 ∈ Thus, the relation (13) becomes

ψ(y1) + ψ(y2) = t subject to y1 + y2 = 0 (yi J). ∈ Let A : J (c t/2, d t/2) be an odd function. We claim that ψ(y) = A(y) + t/2 is a solution→ of (13).− Since−

t t ψ(y) + ψ( y) = A(y) + + A( y) + = t. − 2 − 2 From the deﬁnition of ψ, we get

T t φ(x) = A x + (x I1). − 2 2 ∈ Ponpetch et al. 285

We show that another pair solutions to (10) is given

T t T t U(x) = S k x + ,V (x) = C k x + (x I1). − 2 2 − 2 2 ∈ Since

(U(α1),U(α2),V (α1),V (α2)) F t t t t = S kα¯1 + ,S kα¯2 + ,C kα¯1 + ,C kα¯2 + F 2 2 2 2 where α¯i = αi T/2 (i = 1, 2), using (9) and (12), we have − t t t t S kα¯1 + ,S kα¯2 + ,C kα¯1 + ,C kα¯2 + F 2 2 2 2 t t = S kα¯ + + kα¯ + = S(t). 1 2 2 2

EXAMPLE 5. Let I1 := (0, π/2),I2 := (0, 1); p I1,T R. The trigonometric ∈ ∈ sine and cosine functions sin, cos : I1 I2 are bijections satisfying →

(sin(x1), sin(x2), cos(x1), cos(x2)) := sin(x1) cos(x2)+cos(x1) sin(x2) = sin (x1 + x2) . F

Suppose the functions U, V : I1 I2 satisfy →

(U(α1),U(α2),V (α1),V (α2)) = sin(p) (14) F subject to the two conditions

1 1 sin− U = cos− V, ◦ ◦

α1 + α2 = T (α1, α2 I1). ∈ Then one pair of solutions to (14) is given

T p T p U(x) = sin A x + ,V (x) = cos A x + (x I1), − 2 2 − 2 2 ∈ where A : J ( p/2, 1 p/2) is an odd function on J := ( T/2, π/2 T/2). Moreover, another pair→ of solutions− − to (14) is given by − −

T p T p U(x) = sin m x + ,V (x) = cos m x + (x I1), − 2 2 − 2 2 ∈ for some ﬁxed m R lying in the range ∈ p (2 p) p 2 p max − , − − < m < min , − . π T T T π T − − 286 Functional Equations Characterizing

EXAMPLE 6. Let I1 := (0, π/2),I2 := (0, 1); q I1,T R. The trigonometric ∈ ∈ sine and cosine functions sin, cos : I1 I2 are bijections satisfying →

(cos(x1), cos(x2), sin(x1), sin(x2)) := cos(x1) cos(x2) sin(x1) sin(x2) = cos (x1 + x2) . F −

Suppose the functions U, V : I1 I2 satisfy →

(V (α1),V (α2),U(α1),U(α2)) = cos(q) (15) F subject to the two conditions

1 1 sin− U = cos− V and α1 + α2 = T (α1, α2 I1). ◦ ◦ ∈ Then one pair of solutions to (15) is given by

T q T q U(x) = sin A x + ,V (x) = cos A x + (x I1), − 2 2 − 2 2 ∈ where A : J ( q/2, 1 q/2) is an odd function on J := ( T/2, π/2 T/2). Moreover, another pair→ of solutions− − to (15) is given by − −

T q T q U(x) = sin m x + ,V (x) = cos m x + (x I1), − 2 2 − 2 2 ∈ for some ﬁxed m R lying in the range ∈ q (2 q) q 2 q max − , − − < m < min , − . π T T T π T − −

Regarding the case n = 1, we make the following two remarks.

I) If n = 1, then the equation (3) in Theorem 2 becomes

(S(x1),C(x1)) = S (x1) , F showing C is constant function, contradicting the fact that C is a bijection. II) If n = 1, then there is no valid equation (6) in Theorem 2, while if n = 2, then the equation (6) in Theorem 2 becomes

(S(x1),C(x1)) = C (x1) , H yielding C to be a constant function, which again contradicts its being bijective.

References

[1] T. Davison, Report of Meeting: The Fortieth International Symposium on Func- tional Equations, Aequationes Math., 65(2003), 292. Ponpetch et al. 287

[2] W. Benz, The functional equation f(x)f(y)f(z) = f(x) + f(y) + f(z), Aequationes Math., 68(2004), 117—120. [3] C. Hengkrawit, V. Laohakosol and K. Ponpetch, Functional equations characterizing the tangent function over a convex polygon, Aequationes Math., 88(2014), 201—210.

[4] C. Hengkrawit and K. Ponpetch, Functional equations characterizing the tangent function over a convex polygon II, Thai Journal of Science and Technology, 5(2016), 113—118. [5] K. Ponpetch, S. Mavecha and V. Laohakosol, Functions with constant sums over a hyperplane and applications, Publ. Inst. Math., 105(2019), 65—80. [6] J. Acze´l and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989. [7] Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, Heidelberg, 2009.