On a Functional Equation of Trigonometric Type Soon-Mo Jung, Michael Th. Rassias and Cristinel Mortici Mathematics Section, College of Science and Technology, Hongik University, 339—701 Sejong, Republic of Korea E-mail: smjung@hongik.ac.kr Department of Mathematics, ETH—Zürich, Ramistrasse 101, 8092 Zürich, Switzerland & Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000, U.S.A E-mail: michail.rassias@math.ethz.ch, michailrassias@math.princeton.edu Valahia University of Târgovi¸ste, Bd. Unirii 18, 130082 Târgovi¸ste, Romania and Academy of Romanian Scientists, Splaiul Independen¸tei 54, 050094 Bucharest, Romania E-mail: cristinel.mortici@hotmail.com Abstract In this paper, we study the functional equation, f (x+y)f (x)f (y)= d sin x sin y. Some generalizations of the above functional equation are also considered. 1 Introduction In the fall of 1940, S. M. Ulam gave a wide-ranging talk before a Mathematical Col- loquium at the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the following question concerning the stability of homomorphisms (cf. [15]): Let G 1 be a group and let G 2 be a metric group with a metric d(·, ·). Given " > 0, does there exist a  > 0 such that if a function h : G 1 ! G 2 satisfies the inequality d(h(xy),h(x)h(y)) <  for all x, y 2 G 1 , then there is a homomorphism H : G 1 ! G 2 with d(h(x),H(x)) < " for all x 2 G 1 ? If the answer is a¢rmative, we say that the functional equation for homomorphisms is stable. D. H. Hyers was the first mathematician to present the result concerning the stabil- ity of functional equations. He brilliantly answered the question of Ulam for the case where G 1 and G 2 are assumed to be Banach spaces (see [5]). This result of Hyers is stated as follows: 0 2010 Mathematics Subject Classification: Primary 39B82; Secondary 26D05 0 Keywords: Hyers-Ulam stability, functional equation, trigonometric type functional equation, group, homomorphism, functional inequality, Banach spaces. 1