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Proofs of a Trigonometric Inequality

Cameron Butler & Wei-Kai Lai



A trigonometric inequality is introduced and proved using Hölder’s inequality, Cauchy-Schwarz inequality, and Chebyshev’s order inequality.



To preserve the accuracy of the equations, this article is available as a [pdf].

About the Authors


Cameron ButlerCameron Butler

I am Cameron Butler, an undergrad here at USC. I worked along-side with Dr. Wei-Kai Lai to create: Proofs of a Trigonometric Inequality, a math article written to show proof of the inequality (See article for proof) . My motivation for this proof came from numerous reasons, beginning with my personal desire for solving difficult tasks. I also crave the chance to use my problem solving skills, especially when it requires abstract thinking. When the opportunity to work with Dr. Lai arises during a SC STEPs to STEM Summer Internship Program over the summer of 2013, I quickly accepted. Since working on this project, I have been enlightened to the importance of teamwork and research. It also encouraged me to pursue a career that uses math and problem solving skills. Before participating in this proof, I started my first year of college at USC Salkehatchie Walterboro campus,which was conveniently located in my hometown. There, I received the Dean’s Honors List for the 2012-2013 academic year, and met Dr. Wei-Kai Lai. I now study as a sophomore at USC Columbia. I major in Civil Engineering and wish on working toward a Master degree in the field. Though, in light of opening my own engineering firm, I want to expand my education into business and receive a MBA as well. I would like to thank Dr. Wei-Kai Lai. He has been an inspiring, genuine, and caring mentor. Not enough words could express the value I have found in his ability to share his own knowledge and wisdom. It has truly been a pleasure to work on such an intriguing proof with the help and support of such a magnificent mind.

Wei-Kai Lai

Dr. Wei-Kai Lai is an Assistant Professor of Mathematics at USC Salkehatchie. He holds a B.A. from National Taiwan University, a M.S. from National Taiwan Normal University and a Ph.D. from the University of Mississippi.



[1] G. H. Hardy, J. E. Littlewood, G. Polya, Inequality, 2nd edition, Cambridge Uni­versity Press, Cambridge, 1973. 

[2] A. Lohwater, Introduction to Inequalities, Online e-book in PDF format (, 1982.

[3] J. M. Steel, The Cauchy-Schwarz Master Class : An Introduction to the Art of Mathematical Inequalities, Cambridge University Press, New York, 2004.

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