Trigonometric Ratios Using Algebraic Methods

Abstract

The main aim of this article is to start with an expository introduction to the trigonometric ratios and then proceed to the latest results in the field. Historically, the exact ratios were obtained using geometric constructions. The geometric methods have their own limitations arising from certain theorems. In view of the certain limitations of the geometric methods, we shall focus on the powerful techniques of equations in deriving the exact trigonometric ratios using surds. The cubic and higher-order equations naturally arise while deriving the exact trigonometric ratios. These equations are best expressed using the expansions of the cosines and sine of multiple angles using the Chebyshev polynomials of the first and second kind respectively. So, we briefly present the essential properties of the Chebyshev polynomials. The equations lead to the question of reduced polynomials. This question of the reduced polynomials is addressed using the Euler's totient function. So, we describe the techniques from theory of equations and reduced polynomials. The trigonometric ratios of certain rational angles (when measured in degrees) give rise to rational trigonometric ratios. We shall discuss these along with the related theorems. This is a frontline area of research connecting trigonometry and number theory. Results from number theory and theory of equations are presented wherever required.