Conjecturing Trigonometric Formulas via a Computer Algebra System

Tilak de Alwis
Department of Mathematics
Southeastern Louisiana University
Hammond, LA 70402, USA

Abstract. In this paper, we will show how to use the computer algebra system (CAS) Mathematica to conjecture certain formulas in Trigonometry. One of our goals is to discover general closed-form formulas for powers of sine or cosine functions using sines or cosines of multiple angles. For smaller values of the powers, such calculations can be performed by hand using standard formulas in trigonometry such as Addition Formulas and Double Angle Formulas. However, as the powers increase, the calculations become tedious, and a CAS such as Mathematica becomes a valuable tool. As a specific example, suppose we want to find a general closed-form formula for (cos(x))^n in terms of sines or cosines of multiple angles, where n is a positive integer. The analysis would require two separate cases, one for odd n, and the other for even n. For instance, for even integer values n=2,4,6,... one can use the "TrigReduce" command of Mathematica to obtain expressions for (cos(x))^n, in terms of cosines of multiple angles such as cos(2x), cos(4x), cos(6x), etc. Then one can use the coefficients of these different expressions corresponding to n=2,4,6,... to form an array similar to the Pascal Triangle. We can use the capabilities of Mathematica together with the method of successive differences to conjecture a general formula to describe the coefficients in this array. In this way, one can obtain a general formula for (cos(x))^n where n is even. Using similar methods, one can conjecture a variety of other formulas in trigonometry with the help of Mathematica. Some of these formulas may or may not be found in standard mathematical handbooks. The mathematical induction can be used to prove some of these conjectures.