## 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.