In this article, we considered application of complex analysis to series and generalized Chebyshev polynomials. DeMoivre’s theorem and the general Binomial expansion were used to establish alternative method for generating Chebyshev polynomials. The usual triple recursion formula derived from trigonometric identities was used to generate Chebyshev polynomials which yields same results with the alternative method. Some illustrative examples were given to show ability of the method. The polynomials obtained can be used to demonstrate approximation where maximum component of error is minimized in numerical analysis.