Eulers Formula and Its Applications on Trigonometry and Calculus

Eulers formula, the combination of trigonometric and exponential forms, plays a crucial role in simplifying expressions and solving integrals across mathematics, physics, and engineering. Although its theoretical gravity has been well acknowledged, its practical applications in simplifications and integrations are neglected in the early stages. This paper specializes in reintroducing the applicability of Eulers formula in solving problems, including trigonometric functions, definite integrals, and simple indefinite integrals. This paper aims to demonstrate corresponding examples with various difficulties and rationalize Eulers formula in different forms to solve them clearly. The method of this paper focuses on transforming trigonometric expressions into exponential form to reduce the computational complexity. Hence, this method shows the symmetry in expressions, which helps derive the Fourier series by applying complex exponentials. The final goal of the paper is to reinforce problem-solving skills through the further analysis of Eulers formula and lay the foundation for the upcoming nonlinear partial differential equations.