A Geometric View of Complex Trigonometric Functions

University. When not teaching or doing graph theory, he spends time with his wife Micol and young daughters Adriana and Sage. Students of trigonometry and calculus learn to interpret sin α and cos α as the y and x coordinates of a point located by radian measure α on the unit circle. In this geometric setting, the unit circle is a natural protractor that measures angles on the x-y plane, and sine and cosine are functions that resolve angles into vertical and horizontal components. In more advanced courses, analytic reasoning is used to extend these functions to the complex domain: cos(α + βi) = cos α cosh β − i sin α sinh β (1) sin(α + βi) = sin α cosh β + i cos α sinh β. (2) (See for instance [3, p. 78].) Visual thinkers may accept this with a degree of reluc-tance. They may wonder how—or if—these equations could be understood visually. Can they be seen as arising from a complex version of the unit circle? Can an angle with imaginary or complex radian measure be visualized? This article describes how this can be done. We define the sine and cosine of a complex number through purely geometric means, and get (1) and (2) as a consequence. The complex unit circle While trigonometry’s familiar unit circle lies in R2, our goal suggests that we investi-gate the unit circle as a set of points in complex two-dimensional space