Two-parameter Taxicab Trigonometric Identities

Annual Meeting in Mathematics 2023

Abstract

The metric on $\mathbb{R}^2$ defined by $d((x_1, x_2), (y_1, y_2)) = |x_1 - y_1| + |x_2 - y_2|$ is known as the $\ell^1$ or the taxicab metric. Delp and Filipski define and provide explicit formulas for sine and cosine functions for the taxicab space. Their version agrees with the right-triangle definition of the standard trigonometric functions. In particular, the sine (cosine) of an acute angle in a right triangle is equal to the ratio of the length of its opposite (adjacent) side and the length of the hypotenuse. These functions must have two parameters because a general rotation is not an isometry in the taxicab metric. We derive new identities for the taxicab sine and cosine functions. Specifically, we derive the Pythagorean, angle sum, double-angle, half-angle, and negative-angle identities. Additionally, we derive derivative identities for the taxicab tangent, secant, cotangent, and cosecant functions.
We find that the derivatives of these functions behave similarly to their Euclidean counterparts.