Two-parameter Taxicab Trigonometric Identities

Abstract

The metric on R2 defined by d((x1, x2), (y1, y2)) = |x1 − y1 | + |x2 − y2 | is known as the ℓ1 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. functions behave similarly to their Euclidean counterparts. We find that the derivatives of these.