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Kevin's Corner

Angles and Trigonometry > Trigonometry

We now turn to the definition of the trigonometric functions sine and cosine in taxicab geometry. If you are not familiar with taxicab angles, please review the Angles page prior to continuing.

In the first attempt at defining taxicab trig functions (see [3]), the functions were defined in terms of the Euclidean trig functions. It is worthwhile to establish such relationships, but we will begin with more native definitions that do not rely on Euclidean trigonometry.

DEFINITION: The point of intersection of the terminal side of a taxicab angle θ in standard position with the taxicab unit circle is the point (cos θ, sin θ).

It is important to note that the taxicab sine and cosine values of a taxicab angle do not agree with the Euclidean sine and cosine values of the corresponding Euclidean angle. For example, the angle 1 t-radian, which is equivalent to πt/4 t-radians (see the Circles page for the definition of π in taxicab geometry), has equal taxicab sine and cosine values just as with the "equivalent" Euclidean angle. But, now the sine and cosine values are 0.5 instead of sqrt(2)/2. The range of the cosine and sine functions remains [-1, 1], but the period of these fundamental functions is now 8 = 2πt. In addition, the values of cosine and sine vary (piecewise) linearly with θ:

The structure of the graphs of these functions is similar to that of the Euclidean graphs of sine and cosine. Note that the smooth transition from increasing to decreasing at the extrema has been replaced with a corner. This is the same effect seen when comparing Euclidean circles with taxicab circles. Therefore, the taxicab sine and cosine functions are not everywhere differentiable.

FIGURE 1: Graphs of the taxicab sine and cosine functions.

With a bit of effort, closed forms can be invented for these piecewise functions. On the interval [2k - 1, 2k), the closed-form expressions of sine and cosine are

Tangent

If we define tangent as in Euclidean geometry as the ratio of sine and cosine, we again have a piecewise-defined function. The graph of the taxicab tangent function has a very similar feel to the Euclidean tangent function with asymptotes at odd πt/2 values and a period that is half that of sine and cosine:

FIGURE 2: Graph of the taxicab tangent function.

The taxicab tangent function is differentiable at its "inflection" points with a slope of 0.5.

Secant

Secant is defined as the reciprocal of cosine. Using the reciprocals of the various piecewise branches of the cosine function, we can construct a graph of secant (shown with cosine below).

FIGURE 3: Graphs of the taxicab cosine and secant functions.

The taxicab secant graph exhibits the same characteristics as the Euclidean secant graph: a period equal to that of cosine; asymptotes where cosine is zero; and, a minimum or maximum where the cosine has a maximum or minimum. As with tangent, the secant graph is curved. But, unlike the Euclidean secant function, the taxicab secant function is not differentiable at its extrema (the branches actually meet at a sharp point with the slope of the tangent lines from the left and right being 0.5 and -0.5). This is to be expected since this is where cosine is not differentiable.

Cosecant

Cosecant is defined as the reciprocal of sine. Using the reciprocals of the various piecewise branches of the sine function, we can construct a graph of cosecant (shown with sine below).

FIGURE 4: Graphs of the taxicab sine and cosecant functions.

The taxicab cosecant graph exhibits the same characteristics as the taxicab secant graph in terms of period, differentiability at the extrema, and asymptotes where sine is equal to zero.

Cotangent

Cotangent is simply the reciprocal of tangent. The differences between the taxicab tangent and cotangent graphs will be similar to those seen in Euclidean geometry.

FIGURE 5: Graph of the taxicab cotangent function.

Relationship to triangles

If the taxicab angle is in standard position and a right triangle is formed by dropping a vertical line to the x-axis, then the natural Euclidean trigonometric definitions apply in taxicab geometry. This is true even if the taxicab circle on which the angle is defined is not of radius 1 but of radius r.

If the angle is not in standard position, then a right triangle that it belongs to will not necessarily satisfy these properties. This is because lines are not invariant under rotation in taxicab geometry - the orientation of the angle affects the orientation of the legs of the triangle which affects the taxicab length of those legs. This is well illustrated by the triangles on the Euclid's Axioms page where the angles of size 1 t-radian have varying "sine" and "cosine" values of 0.5 and 1 when using the traditional Euclidean right triangle relationships.

References
[1] Thompson, Kevin and Tevian Dray. Taxicab Angles and Trigonometry, The Pi Mu Epsilon Journal, Worcester, MA. Vol. 11, No. 2 (Spring 2000), pp. 87-96.
[2] Akça, Ziya and Rüstem Kaya. On the Taxicab Trigonometry, Journal of Institute Of Mathematics and Computer Sciences, Vol. 10, No. 3, pp. 151-159.
[3] Brisbin, Ruth and Paul Artola. Taxicab Trigonometry, The Pi Mu Epsilon Journal, Worcester, MA. Vol. 8, No. 2 (Spring 1985), pp. 89-95.

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