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Kevin's Corner

Angles and Trigonometry > Trig Calculus

Before even beginning this discussion, it is important to consider whether the normal application of calculus is valid in taxicab geometry. A derivative is merely a slope of a tangent line. And, slope is merely a change in y (rise) divided by a change in x (run). Since the Euclidean and taxicab metrics agree for line segments parallel to the x- or y-axis (see the Linear Length discussion), the concept of slope, and therefore the concept of a derivative, transfer seamlessly to taxicab geometry.

Although we will not consider it here, the concepts of integration and area also transfer seamlessly to taxicab geometry. See the Concept of Area and Area pages for more information on these topics.

Cosine and Sine

From the definition of the taxicab trig functions, it is clear that the derivatives of both sine and cosine alternate from 1/2 to -1/2 about the extrema. Not only are their derivatives now piecewise constant, the derivative relationship between sine and cosine seen in Euclidean geometry has been lost. Finally, sine and cosine are not everywhere differentiable because of the sharp corners at the extrema.

Foundation for other derivatives

As in Euclidean geometry, the taxicab trig functions tangent, secant, cosecant, and cotangent are defined in terms of the fundamental functions sine and cosine. So, it would not be surprising if the derivatives of these functions are similar to the derivatives of their Euclidean counterparts. But, given the radical differences between the taxicab and Euclidean sine and cosine functions, an investigation is surely warranted.

First, closed form expressions exist for the taxicab sine and cosine functions (see the Trigonometry page). These are a bit unweildy to deal with. And, we should also note that one of the sine and cosine functions changes branches in the piecewise definition every 2 t-radians. So, the expressions we use will be most useful when based on 2 t-radian intervals. This will lend itself nicely when we consider functions that are combinations of both sine and cosine such as the tangent function.

On the interval [2k - 1, 2k), the pseudo closed-form expressions of sine and cosine are

where the imaginary number i is used to create positive and negative terms as needed. As an example, choosing k = 1 for these gives the expressions cosθ = 1 - θ/2 and sinθ = θ/2 on the interval [0, 2) in agreement with our earlier piecewise definition.

Tangent

If we simply take the definition of tangent as the ratio of sine to cosine, use the closed form expressions above, and differentiate with respect to θ, we obtain the derivative of the taxicab tangent function. As explained above, we will take the proof in two cases: on intervals [2k - 1, 2k) with k even and odd.

For k even, we have

For k odd, we have

So, the derivative of tangent is 1/2sec2θ. The factor of 1/2 is the same as the slopes of the sine and cosine functions. It is nevertheless a curious factor not seen in the Euclidean derivative of the tangent function.

From this result we confirm that the taxicab tangent function is differentiable wherever cosine is non-zero.

Cotangent

The derivative of cotangent is found in a similar manner to tangent. The proof will therefore be omitted, but the result is -1/2csc2θ. This again is very reminiscent of the derivative of the Euclidean cotangent function with the added factor of 1/2.

Secant

As we saw on the Trigonometry page, the taxicab secant function is expected to not be differentiable where cosine has an extremum. The basic reason behind this is the fact that cosine is not differentiable at these points. This will prevent us from finding a single closed form expression for the derivative of secant. But, as with tangent we will take the proof in two cases over the intervals [2k - 1, 2k) with k even and odd. For k even, we have

For k odd, we have

This is a very interesting result. Shades of the Euclidean derivative are present, but a whole extra secant term is added or subtracted in the taxicab case. The root of this is the fact that the taxicab sine and cosine do not have a derivative relationship as in Euclidean geometry. As mentioned above, there is no closed form for the derivative of taxicab secant (the two cases exhibit different expressions), and the function has a non-differentiable corner point at 4k-2 t-radians.

While this presentation of the taxicab secant derivative contrasts with Euclidean geometry nicely, if we apply the Pythagorean-style trig identities for taxicab geometry the secant derivative becomes even more interesting. When k is even, θ lives in quadrants II or IV where tangent is negative. In quadrant II, -tanθ + 1 = -secθ and in quadrant IV, -tanθ + 1 = secθ. Using these substitutions in the first result above we have the derivative of secant as ±sec2θ. A similar analysis applies to the second result above. Therefore, on intervals [4k, 4k + 4) (excluding the asymptote) with k even and odd,

Cosecant

The derivative of cosecant is found in a similar manner to secant. The proof will therefore be omitted, but the result is shown below. Over the intervals [2k - 1, 2k) with k even and odd, the derivative of cosecant is

And similarly, over the interval \$[4k-2,4k+2)\$ (excluding the asymptote) with \$k\$ even and odd, the derivative of cosecant can also be expressed as

An Alternate Approach

The derivatives of tangent, cotangent, secant, and cosecant were found above by differentiating combinations of the underlying piecewise functions for sine and cosine. These derivatives can also be found using their basic definitions (such as the ratio of sine and cosine for tangent) and the derivatives of sine and cosine. This approach naturally leads to the secondary results presented above for secant and cosecant. It is also a cleaner approach, but there is benefit in considering the initial results for secant and cosecant found with the presented approach to contrast with the Euclidean derivatives.

References
[1] Thompson, Kevin. Derivatives of Taxicab Trigonometric Functions (unpublished, 2012).
[2] 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.
[3] Akça, Ziya and Rüstem Kaya. On the Taxicab Trigonometry, Journal of Institute Of Mathematics and Computer Sciences, Vol. 10, No. 3, pp. 151-159.

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