ON THE LENGTH OF A CIRCULAR ARC

Richard E. Bayne, James E. Joseph, and Myung H. Kwack

In his note on limits [1], Professor Krantz points out that the limit

(∗)  lim_{x→0} \frac{sin x}{x}

is calculated in the same way in every calculus book that he knows of (and that we know of). The upper estimate

\frac{sin x}{x} ≤ 1

is found by an obvious comparison of a chord against an arc of a circle and the lower estimate

\frac{sin x}{x} ≥ cos x

is computed using the area of a circle, which depends on knowing the limit (∗) itself. To avoid this circular argument, he presented another lower estimate,

\frac{sin x}{x} ≥ \frac{1}{1 + tan x}.

The purpose of this note is to produce one simple proof for both the inequalities

cos x ≤ \frac{sin x}{x} ≤ 1 for 0 < x < \frac{π}{2} ,

using only the definition of length of arc and the obvious comparisons of linear segments of appropriate partitions of the lengths involved.

Figure 1. 2

For every partition \{P_{i}\} of the circular arc AB of length x < \frac{π}{2} on the unit circle, consider the corresponding partitions {Q_{i}} and {R_{i}} of the linear segments AC and BD, respectively, as in Figure 1. Obviously,

|R_{i}R_{i+1}| ≤ |P_{i}P_{i+1}| ≤ |Q_{i}Q_{i+1}|,

where |XY | denotes the length of the linear segment XY. Summing up and taking least upper bounds of the collections of sums, we get

sin x ≤ x ≤ tan x

and finally

cos x ≤ \frac{sin x}{x} ≤ 1.

\underline{Reference}

1. S. G. Krantz, “On the Area Inside a Circle,” Missouri Journal of Mathematical Sciences, 4 (1992), 2–8.

Richard E. Bayne Department of Mathematics Howard University Washington, D.C. 20059 email: bayne@scs.howard.edu

James E. Joseph Department of Mathematics Howard University Washington, D.C. 20059

Myung H. Kwack Department of Mathematics Howard University Washington, D.C. 20059