ma024 - A Direct Proof of the Integral Formula for...

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A Direct Proof of the Integral Formula for Arctangent Arnold J. Insel, Illinois State University, Normal, IL The College Mathematics Journal, May 1989, Volume 20, Number 3, pages 235–237. I n this capsule, we give a direct proof that the Arctangent is an integral of It then becomes possible to use the Arctangent to determine the tangent and the other trigonometric functions. Here (Figure 1) for any real number a , we define Arctan a as the angle (in radians) determined by angle OPR , where is taken as negative if In what follows, we fix a number This will determine two regions, as shown in Figure 2. The region above the x -axis is bounded by the graph of and the x -axis, where Therefore, the total area of this region is Figure 1 Figure 2 The region below the x -axis is a sector of a circle having center and radius 1. The sides of the sector are determined by the y -axis and the line connecting to Thus, these sides determine the angle with value Arctan a . Since the area of a sector of a circle of radius r and angle
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