The Right Theta. 148 c THE MATHEMATICAL ASSOCIATION OF AMERICA. References. The Cartesian plane. P(x, y) θ x

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1 Theorem can be proved much like Theorem. For Theorem 4, we ask the reader to verif the first few cases for k =,,.... It would be interesting to get a closed form for g. Finding other pairs of functions that constitute PP- pairs (for eample in the case when f is a trigonometric function) is also an interesting challenge! Acknowledgment. The authors would like to thank the editor and the anonmous referees for their helpful comments and valuable suggestions. References 1. J. M. Ash, Math Bite: Once in a While, Differentiation Is Multiplicative, Math. Mag. 7 ().. W. E. Boce and R. C. DiPrima, Elementar Differential Equations and Boundar Value Problems, Wile, M. Hurwitz, Is the derivative of a product the product of the derivatives? Math. Teacher 94 (1) L. G. Maharam and E. P. Shaughness, When Does ( fg) = f g? Two-Year College Math. J. 7 (1976) J. Stewart, Calculus: Concepts and Contets, rd ed., Brooks/Cole, 5. The Right Theta William Freed (wfreed@math.concordia.ab.ca) and Athanasios (Tom) Tavouktsoglou (tomtavou@math.concordia.ab.ca), Concordia Universit College of Alberta, Edmonton, Alberta T5B 4E4 A basic concept in trigonometr is the angle associated with a point P(, ) in the plane. The eact elementar formula for the angle function = (, ) has a complicated case-b-case description. In this capsule we derive a more concise formula which applies to the entire Cartesian plane (ecept the origin and points on the negative -ais). P(, ) r The Cartesian plane. As usual, we measure positive counterclockwise from the positive -ais and take π < π radians. The graph of this function is shown in Figure 1 (see Note on how this graph was constructed). 148 c THE MATHEMATICAL ASSOCIATION OF AMERICA

2 π π -ais -ais Figure 1. The angle as a function of and. In elementar courses the value of the angle function is usuall presented using the formula = tan 1. The graph of this formula, shown in Figure, is correct onl in the right half-plane. One must add or subtract π in the second or third quadrants. In addition, it is necessar to make appropriate choices of its values on the negative -ais and the positive and negative -aes. is left undefined at the origin. π/ π/ Figure. An incorrect theta function. The complete form of for all (, ) = (, ) in terms of tan 1 is then VOL. 9, NO., MARCH 8 THE COLLEGE MATHEMATICS JOURNAL 149

3 = tan 1, for > tan 1 tan 1 + π, for < and > π, for < and < π, for = and > π, for < and = π, for = and <. (1) The graph of this piecewise-defined function is shown in Figure. As a visual check, compare with Figure 1 that this is the same function. π π Figure. The piecewise-defined theta function from (1). Let us derive a simpler formula valid in a larger domain. We begin with the halfangle formulas sin = 1 cos and Using a little trigonometr, we deduce that cos = 1 + cos. tan = sin (1 + cos ). Taking a square root (see Note 4), using the relationships sin =, and solving for we get + and cos = + = tan () 15 c THE MATHEMATICAL ASSOCIATION OF AMERICA

4 which is defined for all points in the plane ecept at the origin and along the negative -ais. The complete form of in terms of the preceding formula is therefore { tan 1, ecept for < and = = + + π, for < and =. () The graph of this function agrees with Figure 1, but this time uses formula (). Note that tan 1 is undefined at the origin and also along the negative ais where its argument has the indeterminate form. This agrees with the fact that could reasonabl be defined as either π or π there. We conclude this paper with a few notes. 1. It is unlikel formula () is new because the idea is so elementar. However, in our man ears of teaching, neither of us has met it. Regardless, it is not well known and ma well be useful because, even though the argument in the arctan formula involved is more complicated, the logical structure of the complete formula is much simpler.. The graph of Figure 1 was produced b the Mathematica function ArcTan[, ], not to be confused with tan 1.ArcTan[, ] is numericall evaluated for nonzero real numbers b calling the C librar function atan(, ), which is likel an implementation of (1). (Readers ma wish to check how is calculated in the smbolic algebra programs the use. ). Smbolic algebra programs ma not et use (). For eample, for the point (1, 1), Mathematica gives the correct value of ArcTan 1 as π (likel using a lookup table), but leaves ArcTan 1 unevaluated In our derivation of () it is possible to choose the other square root. If this is done, then the angle range is π <<πdescribed clockwise. A more useful variation, left as an eercise, is = π tan 1. For this formula, + <<6, and is measured positive counterclockwise from the positive - ais. Its graph is shown in Figure tan 1 is a potential function for the line integral + + C + d + + d, where C is an smooth simple closed path about the origin. Then, b Figure 1, in a manner reminiscent of the evaluation of closed contour integrals in comple variables, it is clear that C + d + + d = π ( π + ) = π. tan 1 is also an antiderivative; however, Figure is difficult to interpret for this purpose. The downside of using our formula () is that it is ver difficult to show b hand, either b differentiation or b integration, that it is a potential function. This line integral is an eample in all five calculus tetbooks that we looked at (Stewart; Edwards and Penn; Thomas; Rogawski; and Anton, Bivens, and Davis). VOL. 9, NO., MARCH 8 THE COLLEGE MATHEMATICS JOURNAL 151

5 6 Figure 4. Another theta function useful for some applications. Some tetbooks evaluate it directl as a line integral about a circle in the section on Green s Theorem. Others, in addition, show that tan 1 is a potential function and note it can be interpreted geometricall as the angle function, and that therefore the change of angle along a closed contour about the origin is π. None comment on the difficulties in interpreting this epression in the entire plane or graph this potential function in the entire plane, perhaps to avoid having to eplain the compleities of formula (1). Ken Yanosko (k1@humboldt.edu), Humboldt State Universit, wrote that he enjoed reading the article Commensurable Triangles b Richard Parris in the November issue (8 (7) 45 55), but points out earlier work on the topic. R. S. Luthar classified -commensurable triangles (Integer-Sided Triangles with One Angle Twice Another, College Math. J. 15 (1984) 55 56), and subsequentl, the general case in which one angle is a rational multiple of another was treated b J. Carroll and K. Yanosko, (The Determination of a Class of Primitive Integral Triangles, Fibonacci Quarterl 9 (1991) 6). Robert Cla (rcla@daltonstate.edu), Dalton State College, Dalton, Georgia, writes concerning the Classroom Capsule Conic Sections from the Plane Point of View b Sidne Kung in the November 7 issue (pp. 8 84): The article is correct in stating that equation () is a conic section. However, it is not congruent to the original conic formed b the cutting plane, but rather the projection of the original onto the -plane. The original and the projection will be congruent if the cutting plane is parallel to the -plane. Therefore, one can achieve congruence b rotating the cone and the cutting plane until the plane is parallel to the -plane. 15 c THE MATHEMATICAL ASSOCIATION OF AMERICA