On the Iso-Taxicab Trigonometry

Transcription

1 On the Iso-Taxicab Trigonometry İsmail Kocayusufoğlu and Tuba Ada Abstract. The Iso-Taxicab Geometry is defined in 989. The Iso-Taxicab trigonometric functions cos I (θ), sin I (θ), tan I (θ), and cot I (θ) are defined in []. The aim of this paper is to give the reduction, addition-substraction and half-angle formulas of Iso- taxicab trigonometry. Finally, we will give the Table of Iso-Taxicab Trigonometry. M.S.C. 2000: 5F05, 5F99, 5K05. Key words: Iso-Taxicab Geometry, Non-Euclidean Geometry. Introduction Trigonometric functions are the basic ingredients of elementary mathematics. Although they are elementary concepts, we use them almost in everywhere, all the branches of mathematics, engineer, physics,... So, defining the trigonometric functions on any geometry has an important place. Iso-Taxicab Geometry is a new geometry. It is defined in 989 by K. O. Sowell. It is a non-euclidean geometry. As it is mentioned in [5] that in iso-taxicab geometry three axes occur at the origin: the x-axis, the y-axis and the y -axis. This latter axis forms an angle of 60 with the x-axis and with the y-axis. But, the points will still be named by ordered pairs of real numbers with respect to the x-axis and the y-axis. These three axes separate the plane into six regions, called hextants. These hextants will be numbered from I to V I in a counterclockwise direction beginning with the hextant where the coordinates of the points are both positive (Figure ). At any point in the plane three lines may be drawn parallel to the axes which separate the plane into six regions. Two points, then, may have I IV or II V or III V I orientation to one another. With these orientations, three distance functions arises: (i) x x 2 + y y 2, I IV orientation d I (A, B) = (ii) y y 2, II V orientation (iii) x x 2, III V I orientation If the points lie on a line parallel to x-axis, the formula (iii) holds; parallel to y or y -axes, the formula (ii) holds [5]. Applied Sciences, Vol.8, 2006, pp. 0-. c Balkan Society of Geometers, Geometry Balkan Press 2006.

2 02 İsmail Kocayusufoğlu and Tuba Ada Definitions of iso-taxicab trigonometric functions cos I (θ), sin I (θ), tan I (θ), and cot I (θ) are given in []. We, first, recall them here and then we will define the reduction, addition, substraction and half-angle formulas of Iso- taxicab trigonometry. Finally, we will give the Table of Iso-Taxicab Trigonometry. 2 Iso-Taxicab Trigonometric Functions We first start with defining the iso-taxicab unit circle and then will define iso-taxicab trigonometric functions cos I (θ), sin I (θ), tan I (θ), and cot I (θ). Definition. The set of points such that the iso-taxicab distance from the origin is defines the iso-taxicab unit circle. The equation of Iso-taxicab unit circle can written as (x, y) x + y =, I IV orientation C I = (x, y) y =, II V orientation (x, y) x =, III V I orientation Definition 2. Let f : [0, 2π I ) C I be a function such that f(θ) = P d I (P, A) = θ, θ I = [0, π I ) + d I (P, B) = θ, θ II = [ π I, 2π I ) 2 + d I (P, C) = θ, θ III = [ 2π I, π I) + d I (P, D) = θ, θ IV = [π I, 4π I ) 4 + d I (P, E) = θ, θ V = [ 4π I, 5π I ) 5 + d I (P, F ) = θ, θ V I = [ 5π I, 2π I) f is called the iso-taxicab trigonometric function. It is easy to show that f is one-to-one and surjective function. Using the definition

3 On the Iso-Taxicab Trigonometry 0 of C I, we can write f(θ) = (x, y) y = θ, θ I x = θ, θ II y = θ, θ III y = θ, θ IV 4 + x = θ, θ V 6 + y = θ, θ V I Thus, using the iso-taxicab trigonometric function, each angle θ can be represented by a real number in the interval [0, 6). 2. Iso-Taxicab Cosine and Sine Functions Definition. Let θ be an angle and Q be a point as in Figure. Let P = (x, y) be the intersection point of the line OQ and the unit circle C I. Then, the x coordinate of P defines the Cosine function and the y coordinate of P defines the Sine function. Namely, x = cos I (θ), y = sin I (θ). So, by using Definition and Definition 2, we have the following: As a result, we have θ I cos I (θ) = θ, sin I (θ) = θ θ II cos I (θ) = θ, sin I (θ) = θ III cos I (θ) =, sin I (θ) = θ θ IV cos I (θ) = θ 4, sin I (θ) = θ θ V cos I (θ) = θ 4, sin I (θ) = θ V I cos I (θ) =, sin I (θ) = θ 6 θ 0 cos I (θ) 2 sin I (θ) 0 2 π I π I π I π I 2π I 4π π I I π I π I 2π I

4 04 İsmail Kocayusufoğlu and Tuba Ada 2.2 Iso-Taxicab Tangent and Cotangent Functions Definition 4. Let P be a point on the iso-taxicab unit circle, but not on the y-axis. Let T = (, t) be the intersection point of the line x = and the line OP. The ordinate of the point T is called the iso-taxicab tangent of the angle θ and is denoted by tan I (θ). It is clear that the equation of the line OP is y = mx. Since T = (, t) OP, we have t = m = m. Thus, tan I (θ) = t = m = y x = sin I(θ) cos I (θ).

5 On the Iso-Taxicab Trigonometry 05 Definition 5. Let P be a point on the iso-taxicab unit circle, but not on the x-axis. Let C = (c, ) be the intersection point of the line y = and the line OP. The abcyssa of the point C is called the iso-taxicab cotangent of the angle θ and is denoted by cot I (θ). In this case, we have C = (c, ) OP. So, cot I (θ) = c = m = x y = cos I(θ) sin I (θ). Therefore, from Definition 4 and Definition 5, we have tan I (θ) = and cot I (θ) = / tan I (θ). Thus, θ θ, θ I θ, θ II θ, θ III θ θ 4, θ IV 4 θ, θ V θ 6, θ V I π I π θ 0 I 2π I 5π 6 π I I 2π I tan I (θ) cot I (θ)

6 06 İsmail Kocayusufoğlu and Tuba Ada Reduction Formulas Let θ I. It is not difficult to see that the reduction formulas in Iso-taxicab geometry are as folllows. cos I ( π I θ) = sin I (θ) cos I ( π I + θ) = sin I (θ) cos I ( 2π I θ) = + sin I(θ) cos I ( 2π I + θ) = cos I (π I θ) = cos I (π I + θ) = + sin I (θ) cos I ( 4π I θ) = sin I(θ) cos I ( 4π I + θ) = sin I(θ) cos I ( 5π I θ) = sin I(θ) cos I ( 5π I + θ) = cos I (2π I θ) = cos I (2π I + θ) = cos I (θ) sin I ( π I θ) = cos I (θ) sin I ( π I + θ) = sin I ( 2π I θ) = sin I ( 2π I + θ) = sin I(θ) sin I (π I θ) = sin I (θ) sin I (π I + θ) = sin I (θ) sin I ( 4π I θ) = + sin I(θ) sin I ( 4π I + θ) = sin I ( 5π I θ) = sin I ( 5π I + θ) = + sin I(θ) sin I (2π I θ) = sin I (θ) sin I (2π I + θ) = sin I (θ) We can conclude that cos I (2kπ I + θ) = cos I (θ) sin I (2kπ I + θ) = sin I (θ) for any natural number k. Therefore, the period of the Cosine and Sine functions is 2π I = 6. 4 Addition Formulas There are 42 cases. For each case, we have to apply to Definition and Definition 2 for the vectors θ, θ 2, and θ = θ + θ 2. We give an example, first. Then we will give the table. Example 6. Suppose θ I, θ 2 I and θ = θ + θ 2 I. Let Thus, we have P = (x, y) = (cos I (θ ), sin I (θ )) P = (x, y ) = (cos I (θ 2 ), sin I (θ 2 )) P = (x, y ) = (cos I (θ ), sin I (θ )) For θ : y = θ, x + y = For θ 2 : y = θ 2, x + y = For θ : y = θ, x + y = Solving these equations for x and y, we get x = + x + x y = y + y

7 On the Iso-Taxicab Trigonometry 07 which implies that cos I (θ + θ 2 ) = + cos I (θ ) + cos I (θ 2 ) sin I (θ + θ 2 ) = sin I (θ ) + sin I (θ 2 ). Let s, now, give the table of formulas for 42 cases: θ θ 2 θ cos(θ + θ 2 ) sin(θ + θ 2 ) I I I + cos I (θ ) + cos I (θ 2 ) sin I (θ ) + sin I (θ 2 ) 2 I I II sin I (θ ) sin I (θ 2 ) I II II sin I (θ ) + cos I (θ 2 ) 4 I II III + cos I (θ ) + cos I (θ 2 ) 5 I III III sin I (θ ) + sin I (θ 2 ) 6 I III IV + sin I (θ ) sin I (θ 2 ) sin I (θ ) + sin I (θ 2 ) 7 I IV IV + sin I (θ ) sin I (θ 2 ) sin I (θ ) + sin I (θ 2 ) 8 I IV V + sin I (θ ) sin I (θ 2 ) 9 I V V sin I (θ ) + cos I (θ 2 ) 0 I V V I 2 + sin I (θ ) + cos I (θ 2 ) I V I V I sin I (θ ) + sin I (θ 2 ) 2 I V I I sin I (θ ) sin I (θ 2 ) sin I (θ ) + sin I (θ 2 ) II II III + cos I (θ ) + cos I (θ 2 ) 4 II II IV cos I (θ ) cos I (θ 2 ) + cos I (θ ) + cos I (θ 2 ) 5 II III IV cos I (θ ) sin I (θ 2 ) + cos I (θ ) + sin I (θ 2 ) 6 II III V cos I (θ ) sin I (θ 2 ) 7 II IV V cos I (θ ) sin I (θ 2 ) 8 II IV V I 2 cos I (θ ) sin I (θ 2 ) 9 II V V I cos I (θ ) + cos I (θ 2 ) 20 II V I 2 + cos I (θ ) cos I (θ 2 ) cos I (θ ) + cos I (θ 2 )

8 08 İsmail Kocayusufoğlu and Tuba Ada θ θ 2 θ cos I (θ + θ 2 ) sin I (θ + θ 2 ) 2 II V I I + cos I (θ ) sin I (θ 2 ) 2 cos I (θ ) + sin I (θ 2 ) 22 II V I II cos I (θ ) sin I (θ 2 ) 2 III III V 2 sin I (θ ) sin I (θ 2 ) 24 III III V I sin I (θ ) sin I (θ 2 ) 25 III IV V I sin I (θ ) sin I (θ 2 ) 26 III IV I + sin I (θ ) + sin I (θ 2 ) sin I (θ ) sin I (θ 2 ) 27 III V I sin I (θ ) cos I (θ 2 ) sin I (θ ) + cos I (θ 2 ) 28 III V II sin I (θ ) cos I (θ 2 ) 29 III V I I 2 + sin I (θ ) cos I (θ 2 ) sin I (θ ) + cos I (θ 2 ) 0 III V I II 2 sin I (θ ) + sin I (θ 2 ) IV IV I + sin I (θ ) + sin I (θ 2 ) sin I (θ ) sin I (θ 2 ) 2 IV IV II + sin I (θ ) + sin I (θ 2 ) IV V II sin I (θ ) cos I (θ 2 ) 4 IV V III 2 + sin I (θ ) cos I (θ 2 ) 5 IV V I III sin I (θ ) sin I (θ 2 ) 6 IV V I IV sin I (θ ) + sin I (θ 2 ) sin I (θ ) sin I (θ 2 ) 7 V V III cos I (θ ) cos I (θ 2 ) 8 V V IV cos I (θ ) + cos I (θ 2 ) cos I (θ ) cos I (θ 2 ) 9 V V I IV 2 + cos I (θ ) + sin I (θ 2 ) cos I (θ ) sin I (θ 2 ) 40 V V I V cos I (θ ) + sin I (θ 2 ) 4 V I V I V 2 + sin I (θ ) + sin I (θ 2 ) 42 V I V I V I sin I (θ ) + sin I (θ 2 ) 5 Substraction Formulas There are 6 cases. For each case, we have to apply to Definition and Definition 2 for the vectors θ, θ 2, and θ = θ θ 2. Let s give an example, first. Then we will give the table. Example 7. Suppose θ I, θ 2 I. Then θ = θ θ 2 I. If then, we have P = (x, y) = (cos I (θ ), sin I (θ )) P = (x, y ) = (cos I (θ 2 ), sin I (θ 2 )) P = (x, y ) = (cos I (θ ), sin I (θ )) For θ : y = θ, x + y = For θ 2 : y = θ 2, x + y = For θ : y = θ, x + y = Solving these equations for x and y, we get x = + x x y = y y

9 On the Iso-Taxicab Trigonometry 09 which implies that cos I (θ θ 2 ) = + cos I (θ ) cos I (θ 2 ) sin I (θ θ 2 ) = sin I (θ ) sin I (θ 2 ). We, now, give the table of formulas for 6 cases: θ θ 2 θ cos I (θ θ 2 ) sin I (θ θ 2 ) I I I + cos I (θ ) cos I (θ 2 ) sin I (θ ) sin I (θ 2 ) 2 II II I + cos I (θ ) cos I (θ 2 ) cos I (θ ) + cos I (θ 2 ) III III I + sin I (θ ) sin I (θ 2 ) sin I (θ ) + sin I (θ 2 ) 4 IV IV I cos I (θ ) + cos I (θ 2 ) sin I (θ ) + sin I (θ 2 ) 5 V V I cos I (θ ) + cos I (θ 2 ) cos I (θ ) cos I (θ 2 ) 6 V I V I I sin I (θ ) + sin I (θ 2 ) sin I (θ ) sin I (θ 2 ) 7 II I I + cos I (θ ) cos I (θ 2 ) cos I (θ ) + cos I (θ 2 ) 8 II I II + cos I (θ ) cos I (θ 2 ) 9 III I II + sin I (θ ) cos I (θ 2 ) 0 III I III sin I (θ ) + sin I (θ 2 ) III II I + sin I (θ ) cos I (θ 2 ) 2 sin I (θ ) + cos I (θ 2 ) 2 III II II + sin I (θ ) cos I (θ 2 ) IV I III sin I (θ ) + sin I (θ 2 ) 4 IV I IV + cos I (θ ) + cos I (θ 2 ) sin I (θ ) + sin I (θ 2 ) 5 IV II II + sin I (θ ) cos I (θ 2 ) 6 IV II III cos I (θ ) cos I (θ 2 ) 7 IV III I + sin I (θ ) sin I (θ 2 ) sin I (θ ) + sin I (θ 2 ) 8 IV III II + sin I (θ ) sin I (θ 2 ) 9 V I IV + cos I (θ ) + cos I (θ 2 ) cos I (θ ) cos I (θ 2 ) 20 V I V + cos I (θ ) + cos I (θ 2 ) θ θ 2 θ cos I (θ θ 2 ) sin I (θ θ 2 ) 2 V II III cos I (θ ) cos I (θ 2 ) 22 V II IV cos I (θ ) + cos I (θ 2 ) cos I (θ ) cos I (θ 2 ) 2 V III II cos I (θ ) sin I (θ 2 ) 24 V III III 2 cos I (θ ) sin I (θ 2 ) 25 V IV I cos I (θ ) + cos I (θ 2 ) cos I (θ ) cos I (θ 2 ) 26 V IV II cos I (θ ) + cos I (θ 2 ) 27 V I I V 2 + sin I (θ ) sin I (θ 2 ) 28 V I I V I sin I (θ ) sin I (θ 2 ) 29 V I II IV + sin I (θ ) + cos I (θ 2 ) 2 sin I (θ ) cos I (θ 2 ) 0 V I II V + sin I (θ ) + cos I (θ 2 ) V I III III sin I (θ ) sin I (θ 2 ) 2 V I III IV + sin I (θ ) + sin I (θ 2 ) sin I (θ ) sin I (θ 2 ) V I IV II 2 sin I (θ ) sin I (θ 2 ) 4 V I IV III sin I (θ ) sin I (θ 2 ) 5 V I V I sin I (θ ) + cos I (θ 2 ) 2 + sin I (θ ) cos I (θ 2 ) 6 V I V II sin I (θ ) + cos I (θ 2 )

10 0 İsmail Kocayusufoğlu and Tuba Ada Using the addition formulas, we can easily find the half-angle formulas in Isotaxicab geometry. 6 Half-angle Formulas The half-angle formulas in Iso-taxicab geometry are as follows: θ 2θ cos I (2θ) sin I (2θ) I I cos I (θ) sin I (θ) 2 sin I (θ) 2 I II cos I (θ) sin I (θ) II III + 2 cos I (θ) 4 II IV 2 2 cos I (θ) + 2 cos I (θ) 5 III V 2 2 sin I (θ) 6 III V I 2 sin I (θ) 7 IV I + 2 sin I (θ) 2 sin I (θ) 8 IV II + 2 sin I (θ) 9 V III 2 cos I (θ) 0 V IV cos I (θ) 2 cos I (θ) V I V sin I (θ) 2 V I V I 2 sin I (θ) Before giving the final concept of this paper, we note that these results may also be written in different way: It is clear that So, in the case of 4, above, we also have cos I (θ) + sin I (θ) =, θ I IV sin I (θ) =, θ II V cos I (θ) =, θ III V I sin I (2θ) = + 2 cos I (θ) = + 2 sin I (θ) 7 Table of Iso-Taxicab Trigonometry Finally, let s give the table of values of Iso-taxicab trigonometric functions. Degree Cos I Si n I T an I Cot I ,98 0,06 0,06 59, ,966 0,0 0,04 29, ,950 0,050 0,052 9, ,9 0,066 0,07 4, ,96 0,08 0,090, ,900 0,00 0, 9, ,88 0,6 0,2 7, ,866 0, 0,5 6, ,850 0,50 0,76 5,666 5

11 On the Iso-Taxicab Trigonometry References 0 0,8 0,66 0,200 5, ,86 0,8 0,224 4, ,800 0,200 0,250 4, ,78 0,26 0,276, ,766 0,2 0,04, ,750 0,250 0,, ,7 0,266 0,6 2, ,76 0,28 0,95 2, ,700 0,00 0,428 2, ,68 0,6 0,46 2, ,666 0, 0,500 2, ,650 0,50 0,58, ,6 0,66 0,579, ,66 0,8 0,62, ,600 0,400 0,666, ,58 0,46 0,7, ,566 0,4 0,765, ,550 0,450 0,88, ,5 0,466 0,874, ,56 0,48 0,96, ,5 0,5 0 Si n I Cos I Cot I T an I Degree [] İ. Kocayusufoglu, Trigonometry on Iso-Taxicab Geometry, Mathematical and Computational Applications, Association for Scientific Research, 5, (999), [2] E.F. Krause, Taxicab Geometry, Addison-Wesley, Menlo Park, NJ, 975. [] K. Menger, You will like Geometry, A Guide Book for the Ill. Inst. of Tech. Geometry Exhibition at the Museum of Science and Industry, Chicago, 952. [4] S.L. Salas, E. Hille, T.A. Anderson, Calculus - One and Several Variables, John Wiley & Sons, 986. [5] K.O. Sowell, Taxicab Geometry -A new slant, Mathematics Magazine 62 (989), [6] K. Thompson, T. Dray, Taxicab angles and trigonometry, Pi Mu Epsilon (2000), Authors addresses: İsmail Kocayusufoğlu Eskişehir Osmangazi University, Dept. of Mathematics, Eskişehir -Turkey. ikyusuf@ogu.edu.tr Tuba Ada Anadolu University, Education Faculty, Eskisehir - Turkey. tyuzugul@anadolu.edu.tr