3.3. SPHERICAL TRIGONOMETRY 73 TABLE 3.5 Basic properties and relations characterizing spherical triangles No. The name of property Properties and relations 1 Triangle inequality The sum of lengths of two sides is greater than the length of the third side. The absolute value of the difference between the lengths of two sides is less than the length of the third side, a + b > c, |a – b| < c 2 Sum of two angles of a triangle The sum of two angles of a triangle is greater than the third angle increased by π, α + β < π + γ 3 The greatest side and the greatest angle The greatest side is opposite the greatest angle, a < b if α < β; a = b if α = β 4 Sum of angles of a triangle The sum of the angles lies between π and 3π, π < α + β + γ < 3π 5 Sum of sides of a triangle The sum of sides lies between 0 and 2π 0 < a + b + c < 2π 6 The law of sines sin a sin α = sin b sin β = sin c sin γ 7 The law of cosines of sides cos c =cosa cos b +sina sinbcos γ 8 The law of cosines of angles cos γ =–cosαcos β +sinα sin β cos c 9 Half-angle formulas sin γ 2 =  sin(p – a)sin(p – b) sin a sinb ,cos γ 2 =  sin p sin(p – c) sin a sinb , tan γ 2 =  sin(p – a)sin(p – b) sin p sin(p – c) 10 Half-side theorem sin c 2 =  –sinP sin(P – γ) sin α sin β ,cos c 2 =  sin(P – α)sin(P – β) sin α sin β , tan c 2 =  –sinP sin(P – γ) sin(P – α)sin(P – β) 11 Neper’s analogs tan c 2 cos α – β 2 =tan a + b 2 cos α + β 2 , tan c 2 sin α – β 2 =tan a – b 2 sin α + β 2 , cot γ 2 cos a – b 2 =tan α + β 2 cos a + b 2 , cot γ 2 sin a – b 2 =tan α – β 2 sin a + b 2 12 D’Alembert (Gauss) formulas sin γ 2 sin a + b 2 =sin c 2 cos α – β 2 ,sin γ 2 sin a + b 2 =cos c 2 cos α + β 2 , cos γ 2 sin a – b 2 =sin c 2 sin α – β 2 ,cos γ 2 cos a – b 2 =cos c 2 sin α + β 2 13 Product formulas sin a cos β =cosbsin c –cosα sin b cos c, sin a cos b =cosβ sin c –cosa sin β cosγ 14 The “circumradius” R cot R =  sin(P – α)sin(P – β)sin(P – γ) sin P =cot α 2 sin(α – P ) 15 The “inradius” r tan r =  sin(p – α)sin(p – β)sin(p – γ) sin p =tan α 2 sin(p – α) 74 ELEMENTARY GEOMETRY TABLE 3.5 (continued) Basic properties and relations characterizing spherical triangles No. The name of property Properties and relations 16 Willier’s formula for the spherical excess ε tan P 2 =tan ε 4 =  tan p 2 tan p – a 2 tan p – b 2 tan p – c 2 17 L’Huiller equation tan  γ 2 – ε 4  =  tan p–a 2 tan p–b 2 tan p 2 tan p–c 2 TABLE 3.6. Solution of spherical triangles No. Three parts specified Formulas for the remaining parts 1 Three sides a, b, c The angles α, β,andγ are determined by the half-angle formulas and the cyclic permutation. Remark. 0 <a+b+c<2π. The sum and difference of two sides are greater than the third. 2 Three angles α, β, γ The sides a, b,andc are determined by the half-side theorems and the cyclic permutation. Remark. π <α+β+γ <3π. The sum of two angles is less than π plus the third angle. 3 Two sides a, b and the included angle γ First method. α + β and α – β are determined from Neper’s analogs, then α and β can be found; side a is determined from the law of cosines, sin c =sinγ sin a sin α . Second method. The law of cosines of sides is applied, cosc =cosa cos b +sina sin b cos γ, cos β = cos b –sina sin c sin a sin c ,cosα = cos a –sinb sin c sin b sin c . Remark 1. If γ > β (γ < β), then c must be chosen so that c > b (c < b). Remark 2. The quantities c, α,andβ are determined uniquely. 4 Asidec and the two angles α, β adjacent to it First method. a + b and a – b are determined from Neper’s analogs, then a and b can be found; angle γ is determined from the law of sines, sin γ =sinc sin α sin a . Second method. The law of cosines of angles is applied, cos γ =–cosα cos β +sinα sin β cos c, cos a = cos α +cosβ cos γ sin β sinγ ,cosb = cos β +cosα cos γ sin α sin γ . Remark 1. If c > b (c < b), then γ must be chosen so that γ > β (γ < β). Remark 2. The quantities γ, a,andb are determined uniquely. 5 Two sides a, b and the angle α opposite one of them β is determined by the law of sines, sin β =sinα sin b sin a . The elements c and γ can be found from Neper’s analogs. Remark 1.The problem has a solution for sin b sin α ≤ sina. Remark 2. Different cases are possible: 1. If sin a ≥ sin b, then the solution is determined uniquely. 2. If sin b sin α <sina, then there are two solutions β 1 and β 2 , β 1 + β 2 = π. 3. If sin b sin α =sina, then the solution is unique: β = 1 2 π. 6 Two angles α, β and the side a opposite one of them b is determined by the law of sines, sin b =sina sin β sin α . The elements c and γ can be found from Neper’s analogs. Remark 1. The problem has a solution for sin a sin β ≤ sin α. Remark 2. Different cases are possible: 1. If sin α ≥ sin β, then the solution is determined uniquely. 2. If sin β sin α <sina, then there are two solutions b 1 and b 2 , b 1 + b 2 = π. 3. If sin β sin α =sina, then the solution is unique: b = 1 2 π. REFERENCES FOR CHAPTER 3 75 The following basic relations hold for spherical triangles: sin a =cos  π 2 – a  =sinα sin c =cot  π 2 – b  cot β =tanb cot β, sin b =cos  π 2 – b  =sinβ sin c =cot  π 2 – a  cot α =tana cot α, cos c =sin  π 2 – a  sin  π 2 – b  =cosa cos b =cotα cot β, cos α =sin  π 2 – a  sin β =cosa sin β =cot  π 2 – b  cot c =tanb cot c, cos β =sin  π 2 – b  sin α =cosb sin α =cot  π 2 – a  cot c =tana cot c, (3.3.2.3) which can be obtained from the Neper rules:ifthefive parts of a spherical triangle (the right angle being omitted) are written in the form of a circle in the order in which they appear in the triangle and the legs a and b are replaced by their complements to 1 2 π (Fig. 3.38b), then the cosine of each part is equal to the product of sines of the two parts not adjacent to it, as well as to the product of the cotangents of the two parts adjacent to it. References for Chapter 3 Alexander, D. C. and Koeberlein, G. M., Elementary Geometry for College Students, 3rd Edition, Houghton Mifflin Company, Boston, 2002. Alexandrov, A. D., Verner, A. L., and Ryzhik, B. I., Solid Geometry [in Russian], Alpha, Moscow, 1998. Chauvenet, W., A Treatise on Elementary Geometry, Adamant Media Corporation, Boston, 2001. Fogiel, M. (Editor), High School Geometry Tutor, 2nd Edition, Research & Education Association, Englewood Cliffs, New Jersey, 2003. Gustafson, R. D. and Frisk, P. D., Elementary Geometry, 3rd Edition, Wiley, New York, 1991. Hadamard, J., Lec¸ons de g ´ eom ´ etrie ´ el ´ ementaire, Vols 1 and 2, Rep. Edition, H C. Hege and K. Polthier (Editors), Editions J. Gabay, Paris, 1999. Hartshorne, R., Geometry: Euclid and Beyond, Springer, New York, 2005. Jacobs, H. R., Geometry, 2nd Edition, W. H. Freeman & Company, New York, 1987. Jacobs, H. R., Geometry: Seeing, Doing, Understanding, 3rd Edition,W.H.Freeman&Company,NewYork, 2003. Jurgensen, R. and Brown, R. G., Geometry, McDougal Littell/Houghton Mifflin, Boston, 2000. Kay, D., College Geometry: A Discovery Approach, 2nd Edition, Addison Wesley, Boston, 2000. Kiselev, A. P., Plain and Solid Geometry [in Russian], Fizmatlit Publishers, Moscow, 2004. Leff,L.S.,Geometry the Easy Way, 3rd Edition, Barron’s Educational Series, Hauppauge, New York, 1997. Moise, E., Elementary Geometry from an Advanced Standpoint, 3rd Edition, Addison Wesley, Boston, 1990. Musser,G.L.,Burger,W.F.,andPeterson,B.E.,Mathematics for Elementary Teachers: A Contemporary Approach, 6th Edition, Wiley, New York, 2002. Musser, G. L., Burger, W. F., and Peterson, B. E., Essentials of Mathematics for Elementary Teachers: A Contemporary Approach, 6th Edition, Wiley, New York, 2003. Musser, G. L. and Trimpe, L. E., College Geometry: A Problem Solving Approach with Applications, Prentice Hall, Englewood Cliffs, New Jersey, 1994. Pogorelov, A. V., Elementary Geometry [in Russian], Nauka Publishers, Moscow, 1977. Pogorelov, A., Geometry, Mir Publishers, Moscow, 1987. Prasolov, V. V., Problems in Plane Geometry [in Russian], MTsNMO, Moscow, 2001. Prasolov, V. V. and Tikhomirov, V. M., Geometry, Translations of Mathematical Monographs, Vol. 200, American Mathematical Society, Providence, Rhode Island, 2001. Roe, J., Elementary Geometry, Oxford University Press, Oxford, 1993. Schultze, A. and Sevenoak, F. L., Plane and Solid Geometry, Adamant Media Corporation, Boston, 2004. Tussy,A.S.,Basic Geometry for College Students: An Overview of the Fundamental Concepts of Geometry, 2nd Edition, Brooks Cole, Stamford, 2002. Vygodskii, M. Ya., Mathematical Handbook: Elementary Mathematics, Rev. Edition , Mir Publishers, Moscow, 1972. Chapter 4 Analytic Geometry 4.1. Points, Segments, and Coordinates on Line and Plane 4.1.1. Coordinates on Line 4.1.1-1. Axis and segments on axis. A straight line on which a sense is chosen is called an axis. If an axis is given and a scale segment, i.e., a linear unit used to measure any segment of the axis, is indicated, then the segment length is defined (see Fig. 4.1). ACB Figure 4.1. Axis. A segment bounded by points A and B is called a directed segment if its initial point and endpoint are chosen. Such a segment with initial point A and endpoint B is denoted by −−→ AB. Directed segments are usually called simply “segments” for brevity. The value of a segment −−→ AB of some axis is defined as the number AB equal to its length taken with the plus sign if the senses of the interval and the axis coincide, and with the minus sign if the senses are opposite. Obviously, the length of a segment is its absolute value. The segment length is usually denoted by the symbol |AB|. It follows from the above that AB =–BA, |AB| = |BA|.(4.1.1.1) Main identity. For any arbitrary arrangement of points A, B,andC on the axis, the values of the segments −−→ AB, −−→ BC,and −→ AC satisfy the relation AB + BC = AC.(4.1.1.2) 4.1.1-2. Coordinates on line. Number axis. One says that a coordinate system is introduced on an axis if there is a one-to-one corre- spondence between points of the axis and numbers. Suppose that a sense, a scale segment, and a point O called the origin are chosen on a line. The value of a segment −→ OA is called the coordinate of the point A on the axis. It is usually denoted by the letter x. The coordinates of different points are usually denoted by subscripts; for example, the coordinates of points A 1 , , A n are x 1 , , x n . The point A n with coordinate x n is denoted by A n (x n ). An axis with a coordinate system on it is called a number axis. 77 78 ANALYTIC GEOMETRY 4.1.1-3. Distance between points on axis. The value A 1 A 2 of the segment −−→ A 1 A 2 on an axis is equal to the difference between the coordinate x 2 of the endpoint and the coordinate x 1 of the initial point: A 1 A 2 = x 2 – x 1 .(4.1.1.3) The distance d between two arbitrary points A 1 (x 1 )andA 2 (x 2 ) on the line is given by the relation d = |A 1 A 2 | = |x 2 – x 1 |.(4.1.1.4) Remark. If segments do not lie on some axis but are treated as arbitrary segments on the plane or in space, then there is no reason to assign any sign to their lengths. In such cases, the symbol of absolute value is usually omitted in the notation of lengths of segments. We adopt this convention in the sequel. 4.1.2. Coordinates on Plane 4.1.2-1. Rectangular Cartesian coordinates on plane. If a one-to-one correspondence between points on the plane and numbers (pairs of numbers) is specified, then one says that a coordinate system is introduced on the plane. A rectangular Cartesian coordinate system is determined by a scale segment for mea- suring lengths and two mutually perpendicular axes. The point of intersection of the axes is usually denoted by the letter O and is called the origin, while the axes themselves are called the coordinate axes. As a rule, one of the coordinate axes is horizontal and the right sense is positive. This axis is called the abscissa axis and is denoted by the letter X or by OX.On the vertical axis, which is called the ordinate axis and is denoted by Y or OY , the upward sense is usually positive (see Fig. 4.2a). The coordinate system introduced above is often denoted by XY or OXY . ()c ()a ()d ()b O O O O A A A II I Left Right Upper half-plane Lower half-plane III IV half-planehalf-plane X Y X X X X Y Y Y Y Figure 4.2. A rectangular Cartesian coordinate system. 4.1. POINTS,SEGMENTS, AND COORDINATES ON LINE AND PLANE 79 The abscissa axis divides the plane into the upper and lower half-planes (see Fig. 4.2b), while the ordinate axis divides the plane into the right and left half-planes (see Fig. 4.2c). The two coordinate axes divide the plane into four parts, which are called quadrants and numbered as shown in Fig. 4.2d. Take an arbitrary point A on the plane and project it onto the coordinate axes, i.e., draw perpendiculars to the axes OX and OY through A. The points of intersection of the perpendiculars with the axes are denoted by A X and A Y , respectively (see Fig. 4.2a). The numbers x = OA X , y = OA Y ,(4.1.2.1) where OA X and OA Y are the respective values of the segments −→ OA X and −→ OA Y on the abscissa and ordinate axes, are called the coordinates of the point A in the rectangular Cartesian coordinate system. The number x is the first coordinate, or the abscissa,ofthe point A,andy is the second coordinate, or the ordinate, of the point A. One says that the point A has the coordinates (x, y) and uses the notation A(x, y). Example 1. Let A be an arbitrary point in the right half-plane. Then the segment −→ OA X has the positive sense on the axis OX, and hence the abscissa x = OA X of A is positive. But if A lies in the left half-plane, then the segment A X has the negative sense on the axis OX, and the number x = OA X is negative. If the point A lies on the axis OY , then its projection on the axis OX coincides with the point O and x = OA X = 0. Thus all points in the right half-plane have positive abscissas (x > 0), all points in the left half-plane have negative abscissas (x < 0), and the abscissas of points lying on the axis OY are zero (x = 0). Similarly, all points in the upper half-plane have positive ordinates (y > 0), all points in the lower half-plane have negative ordinates (y < 0), and the ordinates of points lying on the axis OX are zero (y = 0). Remark 1. Strictly speaking, the coordinate system introduced above is a right rectangular Cartesian coordinate system.Aleft rectangular Cartesian coordinate system can, for example, be obtained by changing the sense of one of the axes. There also exist right and left oblique Cartesian coordinate systems,wherethe coordinate axes intersect at an arbitrary angle. Remark 2. A right rectangular Cartesian coordinate system is usually called simply a Cartesian coordinate system. 4.1.2-2. Transformation of Cartesian coordinates under parallel translation of axes. Suppose that two rectangular Cartesian coordinate systems OXY and  O  X  Y are given and the first system is taken to the second by the translation of the origin O of the first system to the origin  O of the second system. Under this translation, the axes preserve their directions (the respective axes of the systems are parallel), and the origin moves by x 0 in the direction of the OX-axis and by y 0 in the direction of the OY -axis (see Fig. 4.3a). Obviously, the point  O has the coordinates (x 0 , y 0 ) in the coordinate system OXY . Let an arbitrary point A have coordinates (x, y) in the system OXY and coordi- nates (ˆx,ˆy) in the system  O  X  Y . The transformation of rectangular Cartesian coordinates by the parallel translation of the axes is given by the formulas x =ˆx + x 0 , y =ˆy + y 0 or ˆx = x – x 0 , ˆy = y – y 0 . (4.1.2.2) 4.1.2-3. Transformation of Cartesian coordinates under rotation of axes. Suppose that two rectangular Cartesian coordinate systems OXY and O  X  Y are given and the first system is taken to the second by the rotation around the point O by an angle α (see Fig. 4.3b). . cosine of each part is equal to the product of sines of the two parts not adjacent to it, as well as to the product of the cotangents of the two parts adjacent to it. References for Chapter 3 Alexander,. 3.6. Solution of spherical triangles No. Three parts specified Formulas for the remaining parts 1 Three sides a, b, c The angles α, β ,and are determined by the half-angle formulas and the cyclic permutation. Remark value of the difference between the lengths of two sides is less than the length of the third side, a + b > c, |a – b| < c 2 Sum of two angles of a triangle The sum of two angles of a triangle