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INTRODUCTION TO ANALYTIC GEOMETRY BY PEECEY R SMITH, PH.D. N PROFESSOR OF MATHEMATICS IN THE SHEFFIELD SCIENTIFIC SCHOOL YALE UNIVERSITY AND AKTHUB SULLIVAN GALE, PH.D. ASSISTANT PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF ROCHESTER GINN & COMPANY BOSTON NEW YORK CHICAGO LONDON COPYRIGHT, 1904, 1905, BY ARTHUR SULLIVAN GALE ALL BIGHTS RESERVED 65.8 GINN & COMPANY PRIETORS BOSTON PROU.S.A. PEE FACE In preparing this volume the authors have endeavored to write a drill book for beginners which presents, in a manner conforming with modern ideas, the fundamental concepts of the subject. The subject-matter is more than the minimum required much more as is necessary to permit slightly for the calculus, but only as of some choice on the part of the teacher. text is It is believed that the complete for students finishing their study of mathematics with a course in Analytic Geometry. The authors have form of a treatise intentionally avoided giving the book the on conic sections. Conic sections naturally appear, but chiefly as illustrative of general analytic methods. Attention is called to the method of treatment. The subject is developed after the Euclidean method of definition and theorem, without, however, adhering to formal presentation. tage is obvious, for the student is of each acquisition. examples are This is worked out The advan- sure of the exact nature Again, each method stated in consecutive steps. illustrative made is summarized in a rule a gain in clearness. Emphasis has everywhere been put upon the analytic is, the student is taught to start from the equation. that shown how to to use it in any other way. work with the Many in the text. figure as a guide, but is Chapter III may side, He is warned not be referred to in this connection. The etry is object of the two short chapters on Solid Analytic Geom- merely to acquaint the student with coordinates in space iii 20206fi PREFACE iv and with the relations between surfaces, curves, and equations in three variables. Acknowledgments are due to Dr. W. A. Granville for many helpful suggestions, and to Professor E. H. Lockwood for suggestions regarding NEW some of the drawings. HAVEN, CONNECTICUT January, 1905 CONTENTS CHAPTER I REVIEW OF ALGEBRA AND TRIGONOMETRY PAGE SECTION 1. Numbers 2. Constants ........... ....... 3. The 4. Special quadratics 5. Cases 6. Variables 7. Equations in several variables Functions of an angle in a right triangle 8. quadratic. when Typical form . . 5 . 9 9 . . 11. 12. Rules for signs 13. Greek alphabet 10. 1 2 4 the roots of a quadratic are not independent Angles in general Formulas and theorems from Trigonometry Natural values of trigonometric functions 9. 1 ..." ..... . . . . 11 11 .... . . . . 12 . 14 15 15 CHAPTER II CARTESIAN COORDINATES 14. Directed line 16 15. Cartesian coordinates 17 16. Rectangular coordinates 18 17. Angles 21 18. Orthogonal projection 22 19. Lengths 20. Inclination 24 and slope 27 21. Point of division 31 22. Areas 35 23. Second theorem of projection . 40 CONTENTS vi CHAPTER III THE CURVE AND THE EQUATION SECTION PAGE 24. Locus of a point satisfying a given condition 25. 26. Equation of the locus of a point satisfying a given condition First fundamental problem 27. General equations of the straight line and 28. Locus of an equation ^ Second fundamental problem 29. 44 circle . 44 . 46 . 50 . 52 ......... 30. Principle of 31. comparison Third fundamental problem. 32. Symmetry 33. Further discussion 34. Directions for discussing an equation 35. Points of intersection 36. Transcendental curves Discussion of an equation . . . . . . . . 55 60 . . 53 .65 66 . .67 69 t 72 CHAPTER IV THE STRAIGHT LINE AND THE GENERAL EQUATION OF THE FIRST DEGREE 37. Introduction . .... . . 76 39. The degree of the equation of a straight line The general equation of the first degree, Ax 40. Geometric interpretation of the solution of two equations of the 38. first -f By + C = 76 . . 80 degree determined by two conditions 42. The equation of the straight line in terms of its slope and the coordinates of any point on the line 41. Straight lines 43. 44. 45. 46. 47. The equation 91 the straight line in terms of its intercepts The equation/of the straight line passing through two given points The normal form of the equation of the straight line ... ........ The distance from a line to a point The angle which a line makes with a second line The system of lines parallel to a given line The system of lines perpendicular to a given line The system of lines passing through the intersection lines . . 86 87 88 92 96 100 107 . . lines 83 104 Systems of straight 49. 61. . . 48. 50. 77 of . 108 two given HO CONTENTS vii CHAPTER V THE CIRCLE AND THE EQUATION SECTION 52. The 53. Circles determined 54. general equation of the circle Systems a; 2 + y 2 + Dx + Ey + F = PAGE ... .115 116 by three conditions 120 of circles CHAPTER VI POLAR COORDINATES 125 55. Polar coordinates 56. 67. Locus of an equation Transformation from rectangular to polar coordinates 58. Applications 132 59. Equation of a locus 133 CHAPTER 126 . . . 130 VII TRANSFORMATION OF COORDINATES 60. Introduction 61. . 136 . ... 136 Translation of the axes 62. Rotation of the axes . 63. General transformation of coordinates 64. Classification of loci- 65. Simplification of equations 66. Application to 138 . . 139 140 by transformation of coordinates equations of the first and second degrees . . CHAPTER 141 144 . VIII CONIC SECTIONS AND EQUATIONS OF THE SECOND DEGREE .... 68. Equation in polar coordinates Transformation to rectangular coordinates 69. Simplification and discussion 70. Simplification and discussion of the equation in rectangular coordinates. Central conies, e ^ 1 71. Conjugate hyperbolas and asymptotes 72. The 67. . . nates. The parabola, e = 1 . . . . . 149 154 of the equation in rectangular coordi. . equilateral hyperbola referred to . : . .154 . 158 165. its asymptotes . . . 167 CONTENTS viii PAGE SECTION 74. Focal property of central conies Mechanical construction of conies 75. Types of 73. 168 .... 76. loci of equations of the second degree Construction of the locus of an equation of the second degree 77. Systems of conies 168 170 173 . 176 CHAPTER IX TANGENTS AND NORMALS ........ The slope of the tangent Equations of tangents and normals 80. Equations of tangents and normals to the conic sections 78. 180 182 79. 81. 82. y 83. Tangents to a curve from a point not on the curve Properties of tangents and normals to conies .... Tangent 184 . . . . . 186 188 in terms of its slope 192 CHAPTER X CARTESIAN COORDINATES IN SPACE 196 84. Cartesian coordinates 85. 198 Lengths Orthogonal projections. CHAPTER XI SURFACES, CURVES, AND EQUATIONS 86. Loci in space 87. Equation of a surface. 201 First fundamental 90. problem Equations of a curve. First fundamental problem Locus of one equation. Second fundamental problem Locus of two equations. Second fundamental problem 91. Discussion 88. 89. . the . . . curve. Third fundamental equation of a surface. Third fundamental equations of a 206 problem 92. Discussion of the 207 problem __ 205 205 . . 201 202 . . . of . 93. Plane and straight line 94. The sphere 95. Cylinders 96. Cones 97. Non-degenerate quadrics 210 211 . . . 213 . 214 . . . . . . . .215 UNIVERSITY', OF ANALYTIC GEOMETRY CHAPTER I REVIEW OF ALGEBRA AND TRIGONOMETRY 1. Numbers. The numbers arising in carrying out the operatwo kinds, real and imaginary. real number is a number whose square is a positive number. tions of Algebra are of A Zero also is a real number. A pure tive imaginary number is a number whose square is a neganumber. Every such number reduces to the square root of a negative number, and hence has the form b V 1, where b is a 2 and number, (V I) An imaginary or complex number is a number which may be written in the form a + b 1, where a and b are real numbers, and b is not zero. Evidently the square of an imaginary number is in general also an imaginary number, since =1. real V + b V^T) = 2 (a which 2. is imaginary Constants. A if a is a2 - b + 2 ab V^l, 2 not equal to zero. quantity whose value remains unchanged is called a constant. Numerical or absolute constants retain the same values in problems, as 2, 3, Vl, all TT, etc. Arbitrary constants, or parameters, are constants to which any one of an unlimited set of numerical values may be assigned, and these assigned values are retained throughout the investigation. Arbitrary constants are denoted by letters, usually by letters from the part of the alphabet. In order to increase the number of symbols at our first 1 ANALYTIC GEOMETRY 2 it disposal, example is convenient to use primes (accents) or subscripts or both. For : Using primes, "a prime or a first"), of' (read "a double prime or a second"), third"), are all different constants. a' (read a'" (read "a Using subscripts, &! (read "6 " one"), 6 a (read 6 two"), are different constants. Using both, c/(read "c one prime"), c8 " (read "c three double prime"), are different constants. 3. The quadratic. Typical form. Any quadratic equation in the may by transposing and collecting the terms be written Typical Form Ax 2 + Bx (1) + C = 0, in which the unknown is denoted by x. The coefficients A, B, C are arbitrary constants, and may have any values whatever, except that A cannot equal zero, since in that case the equation would be no longer of the second degree. C is called the con- stant term. The left-hand member Ax* (2) + Bx + C called a quadratic, and any quadratic may be written in this Typical Form, in which the letter x represents the unknown. 4 A C is called the discriminant of either (1) The quantity B 2 is or (2), and That is, is denoted by A. the discriminant A of a quadratic or quadratic equa- tion in the Typical Form is equal to the square of the coefficient of the first power of the unknown diminished by four times the product of the coefficient of the second power of the unknown by the constant term. The roots of a quadratic are those numbers which make the quadratic equal to zero when substituted for the unknown. The roots of the quadratic (2) are also said to be roots of the quadratic equation (1). to satisfy that equation. A root of a quadratic equation is said REVIEW OF ALGEBRA AND TRIGONOMETRY In Algebra shown that it is (2) or (1) has two roots, Xi and obtained by solving (1), namely, (3) Adding these values, we have x (4) -f- B = x ^Vw* \& Multiplying gives XiX z (5) = 'A Hence Theorem I. T^e sum of the roots of a quadratic is equal to the power of the unknown with its sign changed coefficient of the first divided by the coefficient of th& second power. The product of the roots equals the constant term divided by the coefficient of the second power. The quadratic + Ax* (6) as (2) may may be written in the form Bx + C =*A(x - x,} (x - * 2), be readily member and shown by multiplying out the right-hand substituting from (4) and (5). For example, since the roots of 3 a; 2 tically 3 32-43 + 1 = 3*(x - 1) (x - The character of the 4 a; +1= are 1 and , w$ have iden- *). roots x l and x z as numbers ( 1) when the C are real numbers evidently depends entirely A, B, the discriminant. This dependence is stated in upon coefficients Theorem and II. If the coefficients of a quadratic are real numbers, if the discriminant be denoted by A, then 4/i *The when A when A when A = is is is positive the roots are real and unequal; zero the roots are real and equal ; negative the roots are imaginary. is read "is identical with," and means that the two expressions sign connected by this sign differ only inform. ANALYTIC GEOMETRY 4 In the three cases distinguished by Theorem II the quadratic may be written in three forms in which only real numbers appear. These are ' C = A (xXi) (x x z ), from (6), if A is positive ; Ax 2 + Bx + C = A(xx ) 2 from (6), if A is zero ; Ax 2 + Bx -f- 1 , AC-B \2 Ax* + Bx + C B = A\[-/(* + o~r) The proved thus last identity is adding and subtracting 4. C 2>A ) L\ L - H 2 ~] Y if 7-7* ^ A J ^ is negative. : within the parenthesis. If one or both of the coefficients Special quadratics. B and in (1), p. 2, is zero, the quadratic is said to be special. CASE C I. Equation = (1) (1) 0. now becomes, by factoring, Ax z + Bx = x(Ax + B) = 0. r> Hence the roots are a quadratic equation is zero. And stant term is x1 = 0, x2 = Therefore one root of yl zero if the constant term of that equation is a root of a quadratic, the con- conversely, if zero must disappear. For we have C = 0. if x = satisfies (1), p. 2, substitution CASE II. Equation B= (1), p. 2, (2) From Theorem (3) 0. now becomes Ax 2 I, p. 3, x^ +C= -f x = 2 0. 0, that !=-* is, by REVIEW OF ALGEBRA AND TRIGONOMETRY if Therefore, the coefficient of the in a quadratic equation but have opposite signs. first 5 power of the unknown the roots are equal numerically Conversely, if the roots of a quadratic is zero, equation are numerically equal but opposite in sign, then the coefficient of the first power of the unknown must disappear. For, sum since the B= of the roots is zero, CASE B=C= III. Equation have, by Theorem I, (1), p. 2, 0. now becomes Ax 2 = (4) Hence the always 0. roots are both equal to zero, since this equation 0, the coefficient A bei ng, by hypothesis, that x 2 requires 5. we must 0. = v different Cases when from zero. the roots of a quadratic are not independent. between the roots Xi and x z of the Typical If a relation exists :Form Ax 9 + Ex + C = 0, then this relation imposes a condition upon the coefficients A, B, and C, which is expressed by an equation involving these constants. B 2 For example, if the roots are equal, that -4: AC = 0, by Theorem II, p. 3. Again, if one root is zero, then x& 2 = Theorem p. I, is, ; if Xi hence =x C 2, then = 0, by 3. This correspondence may be stated in parallel columns thus Quadratic in Typical Relation between the roots : Form Equation of condition by the satisfied coefficients In many problems the coefficients involve one or more arbitrary constants, and it is often required to find the equation of condi- by the latter when a given relation exists between Several examples of this kind will now be worked out. tion satisfied the roots. ANALYTIC GEOMETRY 6 Ex. 1. What must be the value of the parameter k if zero -6z + 2 -3fc-4=0? B = - 6, C = k' - 3 k - 4. By Case is a root of the equation 2x2 (1) Solution. is Here A = 2, fc 2 C= a root when, and only when, Ex. zero or 1. For what values of k are the roots of the equation 2. kx 2 real =4 k Solving, I, p. 4, 0. and equal Solution. + 2kx-4x = 2-3k ? Writing the equation in the Typical Form, we have + fcx2 (2) - k (2 + 4)x (3 k - 2) = 0. Hence, in this case, A = k, B = 2k-4, C = 3k-2. Calculating the discriminant A, A= (2 k - = _8fc2 By Theorem A= II, p. 3, - 2 4) we 4k get k - 2) = -8(A:2 + (3 -8A;-f 16 & - 2). the roots are real and equal when, and only when, ' k* .-. k= Solving, + - k 2 or 2 = 0. Ans. 1. Verifying by substituting these answers in the given equation when fc=-2, when fc= 1, (2) becomes (2) becomes the equation the equation "$Hence, for these values of formed as in ^ Ex. a^6, 3. fc, x2 the left-hand fc, 2 +1=0, member : or -2(x-f 2) 2 =0; or (x 1) may be of (2) m (3) trans- equation of condition must be satisfied by the constants the roots of the equation if (62 + a2 m 2) y2 + + a2fc2 _ 2a? kmy a2 &2 =Q are equal ? Solution. 2 =0. (7), p. 4. What and 2x2 -8x-8=0, (2) The equation already in the Typical (3) is Form ; hence A = 6 + a m B = 2 a km, C = a?k2 - a 6 2 2 2 2 2. 2 , By Theorem II, p. 3, A= the discriminant 4 a4 fc2 m2 - 4 (62 + A must a2 m 2 ) (a2 &2 vanish - a2 62 ) Multiplying out and reducing, a2 &2 (fc 2 - a2 m 2 - 62 ) = 0. Ans. ; hence = 0. REVIEW OF ALGEBRA AND TRIGONOMETRY Ex. 7 For what values of k do the common solutions of the simultaneous 4. equations 3x + 4y = (4) x2 (5) become + y2 fc, = 25 identical ? Solution. (4) for y, Solving we have = t(*-3x), y (6) and arranging Substituting in (5) 25 x (7) Let the roots of 2 - 6 kx +~k* be x\ and x 2 (7) corresponding values y\ and y z of = yi (8) and we shall have two in the Typical Then . y, Form gives - 400 = 0. substituting in (6) will give the namely, i(& common solutions and (xi, y\) y2 ) (x 2 , of (4) and (5). But, by the condition of the problem, these solutions must be identical. Hence we must have = xi (9) If, however, the be equal. first X 2 and y l of these is = true (xi y2 . = x2 ), then from (8) y\ and y^ will also Therefore the two and only when, criminant A common of (7) vanishes .-. Solving, Verification. solutions of (4) and become identical when-, (5) the roots of the equation (7) are equal; that (Theorem is, when A; Substituting each value of k in (7), when A; =25, the equation (7) becomes x 6x + 9=0, or (x 3) 2 =0 when k= -25, the equation (7) becomes x2 + 6x + 9=0, or (x-f 3) 2 =0; ; (6), 3, 25 = = and x 3, Therefore the two common when k = = of these values of if k if k = = .. x=3 .-.x= ; 3. substituting corresponding values of k and x, 25 and x when k dis-. 36 2 - 100 (fc* - 400) = 0. = A* 625, 25. Ans. k = 25 or A= 2 Then from the II, p. 3). fc, we have y = ^(25 9)=4; we have y = $ ( 25 -f 9) = solutions of (4) and (5) 4. are identical for each namely, 25, the 26, the common common solutions reduce to x solutions reduce to x = = 3, =4 3, y = y ; 4. Q.E.D. ANALYTIC GEOMETRY 8 PROBLEMS 1. Calculate the discriminant of each of the following quadratics, deterof the roots, and write each mine the sum, the product, and the character quadratic in one of the forms (7), p. 4. 2x2 _ 6x + 4. x2 - 9x - 10. 1 - x - x2 (a) (b) (e) . (c) 2. (d) For what (f) 2k - 3x2 + 6x - &2 + (b) 3. For what (a) (c) (d) 5x2 - 3x = 0. (f) (g) Ans. k Ans. k = l. = - I or 3. Verify your answers. Ans. k Ans. k Ans. k 0. + =-f = 6. = ^-. . Ans. None. Ans. k = &. Ans. None. Ans. k=-$. = 0. x2 -f kx + k2 + 2 = 0. x2 - 2 kx - k - I = 0. (e) 4. 0. -3x-l = 0. x2 - kx + 9 = 0. 2 kx2 + 3 kx + 12 = 2 x2 + kx - 1 = 0. (b) 3 real values of the parameter are the roots of the following equations equal ? fcx 2 parameter k will one root of each of the real values of the following equations be zero ?. 2 - 3 k2 + 3 = (a) 6x '+ 5 kx 4x2 - 4x + 1. 5x2 + lOx + 5. 3x2 - 5x - 22. 5fc2 Derive the equation of condition in order that the roots of the following may be equal. equations (a) m x + 2 kmx - 2px = - k2 (b) x2 2 (c) 5. 2 Ans. p (p - 2 km) = 0. Ans. p (m 2p - 2 6) = 0. Ans. fr2 = 2 a 2 m. . + 2 mpx + 2 6p = 0. 2 mx 2 + 2 bx + a2 = 0. For what real values of the parameter do the common solutions of the following pairs of simultaneous equations become identical ? = k, x2 + y2 = 5. -4ns. k = (a) x + 2 y (b) (c) (d) (e) (f) (g) (h) = mx 1, x2 = 4 y. 2 x - 3y = 6, x 2 + 2 x = 3 y. y = mx + 10, x2 + y2 = 10. Ix + y - 2 = 0, x2 - 8 y = 0. x + 4 y = c, x2 + 2 y 2 = 9. 2 y = 0, x + 2 y = c. a2 + y 2 - x x2 + 4 y 2 8 x = 0, mx - y - 2 m = 0. y 6. If the are to (a) (b) (c) Ans. -4ns. .Ans. 5. m= 1. = 0. m= 3. 6 Ans. None. = = ^.ns. c Ans. c ^.ris. None. 9. or 5. common solutions of the following pairs of simultaneous equations become identical, what is the corresponding equation of condition + ay = a&, y2 = 2px. = mx + 6, .Ax2 + .By = 0. y y = m(x- a), By 2 + Dx = 0. bx . ? + ap) = 0. B (mzB - 4 bA) = 2 - D) = D(4 am Ans. ap (2 62 ^Ins. -Ans. 0. 0. REVIEW OF ALGEBRA AND TRIGONOMETRY 6. A Variables. variable is a quantity to which. jri jjfr investigation, an unlimited number of values can be assigned. In a particular problem the variable may,< in general, assume any value within certain limits imposed by the nature of th$ problem^ It is convenient to indicate these limits by inequalities. For example, if the variable x can assume any value between 2 and 5, that x must be greater* than 2 and less than 5, the simultaneous inequalities is, if *>-2, x<5, are written in the more compact form -2<z<5. Similarly, if the conditions of the problem limit the values of the variable x to any negative number less than or equal'to 2, and to any positive number greater than or equal to 5, the conditions 2 OT x = x ^ x< are abbreviated to 2, and x > 5 or x = 5 2 and x ^5. Write inequalities to express that the variable to 5 inclusive. (a) x has any value from 2 or greater than 1. (b) y has any value less than 8 nor greater than (c) x has any value not less than 7. 2. In Analytic Geometry Equations in several variables. we are concerned chiefly with equations in two or more variables. An equation is said to be satisfied by any given set of values of the variables if the equation reduces to a numerical equality these values are substituted for the variables. when For example, x = y 2, = 3 satisfy the equation + 32,2=35, + 3(- 3)2 = 35. 2x2 since 2(2)2 Similarly, x = 1, y = 0, z = 4 satisfy the equation 2z2_3 y 2 + 2;2_i 8 = since 2 ( 1)2 3 X + ( 4) 2 18 o, = 0. * The meaning of greater and less for real numbers ( 1) is defined as follows a is when a - b is a positive number, and a is less than b when a - b is negative. Hence any negative number is less than any positive number and if a and b are both negative, then a is greater than b when the numerical value of a is less than the numer: greater than b ; ical value of 6. Thus 3 < 5, but - 3 > - 5. the inequality sign. Therefore changing signs throughout an inequality reverses ANALYTIC GEOMETRY 10 An equation said to be algebraic in is any number of variables, can be transformed into an equation example each of whose members is a sum of terms of the form ax my n z p, for x, y, 2, if it where a is a constant and m, n, p are positive integers or zero. Thus the equations x* + x 2y 2 - z +2 a: 5 = 0, are algebraic. The equation is x* + y* = a? algebraic. For, squaring, we get + 2 x$y* + y=a. 2 x*y* = a x a2 + x 2 + yz 2 ax x Transposing, Squaring, 4 xy Transposing, x2 The + y2 2 ax 2 xy 2 ay y. + 2 xy. + a2 = 0. 2 ay Q.E.D. an algebraic 'equation is equal to the highest of An algebraic equation is said to its terms.* any with to the variables when all its terms are Ibe arranged respect degree of degree of transposed to the left-hand side and written in the order of descending degrees. For example, to arrange the equation 2x' 2 + 3y' + 6x'-2x'y'-2 + z/3 = x'*if - y'* with respect to the variables z', y', x '3 This equation _ yfvtf + 2 X is '2 we transpose and rewrite the terms in the order 2x'y' + y'2 + 6a/ + 3y' 2 = 0. of the third degree. An equation which is not algebraic is said to be transcendental. Examples of transcendental equations are y = sinx, y = 2*, logy=3x. PROBLEMS the 1. Show that each of the following equations is algebraic; arrange terms according to the variables x, y, or x, y, z, and determine the degree. (e) * The degree of y = 2 + Vx2 -2x-5. any term is the sum of the exponents of the variables in that term, REVIEW OF ALGEBRA AND TRIGONOMETRY + y .= x (f) 6 11 + V2x2 -6x W*=-|j> y (h) 2. Show = ^ix + B + Vix'J 4- JMTx + N. * homogeneous quadratic that the Ax2 + Bxy + Cy* may be written in one of the three forms below analogous to 2 4 satisfies the condition given the discriminant A =B CASE CASE II. CASE III. I. AC Ax + Bxy + Cy = A(x- hy] (x - y), .Ax 2 + Bxy + Cy 2 = A(x- ky) 2 if A = ^1x2 + Bxy + Cy* = A + 2 2 I2 , (7), p. 4, if : if A> ; ; ', if A<0. [(x Functions of an angle in a right triangle. In any right triangle one of whose acute angles is A, the functions of A are 8. defined as follows sin : opposite A = -** side A= (/ , / esc A= - sec adjacent side -=-* , hypotenuse A adjacent side' hypotenuse _ opposite side tan A = adjacent side From cot the above the theorem B ^ opposite side' hypotenuse cos hypotenuse .4 J = adjacent easily derived is side ^-. opposite side : In a right triangle a side is equal to the product of the hypotenuse and the sine of the angle opposite to that side, or of the hypotenuse and the cosine of the angle adjacent to that side. A C b an angle XOA 9. considered is OA erated by the line an initial positive in as general. gen- In Trigonometry < rotating from The angle is position OX. when OA rotates from OX counter-clockwise, the Angles direction of and negative rotation of when OA s clockwise. *The coefficients A, B, Cand the numbers fj, I 2 are supposed real. ANALYTIC GEOMETRY 12 The fixed line terminal OX is called the initial line, the line OA the line. Measurement of angles. There are two important of measuring angular magnitude, that is, there are methods two unit angles. Degree measure. The unit angle tion, and is called a degree. is ^J^ of a complete revolu- The unit angle is an angle whose subtendequal to the radius of that arc, and is called a radian. fundamental relation between the unit angles is given by Circular measure. ing arc The is the equation = IT radians 180 degrees Or also, by solving (IT = 3.14159 ). this, 1 degree = 1 radian = ^= loU 80 1 .0174 radians, = 57.29 degrees. 7T These equations enable us to change from one measurement to In the higher mathematics circular measure is always another. used, and will be adopted in this book. The generating line is conceived of as rotating around through as many Any real conversely, 10. we revolutions as number is any angle is Hence the important choose. result the circular measure of some angle, measured by a real number. Formulas and theorems from Trigonometry. 1. cotx = tanx 2. tanx = sinx ; secx ; cotx cosx 4. ; cscx = smx = cosx sin x cosx 3. = + cos2 x = 1 1 + tan2 x = sec2 x 1 + cot2 x = csca z. esc x sin (- x) = - sin x esc (- x) = = = secx; cosx; x) sec( x) cos( cotx. tan(- x) = tanx cot (- x) = sin2 x ; ; ; ; ; : and REVIEW OF ALGEBRA AND TRIGONOMETRY 5. 13 - x) = sinx sin (?r + x) = - sinx - x) = - cosx cosjtf ._&.= -<?osxj tanx jftan (ie + x) = tanx tan (it x) = sin (if ; cos (x ; ; ; 6. -x) sin(- = cosx; sin (| cos(--x)=sinx; cos^ sin (2 Tt 8. sin (x + 9. sin (x x) = sin ( = x) cosx; + x) = - sinx; + x) = - cotx. tanf- tan(- -x)=cotx; 7. + x) = O sin x, etc. o y) = sinx cosy + cos x sin y) = sin x cos y cos x sin y. 10. cos (x + y} = cos x cos y sin x sin y. 11. cos(x y) = cos x cosy + sinx sin y. 12 tan (x 14. sin 2 x + y) -- + tan y = tanx I - tanx tany = 2 sin x cos x ; cos 2 x y. - y) = tanx tany + tanx tany . 13. tan (x = sin 2 cos 2 x x ; 1 tan 2 x = 2 tanx x + cosx tan-= ; 16. Theorem. Law In any triangle the sides are proportional of sines. to the sines of the opposite angles; a b c Lqw of cosines. In any triangle the square of a side of the squares of the two other sides diminished by twice the product of those sides by the cosine of their included angle ; 17. Theorem. equals the that sum a2 is, 18. Theorem. = 62 + c2 2 be cos A. Area of a triangle. The area of any triangle equals one two sides by the sine of their included angle half the product of that is, area ; = ab sin C= i be sin A = ^ ca sin B. ANALYTIC GEOMETRY 14 11. Natural values of trigonometric functions. Angle in Radians REVIEW OF ALGEBRA AND TRIGONOMETRY Angle in Radians 15