THE UNIVERSITY OF CHICAGO MATHEMATICAL SERIES Eliakim Hastings Moore General Editor THE SCHOOL OF EDUCATION TEXTS AND MANUALS George William Myers Editor THIRD-YEAR MATHEMATICS for SECONDARY SCHOOLS THE UNIVERSITY OF CHICAGO PRESS CHICAGO, ILLINOIS THE BAKER & TAYLOR COMPANY NEW YORK THE CAMBRIDGE UNIVERSITY PRESS LONDON THE MARUZEN-KABUSHIKI-KAISHA TOKYO, OSAKA, KYOTO, FHKUOKA, SENDAI THE MISSION BOOK COMPANY SHANGHAI GOTTFRIED WILHELM LEIBNITZ GOTTFRIED WILHELM LEIBNITZ was born at Leip- zig in 1646 and died at Hanover in 1716. His was the genius that was both precocious and universal. Even from early childhood he overcame the most trying obstacles. He proved himself of extraordinary qualities before he was twelve. Though he had mastered the ordinary texts in mathematics, philosophy, theology, and law before he was twenty, not until he reached the age of twenty-six, when he was sent on a political errand to Paris, where he made the acquaintance of Huygens, did he become strongly interested in mathematics. He achieved eminence of the highest order in mathematics, philosophy, the- ology, law, and languages. For forty years following 1676 he held the post of librarian of the house of Brunswick and Hanover, earning the highest honors and distinctions in the services of his house only to be cast aside in old age by the existing head of the Brunswick family when he became George I of England. Leibnitz' political and religious papers touching affairs of the dynasty from 1673-1713 constitute an indispensable contribu- tion to the history of his time. His place in the history of philosophy is even larger than it is in mathematics. The Leibnitzian system of philosophy con- stitutes a most important epoch in the history of philosophical doctrines. Leibnitz' life denies the prevailing idea that mathe- matical genius is necessarily narrow and specialized. It also furnishes a conspicuous example of a most important contribu- tor to the advance of - mathematics who was not by profession, at any time during his life, a teacher of the science. His chief services to mathematical science consist of his inde- pendent invention of the language, if not the substance, of the differential calculus, his work on osculating curves, his funda- mental work on the theory of envelopes, his explanation of the method of expansion of functions by indeterminate coefficients, and his recognition of the theory of determinants and his devel- opmental work on the theory. His work displays great skill in analysis, but like the work of most geniuses it is unfinished and characterized by frequent errors. But he blazed out many new routes through the mathe- matical regions which men of lesser genius were aided later in converting into comfortable highways and thoroughfares. The later years of his life were embittered by a contest with the overzealous friends of Newton over the question of priority of invention of the calculus as between him and Newton. Sub- sequent times have seemed to settle the controversy on the basis that Leibnitz and Newton were independent inventors of the calculus, and that most certainly the modern notation of the calculus is due to Leibnitz. For more detail about this famous controversy any history of mathematics may be consulted. In character Leibnitz was quick-tempered, intolerant, selfish, and inordinately conceited. But the products of his genius will ever adorn and enrich the pages of mathematical and philo- sophical history. [See Ball's or Cajori's or Tropfke's History.] GOTTFRIED WILHELM LEIBNITZ Third-Year Mathematics for Secondary Schools With Logarithmic and Trigonometric Tables and Mathematical Formulas BY ERNST R. BRESLICH Head of the Department of Mathematics in the University High School, The University of Chicago THE UNIVERSITY OF CHICAGO PRESS CHICAGO, ILLINOIS Copyright 1917 By The University of Chicago All Rights Reserved Published September 191 7 Second Impression January 1920 u -*jl? *b "^ Composed and Printed By The University of Chicago Press Chicago. Illinois. U.S.A. EDITOR'S PREFACE This third unit of Mr. Breslich's course in general mathematics for high schools aims primarily to carry- forward the spirit and the method of the two former volumes. By using chapter xv at the beginning of the year as a syllabus for reviewing the ground covered in the previous work in geometry, this volume may be readily taken up by classes whose previous work has been in standard courses of algebra and geometry. To aid pupils who do not go beyond the high school, and whose first two years of training have been in corre- lated mathematics, to become sufficiently familiar with the standard special mathematical methods and principles of algebra, geometry, and trigonometry, as such, and to command the existing literature of these branches, the type of correlation here used is what may be termed topical. A certain type of subject-matter is allowed emphasis for a sufficient time to enable pupils to master the appropriate type of methodology. This will seem, to a superficial critic a departure from the close type of correlation of the two preceding texts. This variation is, however, intentional, to the end that algebra, geometry, and trigonometry, as such, may be firmly grasped by the pupil while he is yet in high school. The correct descrip- tion of the prevailing procedure here is: Isolation in details, but correlation in major matters. This assures any real benefits of an isolated type of treatment without losing the more important values of correlation. Mathe- matical training must foster both concentration and generalship. Classroom experience has verified the pro- priety of this type of correlation for third-year classes. vii 420821 viii EDITOR'S PREFACE The author would request open-minded teachers to give the form of reconstructed mathematics herewith presented a fair classroom test. He will gladly accept the issue of such a test. Superintendents and principals should feel that here is something of the sort their spokes- men have been urging, and see to it that the text be given a fair test. Better things can hardly be obtained except through the testing of different methods. Of the methods deserving of a classroom test, those that have proved successful in particular instances are most worth while. The material of this volume belongs to this class. May its friends become as numerous as are those of its companion volumes! G. W. Myers Chicago, III. August, 1917 AUTHOR'S PREFACE This book is the third of the series of textbooks on secondary mathematics. It is designed primarily as a third unit of a year's work to follow the first two unit- courses worked out by the author in First-Year Mathe- matics and Second-Year Mathematics. It completes the study of high-school algebra, trigonometry, and solid geometry. In accordance with the general plan of the series, the book aims to teach in combination mathematical topics which are naturally closely related to each other even though drawn from separate mathematical subjects. Such an arrangement has the advantage of developing the subject of secondary-school mathematics in a sequence which is both psychological and logical. Indeed, the student's understanding of the meaning and the utility of the subject is deepened to such an extent that he is better able to appreciate the scientific character of mathe- matics than when he is studying the separate subjects. The result is that he is more disposed to continue the study of mathematics. Through proper- correlation the whole third-year work can be better motivated and becomes more concrete, each subject gaining from the study of others. For example, the student appreciates the need of studying the theory of logarithms because of their usefulness as a tool for solving problems in trigonometry. He further sees that he must master the theory of exponents in order that he may understand the fundamental principles of the theory of logarithms. ix X THIRD-YEAR MATHEMATICS In the study of simultaneous linear and quadratic equations, in connection with intersecting straight lines and curves, the abstract processes of solving the various types of systems of equations are represented concretely and are easily understood and remembered. Through the graphical representation of equations of the form y =f(x), where f(x) is either a polynomial or a trig- onometric function, he learns to appreciate the funda- mental concept of functional correspondence, which is of greatest value to him as a natural introduction to analyti- cal geometry. Thus correlation removes the disadvantages of the topical plan without losing its advantages. It arouses and holds the student's interest. Many students who find third-year algebra as a separate subject too abstract and uninteresting take great delight in a third-year course in which algebra and trigonometry are correlated, because they enjoy the study of trigonometry and its applications. Special attention is directed to the following features of this course : 1. Reviews are carried on at frequent intervals throughout the course. Hence the introductory review found in most third-year texts has been omitted. Since the course begins with new work, the student has at once the exhilaration of taking a step in advance, while at the same time he is reviewing in a way that gives him a new and higher view of the subject-matter. His time and effort are economized through the avoidance of useless repetitions. 2. At the end of each chapter the principal facts are summarized. 3. The last chapter of the book is a syllabus of all the theorems of plane and solid geometry which were studied AUTHOR'S PREFACE xi in the preceding courses. This may be used for reference purposes while the student is taking the course. It will also be found effective as a basis for a final review of plane and solid geometry as such. The list, at the end of the book, which gives all important mathematical formulas so far studied serves the same purpose. 4. The material has been carefully selected with a view to stimulating independent thinking and to prepar- ing the student for collegiate mathematics. The number of supplementary exercises is sufficiently large to furnish the drill needed to enable the student to meet the require- ments of college entrance examinations. For this purpose many problems taken from entrance examinations of various colleges have been included. 5. The historical notes distributed throughout the book add to the interest of the student in genetic phases of the work. The portraits appearing as inserts have been taken from the Philosophical Portrait Series, published by the Open Court Publishing Company, Chicago. The author takes great pleasure in acknowledging his indebtedness to his colleagues in the department of mathematics for many valuable constructive criticisms. He is indebted also to Principal F. W. Johnson, of the University High School, and to Professor Charles H. Judd, Director of the School of Education, for sympathetic appreciation and encouragement during the preparation of the manuscript. Ernst R. Breslich CONTENTS PAGE Study Helps for Students xvii CHAPTER I. Functions. Equations in One Unknown . . 1 Linear Function 4 Direct Variation 6 Quadratic Function 8 Graphical Solution of Equations of Degree Higher than the Second 10 Synthetic Division. Remainder Theorem ... 12 Equations of Degree Higher than the Second Solved by Factoring . . N 16 The Function- 19 x II. Trigonometric Functions 25 Angles in General 25 Values of the Trigonometric Functions Found by Means of a Drawing 29 Changes of the Trigonometric Functions ... 31 Graphs of the Trigonometric Functions .... 37 Trigonometric Functions of Negative Angles . . 44 Trigonometric Functions of \^ a ) m Terms of the Functions of a 46 Trigonometric Functions of (n o a ) m Terms of the Functions of a 49 III. Linear Equations 53 Linear Equations in One Unknown 53 Linear Equations in Two Unknowns .... 61 Solution of a System of Linear Equations by Determinants 62 Linear Equations with Three or More Unknowns . 68 Solution by Determinants ....,., 70 xiti xiv CONTENTS CHAPTEB PAGE .- IV. Quadratic Equations in One Unknown ... 76 Methods of Solving Quadratic Equations ... 76 Square Root of Polynomials 79 Fractional Equations . . . 81 Equations of Quadratic Form 84 Trigonometric Equations 84 Nature of the Roots of a Quadratic Equation . . 86 Relation between the Roots and the Coefficients of a Quadratic 90 Factoring 91 V. Factoring. Fractions 94 The Difference of Two Squares 94 The Sum or Difference of Like Powers .... 95 Trinomials 96 Polynomials 96 Fractions 98 Complex Fractions 99 VI. Exponents. Radicals. Irrational Equations 104 The Fundamental Laws of Positive Integral Ex- ponents 104 Zero-Exponents. Fractional and Negative Expo- nents 108 Radicals 114 Reduction of Radicals 115 Addition and Subtraction of Radicals . . . . 117 Multiplication of Radicals . 118 Division of Radicals 118 - Rationalizing the Denominator 120 Square Root of a Radical Expression . . . . 122 Irrational Equations 123 Trigonometric Equations 126 VII. Logarithms. Slide Rule 128 Labor-Saving Devices 128 Precision of Measurement 128 Logarithms ............ 131 CONTENTS XV CHAPTER PAGE Common Logarithms 134 The Table of Logarithms 136 Properties of Logarithms 142 -^Exponential Equations 146 The Slide Rule 147 VIII. Logarithms of the Trigonometric Functions. Solution of Triangles . . 153 Use of the Table of Logarithmic Functions . . 153 Use of Logarithms in the Solution of Right Tri- angles 156 Relations between the Sides and Angles of Oblique Triangles 160 Solution of Oblique Triangles 168 Area of an Oblique Triangle 176 IX. Relations between Functions of Several Angles 183 Addition and Subtraction Theorems .... 183 Functions of Double an Angle 189 Functions of Half an Angle 191 Trigonometric Equations 193 X. Binomial Theorem. Arithmetical and Geomet- rical Progressions 196 Binomial Theorem 196 Arithmetical Progression* 202 Geometrical Progression 207 Infinite Geometrical Series 212 - XL Systems of Equations in Two Unknowns In- volving Quadratics 218 Graphs of Quadratic Equations in Two Unknowns 218 Solution of Simultaneous Quadratics .... 225 Solution of Equations of Degree Higher than the Second 231 Solution of Irrational and Fractional Equations . 233 xvi CONTENTS CHAPTER PAGE XII. Areas of Surfaces 239 Polyedrons. Cylinders. Cones 239 Sections Made by a Plane 245 Areas 255 Surfaces of Revolution 264 Area of the Surface of a Sphere 267 XIII. Volumes 272 Volume of a Rectangular Parallelopiped . . . 272 Comparison of Volumes 275 Volume of a Prism 276 Volume of a Cylinder 281 Volume of a Pyramid 284 Volume of a Frustum of a Pyramid 290 Volume of a Circular Cone 291 Volume of a Frustum of a Cone 292 Volume of a Sphere 293 Volume of a Spherical Segment 295 xiv. polyedral angles. tetraedrons. spherical Polygons 302 Polyedral Angles 302 Tetraedrons 308 Spherical Angles 312 Polar Spherical Triangles 314 Symmetry and Congruence 317 Area of a Spherical Triangle 323 XV. Assumptions and Theorems of Geometry Given in the Courses of the First and Second Years 33 1 Logarithms of Numbers 353 Table of Powers and Roots 355 Table of Sines, Cosines, and Tangents of Angles from l-90 356 Formulas 357 Reductions 362 Index 365 STUDY HELPS FOR STUDENTS 1 The habits of study formed in school are of greater impor- tance than the subjects mastered. The following suggestions, if carefully followed, will help you make your mind an efficient tool. Your daily aim should be to learn your lesson in less time, or to learn it better in the same time. 1. Make out a definite daily program, arranging for a definite time for the study of mathematics. You will thus form the habit of concentrating your thoughts on the subject at that time. 2. Provide yourself with the material the lesson requires; have on hand textbook, notebook, ruler, compass, special paper needed, etc. When writing, be sure to have the light from the left side. 3. Understand the lesson assignment. Learn to take notes on the suggestions given by the teacher when the lesson is assigned. Take down accurately the assignment and any references given. Pick out the important topics of the lesson before beginning your study. 4. Learn to use your textbook, as it will help you to use other books. Therefore understand the purpose of such devices as index, footnotes, etc., and use them freely. 5. Do not lose time getting ready for study. Sit down and begin to work at once. Concentrate on your work, i.e., put your mind on it and let nothing disturb you. Have the will to learn. 1 These study helps are taken from Study Helps for Students in the University High School. They have been found to be very valuable to students in learning how to study and to teachers in training students how to study effectively. xvii xvin STUDY HELPS FOR STUDENTS 6. As a rule it is best to go over the lesson quickly, then to go over it again carefully; e.g., before beginning to solve a problem read it through and be sure you understand what is given and what is to be proved. Keep these two things clearly in mind while you are working on the problem. 7. Do individual study. Learn to form your own judgments, to work your own problems. Individual study is honest study. 8. Try to put the facts you are learning into practical use if possible. Apply them to present-day conditions. Illus- trate them in terms familiar to you. 9. Take an interest in the subject. Read the corresponding literature in your school library. Talk to your parents about your school work. Discuss with them points that interest you. 10. Review your lessons frequently. If there were points you did not understand, the review will help you to master them. 11. Prepare each lesson every day. The habit of meeting each requirement punctually is of extreme importance. CHAPTER I FUNCTIONS. EQUATIONS IN ONE UNKNOWN Function. Variable. Constant 1. The formula i = prt may be used to compute the interest, i, of a principal, p, at the rate, r, for t years. If the rate and principal remain the same, the interest, i, de- pends upon the time, t, in the sense that if one is changed the other changes correspondingly. The time and inter- est are variables, the principal and rate constants, and the interest is said to be a function of the time. Dependence of one magnitude upon another is met frequently. For example, the premium of a life-insurance policy depends upon the age of the applicant, the distance passed over by a moving body depends upon the time, the length of a circle depends upon the radius. Sometimes this dependence is expressed in the form of an equation. Thus, the length of a circle is given by the equation c = 2wr. Because of this equation, to every value of c there corresponds a definite value of r. This is often expressed by saying that c is a function of r. The symbols c and r are variables, tt is a constant. 2. Constant. A symbol which represents the same number throughout a discussion, or in a problem, is a constant. EXERCISE In the equations, A = 7rr 2 , A = ^bh, d=rt, s = jgt 2 , v = ^7rr 3 *, one letter depends upon one or more other letters for its value. Which symbols in these equations are constants ? * When such forms as r 2 , \bh, rt, IQt 2 , 2a; +3, and vV-25 first came into mathematics, they were regarded merely as shortened 1 2 TH'Itt L'-YE AK MATHEMATICS 3. Variable. A variable is a symbol representing different numbers in a problem. Name the variables in the foregoing equations. 4. Function. If two variables x and y are so related that to every value of x there corresponds a definite value of y, then y is said to be a function of x. EXERCISES In the following relations show that one symbol is a func- tion of one or more others. 1. s = ^(a+b+c) 4. v=hh o 2. A= cr 5. 7 = ^ o 3. A-tvi 6. C = ~(F-32) 7. Name the constants and the variables in exercises 1-6. 5. Functional notation. Sometimes we are interested in the relation* between two variables rather than in the variables themselves. For example, in uniform motion the distance is equal to the rate multiplied by the time, but with falling bodies the distance is approximately 16 multiplied by the square of the time. The first of these laws is expressed in the form d = rt, the second in the form d= 16Z 2 . In these two cases d is not the same function of t. statements of rules of reckoning, just as in percentage we regarded j) = b-r as a short way of stating: percentage equals base times rate. Later these forms came to be regarded as either (1) rules of reckoning, or (2) the results of the reckoning. As results, they were regarded as numbers, and could be added, subtracted, multiplied, and divided. This gave rise at once to modern algebra. * Relation means any interdependence, and not necessarily ratio. FUNCTIONS. EQUATIONS IN ONE UNKNOWN 3 The equation f=2n expresses the relation between the number of miles and the railroad fare at 2 cents a mile. The relation between the number of eggs and the cost at 2 cents each is represented by the equation c = 2n. In these two examples / and c are the same function of n. The symbol f(x) is used to represent a function of x. It is read function of x, or briefly, / of x. To distinguish between different functions of x other letters are used, as in g(x) or F(x)* Thus, if the function 3z+2 is denoted by f(x) and if, in the same discussion, we wish to refer briefly to some other function, as 1^16 x 2 , we may- denote the latter by the symbol F(x). 6. Evaluation f of functions. To find the value of a function, as z 2 +5z-}-3, for a given value of x, the variable x in the function is replaced by the given value. If x 2 +5x+3 is denoted by f(x), then 3 2 +5 3+3 is denoted * The word "function" was first used in a mathematical sense in 1694 by Leibnitz, though in a different sense from that given here. In October of the same year James Bernoulli employed the word in the Leibnitzian sense. John Bernoulli employed the word in its modern sense in a letter to Leibnitz in June, 1698. Leibnitz' answer at the end of July of the same year shows that he too had given the word "function " its present meaning. The new technical term was first employed in print in a pamphlet by John Bernoulli in 1706. The latter was also the first to define the word. This he did in the Reports of the French Academy of 1718. John Bernoulli and Leibnitz both used special symbols for "function " in 1698. Neither used the symbol defined here. Euler, in a scientific publication of 1734-35, was the first to use the let- ter /, followed by the variable inside of parentheses, as the symbol for function. The French mathematician Clairaut was at the same time using a symbol in which the / of Euler's symbol was replaced by a Greek capital letter (Tropfke, Geschichte der Elementar- Mathematik, Band I, S. 142-43). f Evaluation means to find the value. 4 THIRD-YEAR MATHEMATICS by /(3) . Thus, /(3) is the result obtained by replacing x in fix) by 3, or the functional value of the function fix) for the particular value z=3 of the variable x. EXERCISES 1. If f(x) =z 2 +3z+5, find /(2), /(0), /(- 1), /(a). /(2)=2 2 +3 -2+5 = 15 /(0)=0 2 +3 -0+5=5 /(-l) = (-l)2+3(-l)+5=3 /(a)=a 2 +3a+5 2. If fix)=x>-kc+Z and F(,x) = 2x*-5, find F(6)-/(2). F(6)=2 -6 2 -5 = 67 /(2)=2 2 -4-2+3 = -l .'. F(6)-/(2)=68 3. If/(2/)=2/ 3 -3^+72/-l, find/(l),/(-2),/(0). 4. If /(r) = mr*+nr+p, find /(-3), /(J), /(a). 5. If fix) =x 2 +2x+5 and gix) =z 2 -3a;+2, find/(2) +g(-l) ; find ^3)' Linear Function 7. The function ax+b. Functions like 2x+5, \x 7, 3z+J, are of the form ax+b, where a and 6 are constants and a; is a variable. EXERCISES 1. Which of the following functions are of the form ax+b? |?+32; 2z 2 -4; 3(K; v +gt; i- j?(F-32); 2*-r; cos x; 5*. 2. Give other examples of functions of the form ax+b. 8. Graph* of the function ax+b. The relation be- tween the variable x and the function ax+b may be represented graphically * Graphing was introduced as a systematic mathematical method by Descartes in 1637. FUNCTIONS. EQUATIONS IN ONE UNKNOWN For example, a boy, on a certain date, deposits in a bank the sum of $3. He then deposits $2 regularly at the end of every week. How much money will he have in x weeks ? Show that F(x) = 2x+Z. Find a number of corre- sponding values of x and F(x) and tabulate them as in Fig. 1. Plotting the values of the variable x as abscissas and the values of the function F(x) as ordi- nates, we obtain the straight line AB. F Kxr -j- 7_ ) J '10 ~) 7~ jj / 7 5 L 7 ~t~ ) T_ 7 ' ~ x _1 {_ Q ~1 f X F(X 7 WT' / -5 15 7 2 7 t 3 9 7 4TT ^^ .ifc 42 3[__ 6 IH Fig. 1 /(*) 9. Linear function. The function aa;+ b is a function of the first degree in z. It is also called a linear function of x, because the graph is a straight line. The following shows that, in general, a straight line in a plane can be represented by an equation of the form /(a;) = mx+b: 1. Let P be any point on the straight line ABC, not passing through the origin 0, Fig. 2. Then OQ = x and PQ=f(x). Denote the distances AO and BO by a and b, respectively. Draw BDPQ and denote angle DBP by the letter s. Fig. 2 Show that tan s DP BD J(x)-b X THIRD-YEAR MATHEMATICS Denoting the value of tan s by m, we have ifcO fix) -b X .'.f(x)=mx+b 2. When A B passes through 0, Fig- 3, A / x x .'.f(x)=mx FlG - 3 3. When a line is parallel to the x-axis or to the f(z)- axis, show that its equation is of the form f(x)=c, or of the form x = c, respectively. 10. Intercepts. The numbers a and b, Fig. 2, are the intercepts made by the line A C on the z-axis and the F(x)-a,xk, respectively. EXERCISES Construct the graphs of the linear functions in the following examples: 1. The length of a circle is approximately equal to 3.14 multiplied by the diameter, i.e., c = 3. Ud. 2. The temperature in Fahrenheit degrees is 32 degrees greater than of the temperature in Centigrade degrees i e F = J-(7+32. ' ' 3. When a body falls from rest, the velocity, v, at any time, t, is given by the equation v=gt, where = 32, approximately. 4. Graph the equations f(x) =2z+3; f(x) = -4; x = 2. Direct Variation 11. Direct variation. In preceding work (200, Second-Year Mathematics) we have seen that the state- ments y varies as x, y is directly proportional to x, or y FUNCTIONS. EQUATIONS IN ONE UNKNOWN 7 varies directly as x are expressed algebraically by means of the equation y = ex. Thus, the statements : A man's pay is directly proportional to the number of days he works, the distance of a body moving at a uniform rate varies directly as the time, the length of a circle varies as the radius, are written respectively p = ct, d = ct, l = cr. The constant c is the constant of variation. The vari- ables p, d, and I are linear functions of t } t, and r, respec- tively. EXERCISES Express the following statements in the form of equations: 1. The area of a sphere varies as the square of the radius. 2. The volume of a cylinder varies directly as the square of the diameter of the base if the altitude remains constant. 3. The area of an equilateral triangle varies as the square of a side. 4. When a body falls. from rest (in a vacuum), the velocity varies directly as the time of falling. 5. When a spring is stretched by a force, /, the distance the spring is stretched (elongation) varies as the force (Hooke's law) . 6. The average consumption of coal for steam boilers varies directly as the number of square feet of grate surface. 7. The diagonal of a cube varies as the edge. The edge is 5 when the diagonal is 8.5. Find the diagonal when the edge is 10. 1. Show that d = c e. 2. Determine the constant of variation : 8 5 Since d = c e, 8. 5 = c 5; .*. c = -^- = 1.7. 3. The diagonal may now be determined : d = (1 . 7) (10) = 17. 8. Represent graphically the change in the diagonal, exer- cise 7, as the edge changes. 8 THIRD-YEAR MATHEMATICS 9. The area of a circle varies as the square of the radius and the area is 113 sq. ft. when the radius is 6 feet. Find the area of a circle whose radius is 2\ feet. 10. The time required by a pendulum to make one vibration varies as the square of the length. A pendulum 100 cm. long vibrates once in a second. What is the time of vibration of a pendulum 49 cm. long ? 11. The weight of a liquid is directly proportional to the volume. If 10 cu. ft. of water weigh 625 lb., what is the weight of 25 cubic feet ? 12. The pressure in pounds per square inch of a column of water varies directly as the height of the column in feet. A column of water 2.5 ft. high exerts a pressure of 1 08 lb per square inch. Find the height of a column exerting 1 84 pounds. Quadratic Function 12. Quadratic function. The functions tit 2 , s +gt 2 , 3x 2 -4:X+2, are of the second degree. Show that they are of the form ax 2 +bx+c. Functions of the form ax 2 +bx+c in which a^0 are called quadratic functions. EXERCISES Show that the following functions are of the form ax 2 +bx+c and determine in each case the values of a, b, and c. 1. 2-f 4x 2 -x 4. 2_2 2+5* x 5 6. _ 3. ax 2 -mx+2x 2 -12 6. 4 sin 2 x+3 sin x-7 13. Graph of the function ax 2 +bx+c. In 9 it was shown that the graph of a linear function is a straight line. It will be seen that the graph of the quadratic function ax 2 +bx+c is a smooth curve no three points of which lie on the same straight line. FUNCTIONS. EQUATIONS IN ONE UNKNOWN EXERCISES 1. Graph the function f(x) = x 2 . Tabulate values of f(x) corresponding to values of x between 3 and +3 (Fig. 4). By plotting the points corresponding to the pairs of numbers in the table and drawing the curve passing through these points, the graph of x 2 is obtained. This curve is called a parabola. If # is a negative num- ber and very large, f(x) is positive and very large, e.g., for x= 1,000, f(x) = +1,000,000. As x increases, remaining negative, f(x) decreases. As x approaches zero, f(x) also approaches zero. As x continues to increase indefinitely, fix) increases without bound. This may be represented in a table as follows: /(; X fix) T Xjc _.. 3 9 v " 7 3 f 1 L 5 ~i + 1 +2 +3 1 T, L. ~J_J \ f~ nV ( z.iJX,^ ^ k v\ -3 -2 - O + \\2 t3 X X - n Fig. 4 X -00* negative, increasing positive, increasing + 00 fix) + oof positive, decreasing positive, increasing + 00 *The symbol oo means increasing without bound in the negative direction. t The symbol + oo means increasing without bound in the positive direction. The value x = is said to be a zero of /(x), i.e., it is a value of x such that the corresponding value of f(x) is zero. 2. Graph the positive side of the curve /(x) =z 2 , using on the z-axis a unit equal to 2 cm., and on the f(x) -axis a unit equal to centimeter. Show how this graph may be used as a device for finding square roots of numbers. 3. Show that the graph of the f unction f(x) = x 2 is symmetric with respect to the /(z)-axis. Show that the /(x)-axis is the perpendicular bisector of the line-segment joining any two points on the graph which have equal ordinates. 10 THIRD-YEAR MATHEMATICS 4. Graph the function ttt 2 , taking tt = 3. 14. 5. The velocity of a ball thrown vertically upward with an initial velocity of 64 ft. per second is given by the formula fl 2 = 64 2 -64A, where A is the height attained at any time. Show that H is a quadratic function of v. Repre- sent graphically the function h=f(v), for values of v varying from 64 to 0. 6. Graph the function -, \ . _ f(x) = x 2 2x -3. Compute the values of f(x) as in the table, Fig. 5. Plot the points, as in Fig. 5, and draw the graph. What are the zeros of the function f(x)=x 2 -2x-3? Show how the graph may be used to solve the equation z 2 -3a;-3=0. Find the axis of symmetry of the function x 2 2xZ. ^ Graph the following functions and in each, case locate the axis of symmetry : 7. x 2 -6x+5 9. x 2 +4x 8. 3z 2 -llz-4 10. -x 2 +6x-5 Solve graphically the following quadratic equations: 11. x 2 +4z+2 = 13. z2 +a ._ 6 = 12. x 2 -5x+4: = 14. 4-5x-x 2 = Q Graphical Solution of Equations of Degree Higher than the Second 14. Graph of a cubic function. Functions like x 3 x, x 3 -6z 2 +llz-6, 2z 3 -3z+2 are functions of the third degree or cubic functions. FUNCTIONS. EQUATIONS IN ONE UNKNOWN 11 EXERCISES 1. Graph the function f(x)= x 3 x and locate the zeros. Plot the points corresponding to the Draw these . Compute the table, Fig. 6. pairs of numbers in the table, the curve passing through points. What -are the zeros of x 3 xf State how the graph may be used to solve the equation x 3 x = 0. 2. Graph the function f{x) = x z 6x 2 + 1 lx 6 and use the graph to illustrate the roots of the equation x 3 6z 2 +Hz 6 = 0. 3. The volume, v, of a sphere is ^-n-d 3 . f Show by means of a graph the changes *-L of the volume as the diameter changes. -3 -2 -1 1 2 1 2 1 2 3 -24 - 6 4-1 3 ~8 b 24 I 'fix m -10 I Fig. 6 15. Graphical solution of equations of degree higher than the second. Exercises 1 and 2, 14, indicate how the graph may be used to solve some cubic equations. Equations of degree higher than the third may be solved in a similar way. EXERCISES Solve the following equations, giving the values of the roots approximately to the first decimal place: 1. 10x 3 +29x 2 -5x-6 = Graph the function f(x) = lux 3 +29x 2 - 5x - 6, Fig. 7, and from the graph determine approximately the required values of x. 2. x 3 -2x 2 -7x-4 = X /(*) -4 -162 -3 -2 + 40 -1 + 18 - 6 1 ? 28 2 180 Fig. 7 12 THIRD-YEAR MATHEMATICS Synthetic Division. Remainder Theorem 16. The work of evaluating functions for different values of x is greatly simplified by means of a process called synthetic division and by the use of a theorem called the remainder theorem. 17. Synthetic division. The process of synthetic divi- sion is illustrated in the following example: Divide 2x 3 -7x 2 -Sx+5 by x-2. The process of long division is as follows: 2x*- r 7x 2 -Zx+b | x-2 2s 3 -4s 2 2x 2 -3z-9 -3z 2 -3z -3z 2 +6z -9z+5 -9x+18 -13 To shorten the work of this division, omit the various powers of x, writing only the coefficients. The work is now arranged as follows: 2- -7-3+5| 1- -2 2- -4 2- -3- -9 -3-3 -3+6 -9+5 -9+18 -13 FUNCTIONS. EQUATIONS IN ONE UNKNOWN 13 Next, omit bringing down the terms of the dividend as parts of the remainders. This gives 2-7-3+5 I 1-2 2-4 2-3-9 -3 -3+6 -9 -9+18 13 The coefficients of the quotient may be omitted, as they are equal, respectively, to the first coefficients of the divi- dend and of the remainders. Similarly, the first coeffi- cients of the partial products may be omitted. Finally, since the first coefficient of the divisor is always 1, it need not be written. This reduces the process to the following : 2-7-3+5] ~ 2 -4 -3 +6 -9 + 18 -13 In the process of division, as given above, the partial products 4, +6, and +18 are subtracted. By chan- ging the 2 in the divisor to +2, and thus changing the signs in the partial products, the subtraction of 14 THIRD-YEAR MATHEMATICS the partial product is changed to addition. This gives products, 2-7-3+5 |_2_ 4 -3 -6 -18 -13 Finally, the work may be condensed into the following form: 2-7-3+5 |_2_ 4-6-18 " . 2-3-9-13 Notice that the first three successive terms in the lowest line are the coefficients of the quotient and the last term is the remainder. Division in this abbreviated form is called synthetic division. 18. Rule for synthetic division. To divide f(x) by x a, f(x) is arranged in descending powers of x, supplying zeros as coefficients of missing terms. The coefficients are written horizontally and the first coefficient is brought down. This coefficient is multiplied by a and the product added to the second coefficient. This process is repeated until a product has been added to the last coefficient. The last sum is the remainder. The preceding sums are the coefficients ofxin the quotient, arranged in descending order. FUNCTIONS. EQUATIONS IN ONE UNKNOWN 15 EXERCISES Divide synthetically 1. z 4 -3z 3 +4z+2byz-3 2. 3z 3 -4z+7by:c-l 3. 5x 3 +2x 2 -3byz-5 4. 4:x 3 +x 2 -3x-l by x+2 Change x+2 to x-{-2). 6. 2x 4 +6x-5byz+l 19. Remainder theorem. The following exercises show that the value of f(x) for x a may be found by dividing f(x) by xa: EXERCISES 1. Divide f(x) = z 2 -6x+3 by z-2. By synthetic division the quotient and remainder are obtained as follows: 1-6+3 \2_ 2-8 1-4-5 Hence the quotient is x 4 and the remainder 5. 2. Find the value of f(x) = x 2 -6z+3 for x = 2. By substitution /(2) =2 2 -6 2+3= -5. Thus the value of f(x) ior x = 2 and the remainder ob- tained by dividing /(x) by x-2 are the same. If f(x) denotes a function of x in the form ax n +bx n - l +cx n ~ 2 .... the result of exercises 1 and 2 may be stated as a theorem as follows: Whenf(x) is divided by xa, the remainder isf(a). This principle is called the remainder theorem. 16 THIRD-YEAR MATHEMATICS 20. Proof of the remainder theorem. Since divi- dend s divisor X quotient + remainder, it follows that f(x) = (x-a)Q(x)+R, where the function Q(x) is the quotient and the constant R the remainder. Substituting a for x, f(a) = (a a)Q(a) +R. .'.f(a)mQ.Q(a)+R. .'.f(a) = R, which was to be proved. 21. Evaluation of /(*). According to the remainder theorem the value off(x), for x = a, may be found by dividing f(x) by x a. EXERCISES Find the values of the following functions, letting x take all integral values from 4 to +3: 1. 10x 3 +29x 2 -5x-6 Dividing synthetically by 4, 10+29- 5- 6 | -4 -40+44-156 162 10-11+39 .'./(-4) = -162 Similarly, find/(-3),/(-2), etc. 2. x*-2x*-7x- 3. 3z 3 -4z+7 Solve the following equations graphically: 4. x 3 -kc 2 -2x+8 = Q 6. z 3 +x 2 -10x-10=0 6. a^-3x 2 -a;+3 = 7. z 3 +3z 2 +2z=0 Equations of Degree Higher than the Second Solved by Factoring 22. Factor theorem. In 19 it was shown that the remainder obtained by dividing f(x) by xa is /(a). Hence, if the remainder, /(a), is zero, the division of f(x) by z a is exact and xa is a factor of f(x). Thus xa is a factor of f{x) if f{a) = 0. This principle is called the factor theorem. FUNCTIONS. EQUATIONS IN ONE UNKNOWN 17 EXERCISES The principle in 22 may be used to factor the following polynomials: 1. x 3 +2x 2 -9z-18 If the polynomial has factors of the form xa, then a must be a divisor of 18. Thus we have the following possibilities for a: =*=1, 2, 3, =*=6, =*=9, 18. To find/(l), divide synthetically by 1, 1 +2 -9 -18 [_1_ 1+3-6 1 +3 -6 -24 Since /(l) = 24, it follows that x 1 is not a factor. Why not ? Similarly, we find /( 1) = 8 .'. x+1 is not a factor. /(2) = -20 .*. 3-2 is not a factor. /(-2) = .-. x+2 is a factor. Why? Show that the other factor is x 2 9, which may be factored as the difference of two squares. 2. tf-faP+llx-Q Since there are no powers of x missing, and since the signs are alternately + and , fix) cannot be for negative values of x. Why? Hence the only values to be tried are 1, 2, 3, and 6. 3. z 3 +6x 2 + llx+6 Show that/(:r) ?^0 for positive values of x. 4. 2^ 5i/ 3 +52/ 2 +52/ 6 6. Show that x y is a factor of x n y n if n is an integer, and find the factors of x 5 y 5 . 7. Show that x+y is a factor of x n y n if n is an even integer, and factor x*y s . 8. Show that x-\-y is a factor of x n -\-y n if n is an odd integer, and factor x b +y h 18 THIRD-YEAR MATHEMATICS 9. Show that for even values of n the function x n \y n has no factors of the form x+y or x y. Solve the following equations: 10. x*-7x+S=0 The factors are (x 1), (x2), and (x +3). .'. (z-l)(z-2)(x+3)=0. x-l=0j x 2=0 > satisfy the given equation. [z+3=0j [*ilt Hence jz 2 =2 |x 3 =-3. 11. ^-192/-30 = 16. ^-4rc-&c 2 +32 = 12. x 3 -5x-2=Q 17. y z -y 2 =y 13. * 3 -3H-2=0 18. ^-1 = 14. 2y*-y 2 -5y-2 = Notice that the roots of this equation are the three 15 . y2+\ +y +l =: 4 : cube roots of 1. y yl* u y 23. Equations of degree higher than the second solved like quadratics. EXERCISES Solve the following equations: 1. (x+2) 2 +3(z+2) = 18 By factoring, we have (x+2-f6)(z+2-3)=0 . (*i=-8 " 1*2 = 1 2. y*+2y*-80 = 3. (5i/-4) 2 -2(5?/-4)-63 = t A letter with a subscript, as Xi, is read "x sub one," or "x one." FUNCTIONS. EQUATIONS IN ONE UNKNOWN 19 4. (x 2 -6:r) 2 +5(z 2 -6:c+20)-136 = Change the equation to the form (x 2 -6z) 2 +5(z 2 -6z)+100-136=0, or (z 2 -6z) 2 +5(x 2 -6:c)-36 = 5. 3(7/+3rFl) 2 -7(?/ 2 +3?/+l)+4 = 6. y 2 +y-2 ^- = * y 2 +y 7. 2G/ 2 -3)+^|3+17 = The Function - x 24. Inverse variation. The equivalent statements y varies inversely as x and y is inversely proportional to x are expressed algebraically in the form y= . Thus the statement, the force of gravity varies inversely as the square of the distance, is written i=~r r In this equa- tion c is the constant of variation and / is a function of d. EXERCISES Express the following statements by means of equations: 1. The number of vibrations a pendulum makes in one second varies inversely as the square root of the length. 2. The amount of heat received from a stove varies inversely as the square of the distance from it. 3. The volume of gas inclosed in a cylinder varies inversely as the pressure. 4. The pressure which a given quantity of air at constant temperature exerts against the walls of a containing vessel is inversely proportional to the volume occupied (Boyle's law). 20 THIRD-YEAR MATHEMATICS Solve the following problems: 5. The intensity of light on an object varies inversely as the square of the distance from the source of light to the object. A screen 15 ft. from a lamp is moved to a distance of 5 ft. from it. How much does this increase the intensity ? 6. A pendulum 39.1 in. long makes one vibration in a second. Find the length of a pendulum vibrating 4 times a second. Use exercise 1. 7. The volume of a gas confined in a cylinder is 1 cu. ft. when the pressure is 5 pounds. What is the volume when the pressure is 20 pounds ? Use exercise 4. 25. Graph of -. Corresponding values of x and f(x) for c= 1 are given in the table, Fig. 8. X /<*> 10 /(*) 4 i 3 I . 2 i 1 1 i 2 5 \ : : : 4 5 :S:: 6 ^ 3 ~Z_ -1 _^^ "' ' f J '*,*. a -I .2. . _Li ^k i -6 \ s -5 H *~ -J -4 "J* -i 2 -1 -1 2 -1 3 -4 -1 ' 1 II II 1 1 II 1 II Ml Fig. 8 Plot the pairs of values given in the table, Fig. 8. The curve obtained consists of two branches which do not touch either axis. Since both branches are obtained FUNCTIONS. EQUATIONS IN ONE UNKNOWN 21 from the same equation, /(#)=-, they are said to form x one and the same curve. This curve is called a hyperbola. Show that - decreases if x increases. x x By taking x large enough, - can be made smaller than any number whatsoever. 1 If x is positive and decreases, - increases. By taking x small enough, - can be made larger than any number x however great. Often this fact is expressed briefly by the statement ^= oo. However, this does not mean that 1 1 divided by has a value. It means that - increases without bound as x approaches zero as a limit. EXERCISES 12 1. Graph the function f(x) = . x g 2. Graph the function f(x) = - . Q 3. By means of the graph of the function - find a meaning x for the expression ~. 4. Discuss the changes of - as x changes from oo to + oo. x 26. Joint variation. The area of a triangle varies as the product of the base by the altitude. It is said to vary jointly as the base and altitude. If a train moves with a uniform speed the distance varies j ointly as the rate and time. This may be expressed algebraically by means of the equation d=rt, d denoting the distance, r the rate, and t the time. The statement y varies jointly as x and z is expressed in symbols by means of the equation y = cxz. 22 THIRD-YEAR MATHEMATICS EXERCISES Express by an equation each of the following statements: 1. The distance passed over by a train varies jointly as the rate and time. 2. The pressure of water on the bottom of a basin in which it is contained varies jointly as the area of the bottom and the depth of the water. 3. The pressure of wind on a wall varies jointly as the area of the surface and the square of the velocity of the wind. 4. The volume of a cylinder varies jointly as the area of the base and altitude. Compare the volumes of two cylinders whose altitudes are in the ratio 1:2. ex 27. Direct and inverse variation. If y = then y z varies directly as x and inversely as z. EXERCISES Express by an equation each of the following statements : 1. The cost of posts for a fence varies directly as the length of the fence and inversely as the distance between the posts. 2. The resistance to an electric current varies directly as the length of the wire and inversely as the cross-section. 3. The current furnished by different galvanic cells is directly proportional to the electromotive force and inversely propor- tional to the resistance of the circuit (Ohm's law). Solve the following problems : 4. If y varies directly as x and inversely as z, and if y = 14 when x 7 and z = 1, find y when x = 84 and z = 6. 5. The interest on a sum varies jointly as the rate and prin- cipal. If in a certain number of years the interest on $2,000 at 5 per cent is $400, what is the interest on a principal of $2,500 at 5^- per cent in the same time ? FUNCTIONS. EQUATIONS IN ONE UNKNOWN 23 Summary 28. The following exercises summarize the terms, symbols, and processes taught in this chapter: 1. Give the meaning of the following terms: function intercept variable direct variation constant inverse variation evaluation of a function joint variation linear, quadratic, cubic, direct and inverse variation function parabola zero of a function hyperbola abscissa, ordinate synthetic division 2. Explain the meaning of the following symbols: f(x), Fix), -oo, +oo. 3. Tell how to make the graph of f(x), if f(x) is linear; quadratic; cubic. 4. Explain the use of synthetic division in evaluating a function of x. 5. Explain how to solve equations in one unknown of the second degree, or higher, 1. By means of the graph 2. By factoring 6. Show that a straight line can be represented by an equa- tion of the form f(x) = mx+b. 7. State and prove the remainder theorem. 8. State the factor theorem. 9. Represent graphically, y = x 2 2x. (Wisconsin.*) 10. The attraction of gravitation at points outside the earth's surface varies inversely as the square of the distance from the * (Wisconsin) means: taken from an entrance examination given by the University of Wisconsin. 24 THIRD-YEAR MATHEMATICS earth's center. If the attraction on a certain body is 9 lb. at the surface of the earth, at what altitude above the surface would the attraction on the same body be reduced to 4 pounds ? (Take the radius of the earth as 4,000 miles.) (Harvard.) 11. Lights of equal brightness are placed in three corners of a square room. Show that the intensity of the illumination at the fourth corner is T 5 ^ that at the center of the room. Given that the intensity of the illumination varies inversely as the square of the distance from the light. (Harvard.) CHAPTER II TRIGONOMETRIC FUNCTIONS 29. Angles in general. In the preceding course* the sine, cosine, and tangent of acute angles were defined. These functions have been used in the solution of right triangles, 257, S.-Y.M. However, in the general triangle obtuse angles as well as acute angles are found. To work with the general triangle the notion of the trigonometric functions of an angle must be extended to include angles that are greater than 90. 30. Angle as amount of rotation. The angle XOA, Fig. 9, is considered as generated by the rotation of a line from OX around to the position OA. p IG 9 The line OX is called the initial side and OA the terminal side of the angle. If the line revolves from OX in the counter-clockwise direction, the angle XOA is positive ; if it revolves in the clockwise direction, the angle formed is negative. 31. Quadrants. Two lines at right m angles, Fig. 10, divide the plane around Y ' the point of intersection, 0, into four FlG 10 equal parts called quadrants. The quadrants are numbered as follows: XOY is the first quadrant, YOX' the second, X'O Y f the third, and Y'OX the fourth. * Second-Year Mathematics, 248. 25 26 THIRD-YEAR MATHEMATICS An angle is said to be in the first, second, third, or fourth quadrant, according as its terminal side lies in the first, second, third, or fourth quadrant, the initial side having been placed on OX. Hence angles between and 90, 90 and 180, 180 and 270, and 270 and 360 are said to be in the first, second, third, and fourth quad- rants, respectively (Fig. 11). EXERCISES 1. In which quadrant is Z.XOA, Fig. 11? ZXOBf ZXOCf ZXOD? 2. Draw the following angles and in each case state the quadrant in which the angle lies: 20, 160, 240, 315, 545, -40, -220. 32. Trigonometric functions. Let XOA, Figs. 12 to 15, be a given angle. From any point, P, of the terminal b MX Fig. 12 Fig. 14 Fig. 13 Fig. 15 TRIGONOMETRIC FUNCTIONS 27 line OA, drop a perpendicular, PM, to the initial line OX (produced if necessary). This forms a right triangle, MOP. The sine of ZXOA is the ratio of the side of AMOP that lies opposite the vertex to the hypotenuse, i.e., MP . , jyp in each case. The cosine of ZXOA is the ratio of the side of AMOP that lies adjacent to to the hypotenuse, . OM i.e., Qp . The tangent of Z XOA is the ratio of the side opposite MP to the side adjacent, i.e., ^r? . The cotangent of ZXOA is the ratio of the side adjacent to to the side opposite 0, i.e., TTp. The secant of ZXOA is the ratio of the hypotenuse OP to the side adjacent to 0, i.e., 7^7 > The cosecant of Z XOA is the ratio of the hypotenuse OP to the side opposite 0, i.e., y^p . These are called the ratio-definitions of the trigo- nometric functions. Suggest why these ratios are called functions. 33. Signs of the functions in each quadrant. If the side opposite to the vertex extends upward from the initial line OX it is considered positive, if it extends downward it is negative. If the adjacent side extends to the right of it is posi- tive, if to the left it is negative. The hypotenuse is always regarded as positive. 28 THIRD-YEAR MATHEMATICS Denoting the measure of angle XOA by the Greek letter a (alpha) and the lengths of the sides of AM OP by a, b, and c, respectively, show that the statements of 32 take the following form : ^Qeti52^~-~^_Quadrants I ii III IV Sine a +-: +e a c a c Cosine a i c _b c *\ Tangent a +i a ~b +s a ~b Cotangent a +5 _b a 5 _b a Secant a +i c ~b c b +1 Cosecant a +5 a c a c a The signs of the functions in the various quadrants should be thoroughly well known. The following diagrams, sine-cosecant cosine-secant tangent-cotangent Fig. 16 Fig. 16, will be helpful in remembering the correct algebraic signs of the various functions, TRIGONOMETRIC FUNCTIONS 29 34. Values of the trigonometric functions found by means of a drawing. In the following exer- cises construct on squared paper the given angle. Draw the defining triangle MOP as in Fig. 17. Measure the sides of +* the triangle and deter- mine the algebraic signs and the numeri- cal values of the func- tions of the given angle. \ A V s V \ s ^ IT ^,+ 1.7 Ml V \- S \ V s 51 -Jt^v \_S M -1-0 X Fig. 17 EXERCISES 1. Find the value of the sine of 125, Fig. 17. 2. Find the values of the functions of the following angles: 140, 220, 245, 315. 35. Abbreviations* of the names of the functions. The expressions sine of angle a, cosine of angle a, tangent of angle a, etc., are usually written in the following abbre- viated forms: sin a, cos a, tan a, cot a, sec a, and esc a. 36. Given the value of one function of an angle to construct the angle. EXERCISES 1. Given tan a = f. Construct angle a. Show that there are two angles whose tangent is f , one in the first and one in the third quadrant, Fig. 18. Construct these angles and measure them with a protractor. * See note, p. 138, Second-Year Mathematics. Fig. 18 30 THIRD-YEAR MATHEMATICS 2. Construct angles A, x and a, having given cos A = \, tan x = 3, and cot a= V 3. 3. Construct the angles x and y given by tan x = f , cos 2/ = --. 4. Construct angles a, having given cos a= |, sin a= +|-, tan a = 3, tan a= 3, cot a = 4. 37. Inverse functions. According to 36 an angle may be found if the value of one of the trigonometric functions is known. Thus, the equation sin x=^ determines x as an angle, or arc,* whose sine is . The statement x is an angle whose sine is y is usually written briefly: x = sin~ 1 y or x = arc sin y, read inverse sine and arc sine respectively. Give the meaning of the following : = tan -1 3, y = arc cos ( \), A=&rc sin f. 38. Given the value of one function of an angle, to determine the values of the other functions. EXERCISES 1. Given tan a= f, a being in the second quadrant, find the other functions of a. Construct AMOP, Fig. 19, having OM=-3, MP = 4. Com- pute OP. 2. Let cot z = f. To find the other functions of x, x being in the third quadrant. 3. Given tan x = \. If Z is in the third quadrant, give the values of the other functions. M Fig. 19 * Since an angle whose vertex is at the center of a circle has the same measure, in degrees, as the intercepted arc, the trigonometric functions may be regarded as functions of the arc instead of the angle. TRIGONOMETRIC FUNCTIONS 31 4. Let sin a = --, find cot a, Fig. 20. 5. If Z A is in the third quadrant and tan A = ^ 3 , find sec A and sin A. 6. Find the values of the functions of z if sec z = J . \*5 25/ 13 16 -vSoT v^ol Fig. 20 Fig. 21 7. Find cos x if tan x- 2mn m 2 ri' , Fig. 21. 8. If sin a = 2 a 1+a 2 find cos a. 2#w 9. If tan a= n y n , find sin a. z 2 2/ 2 10. If # = arc sin y, find tan x in terms of y. Changes of the Trigonometric Functions as the Angle Changes from to 360 39. Trigonometric functions represented by lines. Since the trigonometric functions are ratios it is possible to represent them graphically by means of line-segments. This simplifies greatly the study of the changes of the functions that depend upon the changes of the angle. 32 THIRD-YEAR MATHEMATICS 40. Line representation of the sine and cosine functions. Let ZXOA, Fig. 22, be any angle. With a radius equal to 1 and the center at draw a circle. A circle of radius 1 is called a unitrcircle. From the point of intersection, P, of the unit-circle and side OA draw PM0X. Denoting ZXOA by a, we have MP MP sin a = jyp- = = M P, i.e. Fig. 22 the measure of MP is the same as sin a. The segment MP represents sina ' OM OM Similarly, cos a = jyp = = OM , i.e., the measure of OM is the same as cos a. Hence, OM represents cos a. EXERCISE Draw angles lying in the second, third, and fourth quadrants and represent by line-segments the values of the sine and the cosine functions. 41. Changes of the sine and the cosine functions. In Fig. 23 MiPi, M 2 P, etc., represent sin a, and OM 1} OM 2} etc., represent cos a. As a decreases, sin a decreases and cos a increases. When OP coincides with OX, a = 0,sin0 = 0,andcos0= 1. As a increases from to 90, sin a increases and cos a decreases, both being positive. When a = 90, MP coincides with OP and sin 90 - -f 1, while cos 90 = 0. Fig. 23 TRIGONOMETRIC FUNCTIONS 33 / As a increases from 90 to 180, sin a decreases from 1 to and cos a decreases from to 1, Fig. 24. As a increases from 180 to 270, sin a decreases from to 1 and cos a increases from 1 to 0, Fig. 25. Fig. 24 Fig. 25 As a increases from 270 to 360, sin a increases from 1 to and cos a increases from to 1, Fig. 26. The following table gives the values of sin a and cos a for special values of a, a being less than, or at most equal to, 360: ~~~~~~~~ __ Angle Function ^ -___^ 30 45 60 90 180 270 360 Sine 1 2 l/2 2. t/3 2 1 -1 Cosine 1 2 V'2 2 1 2 -1 1 The sine and the cosine functions cannot be greater than +1 and not less than 1. Why? EXERCISES 1. Describe the variation of sin a; as re increases from to 360. Illustrate by means of a figure. 2. Describe the variation of cos x as x increases from to 360. Illustrate by means of a figure. 34 THIRD-YEAR MATHEMATICS 42. Line representation of the tangent and secant functions. Let ZXOA, Fig. 27, be any angle. With radius equal to 1 and with as center draw a circle. At X draw XT tangent to circle 0. Denoting Z XOA by a, we have A A. JLJx -it a i tan a = 7TTF = = AA, i.e., the part OX 1 of the tangent at X intercepted by the initial and terminal sides of ZXOA has the same measure as tan a. Hence XA represents tan a. a . ., , OA OA r . A Fig. 27 Similarly, sec a = -^ = - = OA , i.e.. the measure of OA is the same as the value of sec a. Hence, sec a is represented by OA. 43. Changes of the tangent and secant functions. As a decreases, Fig. 28, XA decreases. For a = 0, XA=0, i.e., tan a = 0. As a increases from to 90, tan a increases. As a approaches nearer and nearer to 90, XA increases without bound. Thus, tan a has no definite value when a = 90. Often this fact is expressed symbolically by the statement tan 90 = + a>. How- ever, this statement does not mean that 90 has a tangent. It means p IG 2 8 that tan a, remaining positive, in- creases without bound as a approaches 90 as a limit. Show that sec a increases from 1 to + oo as a changes from to 90. Show that the sign of sec a, Fig. 28, is the same as the sign of cos a. TRIGONOMETRIC FUNCTIONS 35 When a lies in the second quad- rant, Fig. 29, tan a is represented by the part of the tangent at X which is intercepted between the initial side OX and the extension of the terminal side of ZXOA. The fact that XAi, XA2, etc., extend down- ward from OX shows that tan a is negative in the second quadrant. When a lies in the second quadrant and decreases approaching 90, tan a increases without bound, always being negative. This is expressed in symbols by means of the statement tan 90= 00. As a approaches 180 tan a approaches zero. Fig. 29 A, A, EXERCISES 1. Show that sec a changes from 00 to 1 as a changes from 90 to 180. 2. As a changes from 180 to 270 show that tan a changes from to + 00 ; that sec a changes from 1 to 00. 3. As a changes from 270 to 360 show that tan a changes from co to 0; that sec a changes from +00 to +1. The following table gives the changes of tan a and of sec a as a changes from to 360 : ^"~-\^ Angle Function ^**\^^ 90 180 270 360 Tangent =t 00 == GO Secant + 1 =t CO -1 =T=GO + 1 36 THIRD-YEAR MATHEMATICS 44. Line representation of the cotangent and cosecant functions. Let ZXOA, Fig. 30, be any angle. Fig. 30 Draw a unit-circle with the center at 0. At Y draw YT tangent to circle 0. Show that Z XOA = Z YAO. Denote ZXOA by a. YA YA Show that cot a = jyy = r- = YA, i.e., the part of the tangent at Y which is intercepted by OY and OA has the same numerical measure as cot a. Similarly, show that esc a = ^Ty = ~ r~ = ^ . Hence OA has the same numerical value as esc a and represents esc a. When a is obtuse, point A is to the left of Y and there- fore YA is negative. When a is in the third quadrant, show that YA is positive. When a is in the fourth quadrant, show that YA is negative. EXERCISE Describe the variations of cot a and esc a as a increases from to 360. Illustrate by means of figures. TRIGONOMETRIC FUNCTIONS 37 45. Table giving the changes of the functions as the angle changes from to 360. The following table gives changes of functions of a as a changes from to 360 : ^^\^ Angle Function"\^ to 90 90 to 180 180 to 270 270 to 360 Sine to +1 +1 to to -1 -1 to Cosecant + oo to +1 + 1 to +00 - oo to - 1 1 to - CO Cosine + 1 to to -1 -1 to to +1 Secant + 1 to + <* -oo to -a 1 to oo + oo to +1 Tangent to + co - oo to to +oo -co to Cotangent. . . . + oo to to -oo + oo to to -oo Graphs of the Trigonometric Functions 46. Radian measure. There are two current methods of measuring angles, viz., degree measure and radian, or circular, measure. The student is already familiar with the first, in which the unit-angle is a degree consisting of -g-J-jr of a complete revolution. Before con- structing the graphs of the trigono- metric functions we will examine the second method and its advan- tages over the first. Let a be the measure, in degrees, of Z.A0B, Fig. 31. Draw the unit-circle having the center at the vertex 0. Since a is measured by the intercepted arc, the length of AB may be used to express the value of a. In that case the unit of measure is an arc of unit length, or the angle inter- cepting an arc equal in length to the radius of the circle. This Fig. 31 38 THIRD-YEAR MATHEMATICS method of measuring angles is called radian measurement, or circular measurement. 47. Radian. The unit of circular measure, called the radian, is the angle that intercepts an arc equal in length to the radius of the circle, Fig. 32. When the unit of measure is not indicated it is understood to be a radian. Thus LABC=\ means that /.ABC is ^ of a radian. 48. Relation between circular measure and degree measure. Let Z BOA, Fig. 32, be a radian. Since the length of a semicircumference is irr and since AB* = r, it follows that 7T radians = 180, where 7r = 3. 14159 approximately; 180 V / Fig. 32 a radian = f - J 3f- ) = 57 . 3 approximately. Show that 10= (iio) radians - Fig. 33 49. Relation between an angle, the intercepted arc, and the radius of the circle. Let the Greek letter 6 itheta) denote the number of radians in a given angle, Fig. 33. Then T = - , or 8 = - , or, in words : 1 r' r- ' The number of radians in an angle is equal to the length of the intercepted arc divided by the radius of the circle. Show that s=r8. Express the equation s = rd in words. *The symbol AB means arc AB. TRIGONOMETRIC FUNCTIONS 39 EXERCISES 1. Express 10 15' in circular measure. 10 15' = (10 J) = (10j) (j^) radians = . 19, approximately. 2. Express the following angles in circular measure: 10, 8 30', -50, 58. 3. Find the number of degrees in an angle whose circular 5 radians = ( j = 76?39, approximately. 4. Express in degrees the following angles: , \, .752, 3. 14. 6. Express in radians the following angles: 0, 30, 45, 60, 90, 180, 270, 360. 6. Express the following angles in degree measure: -, -, -, O TV TT 7. Give the values of the following functions: sin ^, tan , 3tt l l COS 7T, CSC . 8. The radius of a circle is 3 feet. Find the length of the arc intercepted by an angle at the center equal to 1 J. 9. The radius of a circle is 10 feet. What is the length of an arc intercepted by an angle of 80 ? 10. What is the circular measure of an angle at the center of a circle intercepting an arc equal to of the radius ? 11. Prove that the area of a sector of a circle is , r being the radius and the angle at the center, in radians. 12. Prove that the area of a segment of a circle is - . Z Z 13. The radius of a circle is 10 feet. Find the angle at the center intercepting an arc 2 ft. long. Express the result in degrees and in radians. 40 THIRD-YEAR MATHEMATICS 50. Graphing the trigonometric functions. To graph the trigonometric functions we may plot the correspond- ing values of angle and function as obtained from a table of functions. However, the following will show that the line representation affords a very simple way of making the graphs. 51. The sine curve. Lay off on OX, Fig. 34, to a con- venient scale, distances representing a = --, - . 7T 12 6 2 . ir, etc., where tt = 3.14. At the points thus obtained lay off vertically the corresponding distances representing sin a. Draw a smooth curve through the top points of these vertical lines. Fig. 34 This is the sine curve. EXERCISES From a study of Fig. 34 answer the following questions: 1. As a varies from to 360 how does sin a vary ? 2. At what places is the change in sin a most rapid ? Where is the change slowest ? 3. How does the curve show that the sine function repeats its values at intervals of 2tt, or 360 ? 4. What is the largest value of sin a ? What is the smallest value ? TRIGONOMETRIC FUNCTIONS 41 52. Periodic function. A function whose values are repeated at definite intervals as the variable increases is a periodic function. EXERCISES 1. Show from Fig. 34 that the period of the sine function is 2tt, i.e., show that sin (a+27r) =sin a. 2. Show that sin ( a) = sin a. 3. Show that sin (* a) =sin a. These exercises suggest certain facts about the sine function to be proved in 58-63. 53. The cosine curve. The curve in Fig. 35 is the cosine curve. To construct it follow the directions given s > - :z T-= f / / - - " s< / / / - I // s _ \ r ~\ ) 7T 7 " /.{7T 2TT 57T v \k v \ - 2 ~ i 2 \ \ \ T ^. 2a :^= - _ ^ d ' Fig. 35 for the construction of the sine curve, 51. Notice that the cosine curve has the same shape as the sine curve and differs from it only in position. EXERCISES Give a discussion of the cosine curve similar to that given for the sine curve, 51 and 52. 42 THIRD-YEAR MATHEMATICS 54. The tangent curve. Draw the tangent curve, Fig. 36, and give a discussion as in 51. Fig. 36 55. The cotangent curve. Using Fig. 37, draw the cotangent curve, Fig. 38, and give a discussion simi- lar to that given for the sine, cosine, and tangent curves. I :fc y Fig. 37 TRIGONOMETRIC FUNCTIONS 43 Fig. 38 56. The secant and cosecant curves. These curves, given in Fig. 39, may be con- structed as in 54 and 55 . The solid line represents the secant, the dotted line the cosecant, function. Fig. 39 44 THIRD-YEAR MATHEMATICS The Trigonometric Functions of Negative Angles 57. Positive and negative angles. By rotating AB, Fig. 40, around A until it takes the position AC, angle BAC is formed. By rotating AB in the opposite direc- tion, angle BAC is formed. It is customary to consider an angle posi- tive when it is formed by rotating a line counter-clockwise, and negative when it is formed by clockwise rotation. Fig. 40 58. Trigonometric functions of a in terms of the functions of a. Denoting /.BAC, Figs. 41 to 44, by a, then ZBAC=-a. Fig. 42 Fia. 44 With A as center and radius equal to 1 draw CC. Draw chord CC. Then AD is the perpendicular bisector of CC* * CC means chord, or segment CC TRIGONOMETRIC FUNCTIONS 45 Show that triangles DAC and DAC, Figs. 41 to 44, are congruent. .-.ADAC^ADAC sin a = DC sin (-a) =DC'=-DC= -sin a Similarly, cos ( a) = AD cos a sin ( a) _ sin a cos ( a) cos a .*. tan ( - a) = - tan a Why ? EXERCISES 1. Show that 1. cot (a) = COt a 3. CSC (a) = CSC a 2. sec ( a)= sec a 2. Express in terms of functions of positive angles the values of all functions of the following angles: -30, -45, -60. The exercises in 58 show that any trigonometric function of a is equal numerically to the same function of a, but differs in sign, with the exception of the cosine and secant. " - m 59. In 34 we have seen that the values of the trigo- nometric functions of any angle may be found from a draw- ing. The values of the functions of angles less than 90 may be found by means of special tables (see p. 356). The values of the functions of angles greater than 90 can be found by means of certain relations which enable us to express the functions of any angle in terms of some func- tion of an angle less than 90. These relations are to be worked out in the following sections. 46 THIRD-YEAR MATHEMATICS The Trigonometric Functions of (J^a) in Terms of Functions of a 60. The relations derived from the graphs of the functions. Let the curves in Fig. 45 represent the sine li- f -4* SB' l^ .' .1 j? JT ** yC i v 1 ! i 1 ' 77 V / ir V A 4" 7T \ V yd \ v , 2 / / % / , lif / r S / / t \r f / s 4 / ' f\ Jl 4 / - ^ "' 1 lk ^ ' Fig. 45 and cosine functions drawn to the same scale, and let OA=a and OA' = 90+a = |+a, a being less than 90. _. By moving either curve a distance equal to , it can be 2 made to coincide with the other. Thus AB will coincide with A'B\ Hence, sin (90+a) = cos a. (1) Similarly, the sine curve can be made to coincide with the cosine curve by moving it to the right a distance equal IT to g and then rotating it about OX as an axis. AC will then coincide with A'C Hence, cos (90+ a) = - sin a (2) .'.tan (90+ a) = - cot a Why ? (3) and cot (90+a) = -tan a Why ? (4) Show that sec (90+ a) = -esc a (5) and esc (90+ a) = sec a (6) Since A'B' = A "", it follows that sin (90 -a) =sin (90+a) = cos a (7) TRIGONOMETRIC FUNCTIONS 47 Since A"C" = A'C", it follows that cos (90 -a) = -cos (90+a) = +sin a (8) .*. tan (90 - a) = cot a Why ? (9) and cot (90 - a) = tan a Why ? (10) Show that sec (90- o) = esc a (11) and esc (90 a) = sec a (12) Sketch roughly the graphs of the tangent and cotan- gent functions and verify relations (3) and (4). The relations (1) to (12) may be summarized as fol- lows : any trigonometric functions of (90 a) is equal to the cof unction of a;' and any trigonometric function of (90 -\- a) is equal numerically to the cof unction of a, but differs in sign with the exception of the sine and cosecant functions. 61. The relations found in 60 may be proved as follows : Denoting ZXAB, Fig. 46, by a, then ZABC=^-a. a . tv . - = sm a = cos \~a c \2 b . (t - = cos a = sin a c \2 r = tana = cot ( = a - = cot a = tan a a \2 c /it 7 = sec a = csc (^ a C fir - = csc a = sec - a a \2 Fig. 46 48 THIRD-YEAR MATHEMATICS Let BAD = ^, Fig. 47, then ZXAD=(^+ a y Prove that ACAB^ AEDA. . . sin ( o + a ) =- = +cos a. j /* t \ ~ a a ana cos 77+ a J * = = sin a \2 / c c Fig. 47 EXERCISES 1. Prove relations (3) to (6), 60. 2. Find the exact values of all functions of the following angles: 120, 135, 150. 3. Prove the relations sin (~+ol) =cos a and cos ( J+) = sin a for the following cases: 1. When a lies in the second quadrant, Fig. 48. 2. When a lies in the third quadrant, Fig. 49. 3. When a lies in the fourth quadrant, Fig. 50. Fig. 48 Fig. 49 Fig. 50 TRIGONOMETRIC FUNCTIONS 49 The Trigonometric Functions of In _~ =*= EXERCISES Give reasons for the following: 1. 1. sin (180-a)=sin [90+ (90 -a)] = cos (90-a) = sina 2. cos (180-a) = cos[90+(90-a)]= -sin(90-a) = -cosa 3. tan (180-a) = -tana 5. sec (180-a) = -sec a 4. COt (180-a) = -COta 6. CSC (180-a)=CSCa 2. 1. sin (180+a) = -sina 4. cot (180+a) = cot a 2. cos (180+a) = -cos a 5. sec (180+a) = -sec a 3. tan (180+a) = tan a 6. esc (180+a) = -esc a 3. 1. sin (270-a) = sin[90 o +(180 o -a)] = cos(180-a) = -cosa 2. cos (270 -a) = -sin a 5. sec (270 -a) = -esc a 3. tan (270-a)=cota 6. esc (270-a) = -sec a 4. cot (270-a)=tana 4. 1. sin (270+a) = -cosa 4. cot (270+a) = -tan a 2. cos (270+a) = sina 5. sec (270+a) = csc a 3. tan (270+a) = -cot a 6. esc (270+a) = -sec a 5. 1. sin (360-a)=*-sina 4. cot (360-a) = -cot a 2. cos (360-a) = cosa 5. sec (360-a)=sec a 3. tan (360-a) = -tan a 6. esc (360-a) = -esc a 50 THIRD-YEAR MATHEMATICS 6. 1. sin (360+a) = sina 4. cot (360+a) = cot a 2. cos (360+a) = cosa 5. sec (360+a) = sec a 3. tan (360+a) = tana 6. esc (360+a) = csc a 63. From a study of the preceding exercises we learn the following : 1.* A function of (an even multiple of 90^ a) is equal numerically to the same function of a. 2. A function of (an odd multiple of 90 =*= a) is equal numerically to the corresponding cof unction of a. 3. The sign of the result is the same as the sign of the original function in the quadrant in which the angle (n-0a) lies. EXERCISES Express the following as functions of positive acute angles : 1. sin 580 sin 580 = sin (6 90 +40) = -sin 40 2. cos 315 6. sin 240 3. sin (-196) 7. tan (-410) 4. tan (2 t Vtt) 8. cos 120 5. cos 120 9. sin 300 Express the following as functions of positive angles less than 45: 10. cos (-428) 13. tan (-65) 11. sin (-84) 14. sin 1,420 12. esc (834) 15. cot l,330 c |0 * These principles hold even when a is not an acute angle. In case a is greater than 90 the sign is determined by the quadrant in which the angle (n ^=*=a j would lie if a were acute. TRIGONOMETRIC FUNCTIONS 51 Find the values of the following: 16. cosl80 o -sec 2 45 o -4sin30 o +T/2sin45 o 4-cosl80 o csc90 _ IT 2tT . 7T 2lT 17. cos - cos -sin - sin - oo 6 6 ,, 7T 07T i 7T 07T 18. sin - cos -fcos - sin o 6 bo Simplify 19. tan (-*); cos(^-x); sin (270-x); -cot(90+z); cos (-1,230) Summary 64. The student should know the meaning of each of the following terms : initial side sine cosecant terminal side cosine inverse trigonomet- quadrant tangent ric functions trigonometric cotangent circular measure functions secant radian 65. The following exercises review the main topics taught in the chapter: 1. Give the signs of the trigonometric functions in each of the four quadrants. 2. Show how to find the values of the trigonometric functions of given angles by means of a drawing. 3. Show how to construct an angle when the value of one of the trigonometric functions of the angle is known. 4. Explain how the values of the trigonometric functions of an angle may be found when the value of one of the functions is given. 6. Discuss the changes of the trigonometric functions as the angle changes from to 360, using the straight-line representa- tion. 52 THIRD-YEAR MATHEMATICS 6. Give the exact values of the sine and cosine of the follow- ing angles: 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, 330, 360. 7. Prove the relations between circular measure and degree measure. 8. Show that the length of an arc of a circle is given by the equation s = rO, 6 being the number of radians in the angle. 9. Draw the graph of each of the trigonometric functions and tell what the graphs show. 10. Express the functions of (a) in terms of functions of a. 11. Express the functions of (Z^ -) in terms of functions of a. 12. Show how to express the functions of any angle as func- tions of a positive angle less than 45. CHAPTER III LINEAR EQUATIONS Linear Equations in One Unknown 66. Normal form. We have seen that every linear function of x may be changed to the form ax-\- b, 7. Simi- larly every linear equation in one unknown and many fractional equations may be changed to the form ax-\-b = 0. This is called the normal form of a linear equation in one unknown. EXERCISES Change the following equations to the normal form ax-\-b = 0: Subtracting as indicated, x-\ - = 7. Multiplying by 36, 63z+81-36z+8x-4 = 252. Simplifying, we have the normal form, 35a: 175=0. -I(3-iHHi-*)=i 3. y-\Sy-[y-(Sy-l)]\ = l 4. (2- 2 /)(3-2/) = (l 2/)(5-2/) $5. (2+y)(2y+l) + (2-y)(2y+l)-7y = 0* 6. (82/+3) 2 -32/(132/+9)-(52/+2)2 = 13 3 65 _a_ 6 7 * 2-5" h 4~2z-10 x-6 x-a 8. ax+& 3 = 6:r+a 3 t 11 - (a+6)2/ = 2a- (0-6)3/ _j, 3^_ _ 12 ' (^-a) 2 -(^-6) 2 = c 2 * 9 ' ^+l~H : 2 +2 ~ 13. (x-a)(x-b) = (x-c)(x-d) * All problems and exercises marked % may be omitted. 53 54 THIRD-YEAR MATHEMATICS 67. Solution of linear equations. Subtracting b from both sides of the equation ax+b = Q and then dividing by a, we have _^ jc= a This may be stated in words as follows : The root of a linear equation in normal form is ob- tained by dividing the negative of the constant term by the coefficient of the unknown number. EXERCISES Solve the following equations: 1. 5l(|-3) + .17(2/-2)=3.% 2 . 9 .9+ 7 - 2y ~ 5 - 5 = 3.3 y -.li/ , 2 (^)+^-> M ) 9 9 5 5 5. x-7^ x-2~ x-S~ x+1 5 2 2 H-2 4 2/ 2 ~2/ 2 2 2 Change -; -; to + 4_^2 ^2_4 2x 2m n (mri)xm 2 , m n 6. 2 ^ t, + o m n w?n 2 mn +7 3 _2+l_ 2 2 + '* 2+1 2-1 1-2 2 2(y-7) 2-y y+3 2/ 2 +3 2 /-28 i "4-2/ S 5 \V - ^ ' -p- -K L j 5 ^ Fig. 52 Since it is impossible to divide the numbers 3 and 6 by zero the resulting forms show that the equations have no common solution. graph the system we find that both are 2. If we / *- y= 2 \5x-5y=l0 represented by the same line, Fig. 53. Henoe any solution of either equation is a solution of the other. Such equations are said to be equivalent or dependent. The difficulty that arises here is not that there is no solution, but that there are too many. 51 -7 7 2 7 ~7 xj. -f> Z x 2 / 1 Fig. 53 LINEAR EQUATIONS 67 Solving the same system by determinants, we have 2 -1 10 -5 1 -1 5 -5 -10+10 2 10 -5 + 5 = 0' V = Since a number multiplied by zero always gives zero, the expression ^ may represent any number. Hence the solution is indeterminate. It is easily seen that one equa- tion may be derived from the other by simple multiplica- tion by a constant. The two preceding examples show that a system of the form j ax-\- by = c \aix+biy = ci has one, and only one, solution, if the determinant abi aib is not equal to zero. Hence this fact may be used to determine whether a system of equations has one and only one common solution. EXERCISES Show which of the following systems are equivalent, and which inconsistent. 1. 2. { 3x+|=6 '3x-2y=U < 9x-Qy = 3G 2 x+jV = 2 2+7z, h, b 2) 63, etc., are called the elements. The horizontal lines in the square form are the rows and the vertical lines the columns of the determinant. Each term in the expansion is a product of three ele- ments, no two of which lie in the same row or in the same column. A determinant of the third order may be expanded as follows : Draw the diagonal through the first element, ai, Fig. 54, and the parallels to it through 02 and a 3 respec- tively. This gives the terms ai& 2 C3, a2&3Ci, and a^h. Then draw the diagonal through C\ and the parallels through c% and c 3 . - The signs of the last three products are changed, gives the terms cib 2 a d , c^hai, and CzChbi. Fig. 54 This EXERCISES Evaluate the following determinants: 5 2-6 1 4 7 =5-4- 1+1 -3- (-6R2-7-2 2 3 1 -(-6) -4.2-7- 3- 5-1- 1-2 = 20-18+28+48-105-2 = -29 70 THIRD-YEAR MATHEMATICS 2. 1 3 8 1 1 2 -12 4. 1 1 8 1 -4 5 1 -1 1 1 . 11 1 2 1 1 -1 -1 5. 3 7 3 8 3 4 3 5 76. Solution by determinants. By solving the equa- tions (aix+biy+ciz = di la2X-\-b 2 y+C2Z = (k {a z x+hy+c z z = d z a formula may be obtained for the solution of any system of three linear equations in three unknowns. Eliminating y between the first two equations, we have {aih. avbi)x+ (b 2 Ci 6102)2/ = di&2 _ #261- (i) Eliminating y between the first and third equations, we have (o 3 6i aib 3 )x+ (c 3 6i hd)y = d 3 6i dib 3 . Solving equations (1) and (2), we have d\b 2 Cz + a\bzd + dzc^bi C\b 2 dz C2&3CJ1 Cza\b\ (2) x = O162C3 +0263(4 +O3C261 cib 2 a 3 C2&3O1 c 3 O20i According to 75 this may be written di 61 Ci k b 2 C2 Us 63 Cz ai 61 Ci 02 62 Ol o 3 63 Cz LINEAR EQUATIONS 71 Notice that the denominator is a determinant whose elements are the coefficients of x, y, and z in the given system and that the numerator is derived from the de- nominator by replacing the coefficients of x by the constants. Similarly, ai di Ci ai h di 02 a\ C2 a* 62 a\ a% d% Cz ., z = a z bz dz ai bi Ci ai bi ci 02 62 Ol 2 62 Ol a$ 63 c 3 0*3 bz cz EXERCISES Solve by determinants: [2z+32/+4z=16 1. <5x-8y+2z = l {3x-y-2z=5 16 3 4 1 -8 2 5 -1 -2 2 3 4 j y~ 5 -8 2 3 -1 -2 2 16 4 5 12 3 5-2 2 3 4 5-8 2 3 -1 -2 .-.(*, y, z) = (3, 2,1) 2 3 16 5 -8 1 3 -1 5 2 3 4 5 -8 2 3 -1 -2 (5x+2y-4:Z=-3 2. !4x+5i/+2z = 20 l3x-3!/+5z=12 3. Sx-y+2z=9 z-2?/+3z=2 2x-32/+3=l fa+36+9c = 23 4. we may obtain a solution of every quadratic equation by solving ax 2 +bx+c = 0. This will lead to a formula for finding the roots. Subtracting c from both sides of the equation ax 2 +6x+c = 0, and dividing by a, we have X 2 -\ = b 2 Completing the square by adding ^ to both sides, x 2 +-x+ t a 4a 2 4a 2 ' * This method originated with the Hindus. Aryabhatta (b. 476 a.d.) first used it in a slightly different form from that given here. Brahmagupta (b. 598 a.d.) used it so extensively that it has been given the name Brahmagupta's Rule, and Cridhara later modified it slightly, bringing it to the modern form. (Tropfke, Band I, S. 257.) 78 THIRD- YEAR MATHEMATICS Extracting the square root, , b lb 2 -4ac x+ 2a= * visr Hence Xi= ~b+V^ac f ft _--^Zg EXERCISES Solve the following equations by formula: 1. 3z 2 +5x-2 = Comparing this equation with the equation ax 2 -\-bx+c-0, we find that o=3; 6=5; c= 2 Substituting these values in the formulas, -5= V 25+24 -5=7 1 , . x- ^ = --, and -2 2. 2^+5a;+2 = 10. r 2 -3.50r+2.80=0 3. 2r 2 r 6 = Give values correct to three 4. lAx 2 -\-5x = 2A significant figures. 6. 6p 2 -13p=10p-21 jn. t 2 = .100- .200t $6. my 2 +ny+p = 12> 16. 08^+20^=1,000 7. ax 2 +(b-a)x-b = 13 ( m - n)y 2- m 2 y + m 2 n = 8. a-y 2 =(l-a)y (2 y -l)(y-S) = 2 9. y 2 -l.Gy+0.3 = . . ++ 15. (z-l) 2 (z+3)=z0c+5)(x-2) Give values correct to ' '. two significant figures. J16. ?/ 2 -6mn/+ra 2 (9r 2 -4n 2 ) = 17. m 2 y 2 - (m 2 +mn)y = 2m 2 - 5mn+2n 2 The formula gives: m 2 +mn=/m 4 +2m 3 n+m 2 n 2 +8m 4 -20m 3 n+8m 2 n 2 " V 2m* _ m 2 +mn = /9m 4 - 18m 3 n +9m 2 n 2 2m 2 m 2 +ran = (3m 2 3mn) = ~= -, etc. 2m 2 ' QUADRATIC EQUATIONS IN ONE UNKNOWN 79 In solving exercise 17 it is necessary to find the square root of the polynomial 9m 4 18m 3 n+9m 2 n 2 . This is readily done by inspection. It is, however, not always easy to find by inspection the square root of a polynomial. Hence we shall learn the process of extracting the square root of a polynomial before proceeding with the general solution of quadratic equations. Square Root of Polynomials 84. The process of extracting the square roots of polynomials is suggested by such equations as : (a+b) 2 = a 2 +2ab+b 2 (a+6+c) 2 =(a+6) 2 +2(a+6)c+c 2 = a 2 +2a6+6 2 +2ac+26c+c 2 1. We note that a 2 , the first term of the polynomial, is the square of the first term of the square root. Therefore, if the polynomial is arranged according to powers of a letter, the first term of the root is found by extracting the square root of the first term of the polynomial. 2. The term 2ab, which is the first of the remaining terms of the polynomial, is twice the product of the first term of the root by the next term. Therefore the second term of the root is found by dividing the first term of the remainder by twice the first term of the root. 3. By adding b, the second term of the root, to twice a, the first term, and then multiplying this sum by the second term, b, we obtain 2ab-\-b 2 . Subtracting this from the first remainder, 2ab+b 2 +2ac-\-2bc-\-c 2 , we have the second remainder, 2ac-\-2bc-]-c 2 . 80 THIRD-YEAR MATHEMATICS By dividing 2ac, the first term of this remainder, by 2a we find c, which is the third term of the root. The process in (3) is then continued. If at any time there is no remainder the polynomial is a square. - EXERCISES Find the square roots of the following polynomials: 1. 4-197* 2 +12?i-427i 3 +49n 4 First, the polynomial is arranged according to powers of n, 7n 2 Sn2 = Square root (7n 2 ) 5 49n 4 -42n 3 -19n 2 +12n+4 A2n 3 +9n 2 thus : Polynomial = |49n 4 - 42n 3 - 19n 2 + 12n +4 The first term of the root is V4Qn* = 7n 2 Subtracting the square of 7n 2 from the polynomial, we have the remainder: The first term of the remainder divided by 2 7n 2 gives 3n, which is the second term of the root. Adding this to 2 7n 2 we have 14n 2 -3n. This is multiplied by 3n: * (14n 2 -3n)(-3n) = The product obtained is then subtracted from the preceding remainder, giving : The first term of the last remainder di- vided by 2 In 2 gives 2, which is the third term of the root. Adding this to 2 times the sum of the first two terms of the root we have 14n 2 -6n-2. This is multiplied by -2: (14n 2 -6n-2)(-2) The product obtained is then subtracted from the preceding remainder, leaving no remainder. Hence the given polynomial is a perfect square and the square root has been found exactly. -28n 2 + 12n+4 28n 2 + 12n+4 QUADRATIC EQUATIONS IN ONE UNKNOWN 81 2. 10x 2 +12z 3 + l+4x+9z 4 3. 16z 6 +10z-8x 3 +l-40:r 4 +25x 2 4. 9x 2 -Sxy 2 -30xy+5y 3 +25y*+^ |5. x 2 +9z 2 +4y 2 -4xy-12yz+Gxz 6. z 4 +25+6z 3 -30;r-:r 2 J7. lBa: 6 +25x 4 -24x 5 +10x 2 -20x 3 -4a:+l 8. 12a-23a 4 +8a 5 +5a 2 +4-22a 3 +16a 6 o m2 _i_ 1 1 ,6m , 6a , a 2 a 2 a m m 2 1 10. 1+z, to 5 terms Solve the following equations: 11. y 2 4my+ny-\-3m 2 5mn 2n 2 = 0, for y 12. 2/ 2 -3ai/-3fo/+3a = 2/-9a&, lory fl3. 2 -3aZ-2 = Z-2a 2 -3a, for* 14. x 2 Sx 5x 2 2mx 2 = 3+mx+4m, for z Fractional Equations 85. Solve the following equations : J2. rr+ i = s . f or a; (Harvard) , 4a 2 V 4a 2 -6 2 , _ _ ... 3 -5+2-^=i(i^)' f0ra; (Sheffield) ^JSSSrl ( princeton ) 82 THIRD-YEAR MATHEMATICS * ^+i=6+i=r (Harvard > s~3 . 2s+3 5s-3_3s-l _ 8 - ^i~ + ^ 6 2 3 . x+l-t-^ - = t10 - 3iii)-^T + 3wi) = i (Chicag0) ,, 9a 3 ,, baVa (Za-Va\ 2 1* + ^ =1 / 3a- vV t 2 a 26 a; a 2b Problems Leading to Quadratic Equations 86. Solve the following problems : 1. A rectangular box has a volume of 1,500 cubic inches. Its depth is 5 in. and it is 5 in. longer than wide. Find its dimensions. (Sheffield.) 2. One side of a rectangle is 20 cm. longer than the other. The diagonal is 7 cm. longer than the longer side. Find the area of the rectangle. (Harvard.) %Z. It is shown in plane geometry that the length of the side, s, of a regular decagon inscribed in a circle of radius a is deter- mined from the equation as _s s a Assuming this equation, solve for s in terms of a. Find the value of s correct to three significant figures when a = 100. (Harvard.) 4. On the side AB of a square ABCD a point E is marked at a distance of 10 in. from A. The area of the trapezoid EBCD is less by 22f sq. in. than three-fourths of the area of the square. How long is a side of the square ? (Harvard.) QUADRATIC EQUATIONS IN ONE UNKNOWN 83 5. A man, after having bought an article, sells it for $21. He loses as many per cent as he gave in dollars for the article. What did he pay for it ? (Yale.) 6. A trunk 30 in. long is just large enough to permit an umbrella 36 in. long to lie diagonally on the bottom. How much must the length of the trunk be increased if it is to accom- modate, diagonally, a gun 4 in. longer than the umbrella? (Chicago.) 7. A rectangular piece of tin is 4 in. longer than it is wide. An open box containing 840 cu. in. is made by cutting a 6-inch square from each corner and turning up the ends and sides. What are the dimensions of the box ? (Chicago.) $8. An open box, to be made from a square piece of card- board by cutting out a 4-inch square from each corner and turn- ing up the sides, is to contain 256 cubic inches. How large a square must be used? (Chicago.) 9. The rates of two trains differ by 5 mi. an hour. The faster requires one hour less time to run 280 miles. Find the rate of each. (Yale.) 10. If a body falls from rest, the distance, s, that it falls in t seconds is given by the formula s=16 2 . A man drops a stone into a well and hears the splash after 3 seconds. If the velocity of sound in air is 1,086 ft. a second, find the depth of the well. 11. Find the sides of a right triangle in which the sides of the right angle are respectively 20 in. and 10 in. shorter than the hypotenuse. 12. A rectangular tract of land, 800 ft. long by 600 ft. broad, is divided into four rectangular blocks by two streets of equal width running through it at right angles. Find the width of the streets, if together they cover an area of 77,500 square feet. (M.I.T.) |13. What is the number of sides of a polygon having 170 diagonals ? 84 THIRD-YEAR MATHEMATICS 14. Two men can do a piece of work in 6 hr. 40 minutes. One can do the work alone in 3 hr. less time than the other. In how many hours can he do it alone ? 15. The sum of the two digits of a given number is 5. If the order of the digits is changed, the product of the result by the original number is 736. Find the number. Equations of Quadratic Form 87. Solve the following equations : 1. 2/ 4 -262/ 2 +25 = |4. 6* 4 +6=13* 2 2. z 6 -8 = 7z 3 6. (z 2 -2z) 2 -7(z 2 -2z) + 12 = t = 97 2 9 4 Q7 3. W+-, = -k $6. (4z+5) 2 +2(4:c+5)-15 = Trigonometric Equations 88. Conditional equations containing trigonometric functions of unknown angles are called trigonometric equa- tions. For example, 2 sin x = tan x, 5 sin 2 x-\- cos 2 x = 2. Some trigonometric equations are easily solved by factoring. If the equation contains several trigonometric func- tions of x } it is generally best to change its form so that it contains only one function of x. This is accomplished by use of the following fundamental identities : ., 1 _ cos X 1. sin # csc =1 5. cot x=- sin x 2. coszsec#s=l 6. sin 2 +cos 2 z=l 3. tan x cot x=l 7. tan 2 #+l = sec 2 x 4. tan z= 8. cot 2 z+l = csc 2 x cos x QUADRATIC EQUATIONS IN ONE UNKNOWN 85 EXERCISES Find all the positive values of the angle between and 360 c which satisfy the following equations : 1. cot x = 2 cos x CO* a . cos x Since cot x = sin x cos x sin x Hence, 2 cos x=0 sin x Factoring, cos x ( 2 ) = Vsin x / This equation is satisfied if cosx = = 2 cos x Fig. 56 or if 2 = sin x cos x = 0, x is equal to 90, or 270 c 2 = 0, then sin x = .*. x = 30, or 150, Fig. 56. .*. x=30, 90, 150, and 270 are the positive values of x, less than 360, satisfying the given equation. 2. 5 sin 2 x+cos 2 x= f Since sin 2 x = 1 cos 2 x, it follows that 5 5 cos 2 x+cos 2 x = 2 .'. 4 cos 2 x = 3 cosx = =|"/3. If cos x = + |t/3, it follows that x = 30, or 330. If cos x = - Jt/3, it follows that x = 150, or 210. .'. x = 30, 150, 210, and 330 are the positive values of x, less than 360, satisfying the given equation. 3. tan x sec x V2 1 Put sin x tan x = , sec x cos x cos x , cos 2 x = l sin 2 x Then show that = r-r = V2 1 sin 2 x .'. sin x = V2 V2 sin 2 x Solve this equation for and find the required values of x. 86 THIRD-YEAR MATHEMATICS 4. 2 cos 2 x+3 sin z-3 = 9. 2 sin 2 0+3 cos 0=0 5. 3 sec 2 x 7 tan 2 x = tan a; + . . 1 $10. cos 2 0-sin = t 6. 4 cos 2 z = cot x 7. tanz = cosz 11. tan 0+cot = 2 $8. sin 2 0-cos 0+1 = $12. sin 2 0-2 cos 0=^ Nature of the Roots of a Quadratic Equation 89. Complex numbers. We have seen, 82, that some quadratic equations cannot be solved by graph, e.g., the equation x 2 4a;+5 = 0. However, by means of the quadratic formulas we find that 4i/l6-20 4l/^4 rt , * -^ J =2-l/-l Thus the roots of the equation 2 4z+5 = involve the square root of a negative number. We cannot extract the square root, or any even root, of a negative number, since the square of a real number is always positive. Thus, l/-4 cannot equal +2 or -2, since (+2) 2 = (-2) 2 = +4. An even root of a negative number is called an imaginary number. Expressions of the form of a+V^b, where a is a real number and b a positive real number, are complex num- bers. Thus, l + l/^2, i/+3_i/38 are complex num- bers. They are also called imaginary numbers. However, the latter term is misleading, because complex numbers are not imaginary for the person who has made a study of these numbers. 90. Classification of numbers. Positive integers and fractions are the first numbers with which the pupil becomes acquainted. Later the study of negative num- bers is taken up. Positive and negative integers, the GASPARD MONGE GASP ARD MONGE GASPARD MONGE, the son of a small peddler, was born at Beaune in 1746 and died at Paris in 1818. A plan of his native town, drawn by him, fell into an army offi- cer's hands, and its excellence so impressed the officer that he recommended that Monge be admitted to the training school at Mezieres. His low birth prevented him from receiving a commission in the army, but he was allowed to attend the annexe of the school, where he learned surveying and drawing. But he was not sufficiently well born to be allowed to do calcu- lator problems. A difficult plan for a fortress was to be drawn, and Monge did it by a geometrical construction. This turned the tide of young Monge's fortunes. The officer at first objected to receiving Monge's plan because he had taken less time than etiquette required for such a problem, but the superiority of Monge's method finally won its acceptance. In 1768 Monge was made professor of descriptive geom- etry, though the results of his methods were to be a secret confined to officers above a certain rank. In 1780 he was made professor of mathematics at Paris, and he communicated his earliest paper of importance to the French Academy in 1781. The paper discussed lines of curva- ture drawn on a surface. He found that the validity of solu- tions is not impaired when imaginaries are involved among subsidiary quantities. Euler had treated these questions, but Monge's methods were superior to those of Euler. Monge applied his results to central quadrics in 1795. In 1786 he had published a work on statics. Monge became embroiled in the politics of the Revolution and narrowly escaped the guillotine. In 1798 he was sent to Italy on state business, and thereafter joined Napoleon in Egypt. After Napoleon's defeat, he escaped to France and settled down at Paris. Monge was now made professor and gave lectures at the Polytechnic School of Paris on descriptive geometry, and in 1800 published his text entitled G6om4trie descriptive. In this he treats the theory of perspective and the theory of surfaces in a masterly way. On the restoration he was deprived of his offices, stripped of his honors, and thrown out of the French Academy. These humiliations soon led to his death. [See Ball, pp. 426-27, and Cajori, pp. 286-87.] QUADRATIC EQUATIONS IN ONE UNKNOWN 87 fractions and zero, form the domain of rational numbers. For example, 4, 8, -, 1.75, are rational numbers. A rational number may always be expressed exactly as the quotient of two integral numbers. This includes integers, since they are fractions with 1 as denominator. However, such numbers as Vz, V^ 3^7, cannot be expressed exactly as quotients of integers. They are classed as irrational numbers. The rational and irra- tional numbers form the domain of real numbers. For example, the number t, known to us from the study of the circle, is a real number, but not a rational number, as it cannot be expressed exactly as a quotient of two integral numbers. 91. Graphical representation of real numbers. Posi- tive integers may be represented by equidistant points on a straight line, as OA, Fig. 57, or by the distances of these points c - , '-;-,' -/ ) , * ; ; , from a fixed point, as 0. B o c A Negative numbers are then Fig. 57 represented by equidistant points laid off in the direction opposite to that of OA, as OB. The origin, 0, represents zero. Fractions are represented by intermediate points. Thus the point C represents the fraction -|. Similarly any rational number may be represented by a point on the line. Although between any two rational numbers, however close, other rational numbers may be inserted, there are points on the line which do not represent rational numbers. For example, we know that V~2 is the length of the diagonal of a square whose side is of unit length. Therefore, by laying off on OA a length equal to this diagonal, we obtain a single definite point which represents 88 THIRD-YEAR MATHEMATICS the irrational number l/2. Indeed it can be shown that to any irrational number there corresponds a definite point on OA. Hence any real number can be represented by a point on a straight line. This line is called the axis of real numbers. 92. Nature of the roots of a quadratic. The char- acter of the roots of the equation ax 2 -{-bx-\-c depends upon the number b 2 \ac. The function b 2 4ac is called the discriminant of the equation ax 2 +bx-\-c = 0. In the following we consider the coefficients a, b, and c to be rational numbers. 1. If b 2 4ac = 0, the two values of x are the same. Thus the roots of the quadratic are real, rational, and equal. For example, for the equation x 2 6z+9 = we have 6 2 -4ac = 36-36 = 0. Hence, without solving the equation, we know that the roots are real, rational, and equal. 2. If b 2 4ac>0, the roots are real and unequal. (1) If 6 2 4ac is a square, the roots are rational. (2) If 6 2 4ac is not a square, they are irrational. For example, for the equation x 2 92+14 = we have 6*-4ac = 81 -56 = 25. Hence the roots are real, rational, and unequal. For the equation x 2 +5x+l = we have 6 2 -4ac = 25-4 = 21. Hence the roots are irrational. j 3. If b 2 4ac<0, the expression Vb 2 4ac is imaginary and the roots of the equation are called complex. Thus for 2x 2 +z+l = we have fc-4ac=l-8= -7. Hence the roots of this equation are complex. QUADRATIC EQUATIONS IN ONE UNKNOWN 89 The preceding discussion may be represented in a table as follows : The roots of a quadratic equa- tion are complex, if 6 2 -4ac<0 real, if b 2 4ac is not <0 rational and equal, if & 2 -4ac = unequal, if 6 2 -4ac>0 and a perfect square irrational, if 6 2 -4ac>0, and not a, per- k feet square EXERCISES Without solving determine the nature of the roots of each of the following equations : 1. 3z 2 -8z+5 = 0* 6 2 -4ac = 64-4(3)(5) =64-60 = 4 .*. the roots are real, rational, and unequal. 2. x 2 -4x+8=--0 6. 9x 2 +12x+4 = 3. a 2 +3a-l = 7. 7y 2 +3y = 4. 5z 2 -3z = 2 8. 5z 2 +7:c+3 = 5. 3z 2 = 7z+6 9. z 2 -6z+4 = Find the values of d for which the roots of the following equations are equal: 10. 9x 2 +(l+d)x+4: = 6 2 -4ac = l+2d+d 2 -144 Hence, to make the, roots equal, we put d*+2d- 143=0 .\d-ll, -13 90 THIRD-YEAH MATHEMATICS 11. y*+y+d=0 U4. y 2 +3dy+d+7 = 12. 2x 2 +(l+d)x+2 = 15. (d+l)x' i +dx+d+l = 13. z 2 -4dz+4=0 |16. 2dx 2 +(5d+2):r+(4d+l) = Relation between the Roots and the Coefficients of a Quadratic 93. Denoting the roots of the equation ax 2 +bx+c = by - - -b+VV-ac and r2= -b-VW-iac 2a 2a we have by addition : -2b b by multiplication : (-6) 2 -(j/fr 2 -4ac) 2 _ 6 2 -6 2 +4ac ^c 2a 2a 4a 2 a b n r 2 Hence the sum of the roots is a and the product of the roots is EXERCISES Find the sum and the product of the roots of the equations in exercises 1 to 5. 1. 2x 2 -9z+8 = Since a = 2, b= 9, c = 8, it follows that -9 9 ri+r2 =__ = - and that 8 A nr 2 = 2 = 4. QUADRATIC EQUATIONS IN ONE UNKNOWN 91 2. 2x 2 -9x-5 = 4. x 2 +2x+2 = 3. x 2 -12x-13 = 5. 4-?/-6?/ 2 = 6. One root of the equation x 2 kx+2l = is 7. Find the value of k. n r 2 = 21 n =7 .'. r 2 = 3 and A; = 10 7. One root of the equation Sx 2 kx -f 10=0 is 5. Find k and the other root. 8. Find the values of p and q in the equation x 2 -\-px+q = 0, 1. If the roots are 6 and 4, 2. If the roots are S-V& and 3+ /6. 9. Form the equation whose roots are 2 and 3. b c Since 2 + (3) = and 2( 3)=~, we may substitute these Cb Qi b c values in the equation x 2 -\ x-\ =0. a a This gives z 2 +z-6=0. 10. Form the quadratic equations whose roots are : 1. -3,10 5. 2=^3 2. -8, -3 6. a, -b 3. 6, -i 7. Vd, -VI 2> 4 4. i, x 8. ra-f-n. mn Factoring 94. The solution of a quadratic equation enables us to find the factors of the quadratic trinomial ax 2 +bx-\-c. Show that ax 2 +bx+c = a[ x 2 -\ a a/ = a[x 2 - (ri+r 2 )z+rir 2 ] ,\ ax 2 +bx+c=a(x ri)(x-r 2 ). 92 THIRD-YEAR MATHEMATICS EXERCISES Determine the factors of the quadratic functions given in exercises 1 to 7. 1. 5x 2 -h3z-20 Let5x 2 +3x-20 = Tu m , -3 + ^409 r -3-/409 -3-/4091 5x * +3x -20^[x-=^\ 10 J 2. 2x 2 +5x-7 5. z 2 -2ax+a 2 -6 2 3. 2x 2 +5x+2 |6. z 2 +4a6x-(a 2 -6 2 ) 2 4. Gy 2 +y-l J7. abx 2 (a+b)x+l Reduce the following fractions : 2?/ 2 +7?/-4 . a x 2 -2ax+x+a-\ ' 3y 2 +lh/-4 * ' ax 2 -3ax+x+2a-2 2t 2 +8t-90 . 20xhf+20xyz-21z 2 ' 3 2 -36+105 l ' 10x 2 ?/ 2 +24:n/z-18z 2 Summary 95. The chapter has taught the meaning of the follow- ing terms : quadratic equation real numbers normal form of a quadratic rational numbers equation irrational numbers imaginary numbers nature of the roots of a complex numbers quadratic, discriminant QUADRATIC EQUATIONS IN ONE UNKNOWN 93 96. The following are typical problems of the chapter : 1. To solve quadratic equations in one unknown by the following methods : (1) By factoring, (2) By formula, (3) By completing the square, (4) By graph. 2. To extract the square root of a polynomial. 3. To solve fractional equations which reduce to quadratic form. 4. To solve equations of other degree than the second, but of the form of quadratics. 5. To solve trigonometric equations. 6. To solve problems leading to quadratic equations. 7. To represent real numbers graphically. 8. To determine the nature of the roots of a quadratic equation. 9. To solve quadratic equations, having given a rela- tion between the roots and the coefficients. 10. To factor quadratic trinomials by means of the quadratic formula. CHAPTER V FACTORING. FRACTIONS 97. Purpose of the chapter. The work in factoring given in 22 and 94 completes what is commonly taught about this subject in an elementary course in mathematics. We have had no general method by which all polynomials can be factored, but we have learned to recognize certain forms which suggest a particular method to be used in factoring them. It is the purpose of this chapter to review and summarize what we should know about factoring and to make use of factoring in the operations with alge- braic fractions. 98. The difference of two squares. Find the prime factors of the following: i. x 2 -y 2 7. 16a 2 -(2m-3w) 2 x 2 -4y* = (x+2y)(x-2y) 2. 9a 2 -25 3. 1-r 4 8. {x 2 -y 2 ) 2 -x* 9. 9c 2 - (3a -2c) 2 10. 36(a+6) 2 -25(c-rf) 2 4. 9a 2 x-Wy 2 %lm ( a+6 )2_( a _ 6)I 5. a 6 -6 6 , a 8 -6 8 12. (4a+5) 2 -(2x-3) 5 ) 2 6. a 4 -6 4 , a 12 -b 12 13. (x+y+z) 2 -(x-y-z) 2 Reduce the following fractions to simplest terms : m{x 2 -y 2 ) x 2 +y> 14 * (a+b)(x-y) 16 * x*-y* (x+a)(x-b) (4a-d) 2 -(26-3c) 2 ' (x 2 -a 2 )(x 2 -b 2 ) 17 * (4a-26) 2 -(3c-d) 2 94 FACTORING. FRACTIONS 95 Add and subtract as indicated : x x . 2x-6 18 19. 3x+ 3x 2 2x-3 2x+3 ' 9-4x 2 **" w ' x-3 x+3 ' x 2 -9 Multiply and divide as indicated : 20 (ft+8) 2 a 2 - 16 (ft-4) 2 ' ft 2 -64 21. x 2 -9 (x+3) 2 x-4 * (x-4) 2 99. The sum or differences of like powers. Factor the following : 1. 64ft 3 +276 3 64a 3 +276 3 = (4a +36) (16a 2 - 12a6+96 2 ) 2. 8x 3 +125v 3 7. 8v is +27w is 3. 27x 3 -8?/ 3 8. (a+&) 3 -c 3 4. 343+x 3 9. (w+3) 3 +ft 3 5. p z J 10. x 3 (m+n) 3 6. 512c 3 +27d 3 11. (5m-7i) 3 +c 3 Reduce to the simplest form : 12 1 14 x 2 +x+l Multiply as indicated : X 2 __yi x 6_ y6 X? 16 x 3 if x z -\-y z x 2 +?/ 2 Subtract as indicated : a 2 +ft 1 a 3 -l a-1 Solve for x: 2x-3 . x 13. 15. 17. 64a 6 6 6 +l 4a 2 6 2 +l x 2 9 x 2 +3x+9 x 3 -27 x+3 3 5 a a 1 2ft-3 ft 2 -l 18. x 2 -l ' x-1 x+1 Factor the following: 19. ft 6 +6 6 ; ft 9 -6 9 ; a 9 +6 9 ; ft 12 +6 12 96 THIRD-YEAR MATHEMATICS 100. Trinomials. Factor the following : 1. 4x 2 -12xy+9y 2 7. x*-3x 2 y 2 +y A 2. x*y 2 +2x 2 yz+z 2 8. a 4 +2a 2 6 2 +96 4 3. 5x 2 -38:r+21 9. 6z 2 -17x+5 4. 66 2 -296+35 10. 3z 2 +8z-7 5. 1 6xy+5x 2 y 2 11. ( \l-\-x x / \l-\-x X / FACTORING. FRACTIONS 99 8. Simplify: ( 2 +^)(^Ha-f6+i) ( ShefMd) 9. Reduce: ffiiwSS (SheffieId) 10. Reduce: 1 +*-*-* +*=* (Sheffield) 1 a 2 a- 5 1 11. Simplify: (-s=5H:5=2) <** 12. Simplify: 13. Simplify: 8c 3 -l A 4 \ . /2c-l\ /15 ,, 9c2 - 12c+ 4 -( 1 -^+2)-(9c^i) (B ard) 14. Simplify: ja^_^ + _M.U ^-l) (Board) 15. Find the highest common factor and the least common multiple of x 3 3x 2 +x 3 ,_ . ... and s*-3s*-s+3 ( Columbia ) Miscellaneous Exercises on Complex Fractions 104. Solve the following : 1. Simplify the following expression: a _xa (Harvard) 1 2, \ x+a x 2 a? 100 THIRD-YEAR MATHEMATICS 2. Reduce the following expression to a single fraction; U ^7T (Harvard) 3. Solve: /1+x l-x\ /3.x \ r " h 2x-6/2 X , v , N (l^-I^)(^+- 4 -^) = J5_ (YalG) x 3 4. Reduce to the simplest form: 11 . 2a a-\-x ax a 2 x 2 /rn . N 1 1 2a (ChlCag0) a-\-x ax a 2 x 2 5. Reduce: (Chicago) a 2 2ax , 1 x 2 6. Reduce: 2+acH 1 x \-\-x t ac ~~2~ ; a 2 c 2 -l 1-z 4 7. Simplify: (Chicago) W-pl . J A J_\ (Princeton) 1-2/ 2 x 2 -z i_l x, FACTORING. .FRACTIONS 401 8. Simplify: 6a 3 +7a6 2 +126 3 1 fTX ,, 3a2 - 5 a6-46 2 " 3 5a+46 (HarVard) 196 19a 2 9. Solve: _5 7_ X 3 , 3 * 1 + 1 5 3 10. Solve: (s-1) |(1+*) 35(l-J) 7 1+-^-= 2 . (Harvard) z+2aH a; 11. Solve for z: , 2 12. Show that the equation 2ax 1 x+2a 2 +6a s , a+3 a 2 + 2 -9 x reduces to x 2 (z 2 -2ax+a 2 -9) =0 (Harvard) 13. Simplify: % s | % 2 /i.^Wl-^l x+2y 2y-x^x*-4y\ \ 9/ \ 16/ (YaJe) 4y-:c ' t 7c 4 (2?/-a;) 2 144 14. Simplify: g 2 +& 2 _ 6 e a 2 6 2 ^ /a+6, a 6\ e / a , b \ 1_1 " a^+F 3 ' U^ a+6/ * \a+& a-b) b a 102. THTTlDiYE^ .MATHEMATICS 15. Simplify: a+- 16. Simplify: x-l\ 2 / x+l \ 2 _/ x l \' \x-l) Wu (Cornell) I. Binomials Summary 105. Polynomials to be factored may be classified according to the number of terms they contain. 1. The difference of two squares: (1) x 2 -y 2 =(x+y)(x-y) (2) ( a +b) 2 -(s+t) 2 = (a+b+s+t)(a+b-s-t) 2. The difference of two cubes: x* - y* = (x - y) (x 2 +xy+ y 2 ) 1 3. The sum of two cubes: x* + y z = (x + y) (x 2 - xy + y 2 ) 4. The sum or difference of like powers higher than the fourth: x 5 y b ; x 5 -{-y 5 ; x 7 y 7 ; x 7 -\-y 7 ; etc. 1. The trinomial of the form ax 2 -\-bx-\-c Factored by trial: (1) 10x 2 -17:c+3 = (2z-3)(5:c-l) Factored by formula: (2) ax 2 -\-bx-\-c = a(x n)(x r 2 ),nandr 2 being the roots of the equation ax 2 +bx+c = |2. The trinomial square: x 2 2xy+y 2 =(xy) 2 3. The incomplete trinomial square: z 4 +a;y+y 4 = (x A +2x 2 y 2 +y 4 )-x*y 2 = (x 2 +y 2 +xy) {x 2 +y 2 -xy) II. Trinomials FACTORING. FRACTIONS 103 III. Polynomials, not including the forms given in I and II 11. Polynomials containing a common monomial factor: ax-\-ay-{-az = a(x-\-y-\-z) 2. The perfect cube of a binomial: a 3 3a 2 &+3a& 2 =*6 3 =(a6) 3 13. Polynomials whose terms may be so grouped as to change them to one of the preceding forms: (1) x 2 +2xy+y 2 -k 2 =(x+y) 2 -k 2 (2) x 2 +2xy+y 2 -a 2 +2ab-b 2 = {x+y) 2 -{a-b) 2 (3) x 2 +2xy+y 2 -5x-5y+6 = (x+y) 2 -5(x+y)+6 4. Polynomials containing binomial factors of the form xa: 3x 3 -x 2 -4x+2=(x-l)(3x 2 +2x-2) CHAPTER VI EXPONENTS. RADICALS. IRRATIONAL EQUATIONS The Fundamental Laws of Positive Integral Exponents 106. Base. Exponent. Power. The symbol a 3 means a a a. Similarly, a n means a a a a. . . .(n factors). The number a in a n is the base, n is the exponent, and a n is the nth power of a, or, briefly, a n is a power. Accordingly, a n has a meaning only if n is a positive integer. 107. The product of two powers having equal bases. The product of two powers having the same base may be simplified as shown in the following illustrations : 1. 5 3 - 5 4 =(5- 5- 5)(5- 5- 5- 5) = 5 7 2. (-2)2. (_ 2 )3 = (-2)(-2)(-2)(-2)(-2) = (-2)* 3. a m -a n = [a a- a. . . . (m factors)] [a a a (n factors)] = [a a a a. . . . (m-fn) factors] = a m+n . Q m , a n_ a m+n Express this law in words. EXERCISES Write the following expressions in simplest form: 1. a 9 a 8. (2a) 3 3a 2 2. (-a) 9 - (-a) 9. (-6) 3 -(-6) 5 3. fc 2 <* 6 10. (-2) 3 +(-5) 2 -(-l) 4 4. a** 1 a 2 11. 2(x+?/) 2 S(x+y) 3 5. z 3r x r 12. (-&)" (a-6) 2 "+ 3 6. a x b x a 2 *6 3 * 13. 3a 2w + 3 6 n ~ 4 4a n " 6 6 3n - 2 a?b An 7. 2a 3 3a 2 14. a\a+b)\a-b) r - 1 a^a+fc)'* 1 104 EXPONENTS. RADICALS. IRRATIONALS 105 Find the value for x = 2: 15. 3z 3 -2z 2 +5-4 16. rf+2x?-7x*-Sx+2 Factor the following polynomials: 17. x 4m -5x 2m +6 18. x m + i +2x m + 2 +10x m 108. The quotient of two powers having equal bases. The following examples illustrate how to find the quotient of two powers with equal bases : a 5 _ft ft ft a a _ 2 'a 3 ft & & o (~3) 6 (-3)(-3)(-3)(-3)(-3)(-3) z " (-3) 4 ~(-3)(-3)(-3)(-3) ~ { 6) a m ft -ft -ft -^....(m factors) 3. ^- = - : : : ? * i s ==a * a ' [{mn) factors] a n ft ft ft ft (n factors) LV J =a m ~ n a 1 = a m n ; provided m>n. Express this law in words. EXERCISES Write the following expressions in simplest form: 1. x 10 +x a 2 a 2n ~ l 7. m 12 a n 2 * m 2 (a 2 -6 2 )(s 3 +i/ 3 ) 2 3. (_ a )6^_(_ a ) 2 # (x+y) 2 (a+b) 2 a x+4 x 2 y 2 {x-y) A 4 ' -tf- y ' (x-^)(x-y) ( a+ 5)3x+ y 9a 6 6 3 c 4 ^ 3a 2 6c 3 5 - ( a+ 5)x+j/ 10 * 4zyz : Sx b y 2 z 3 a 2n a^V^+^V 3 6. it 11. a 2 z?/;2 106 THIRD-YEAR MATHEMATICS 109. The power of a product. The following illustra- tion shows how to find the power of a product : (ab) z = ab ab - ab = a a a b b b = a 3 6 3 Similarly, show that (2 5) 3 = 2 3 5 3 ; (3 a) 4 = 3 4 a 4 . Show that (ab) m =a m b m , or a m b m =(ab) m . Express this law in words. EXERCISES Express the following powers as products : 1. (2ab) A 3. (Sxyzy 5. (abxy) 2n 2. (xy) z 4. (-2ab) 2 6. (2ran Sp) : Find the value of each of the following: 7. 2 3 3 3 8. 2 2 5 2 10. 20 2 -5 2 23.3 3 = (2.3) 3 4 3. 25 3 = 63=216 9- 4 '^ 11. 3 4 -2 4 110. The power of a quotient. The following illus- tration shows how to find the power of a quotient : / a V a a a _ fl3 W ~b'b'b = b 3 Similarly, show that: V ~V ; W ~(3?/) 4 Show that () -*; or *() Express this law in words. EXPONENTS. RADICALS. IRRATIONALS 107 EXERCISES Express the following powers as quotients: 1 {W-Y q /lOaby _ f -2mt y X ' \2abJ 3 * \9acJ 5 * \-5pq) /gy\ 4 _ (ay) 4 sV V2a6/ (2a&) 4 16a 4 6 4 * (*)' (- .CO I"* r /(2x)(3y)' 6 ' V(4a)(-6). Find the value of the following: 42 3 6 5 7. 73 8- 35 812 10. ^ 42 3 /42y_ 6 3 8 * P "W/ 9-44 15 2 111. The power of a power. The following example illustrates how to find the power of a power: (a b ) 3 = a 5 a 5 a 5 = o 16 Similarly, show that (2 4 ) 3 = 2 12 ; (a 3 ) 2 = a 6 Show that (O n =a mn = (a n ) m Express this law in words, EXERCISES Simplify the following: 1. [(-2) 2 ] 3 4. (a 4 ) 3 7. (x*-b)+b 2. (-2 2 ) 3 5. (6 2 ) 3 8. (a*- 4 )*- 1 3. (z 6 ) 2 6. (a+i) 2 9. (a?*)*~ 2 108 THIRD-YEAR MATHEMATICS MISCELLANEOUS EXERCISES 112. Write the following in the simplest form: . /2a 2 6 3 \ 3 /3xy\* /9x\* /3a\ /36Y 6. \(_^t) Z V /a?b 4 c n \ n 3# mi 5 7 - vavv (x62)3 8. (a 2 -6 2 ) 2 /_2& 3 x\ 2 " / a 3 +6 3 \ V 5oV 9# Va+6/ Zero Exponents. Fractional and Negative Exponents 113. The symbol a m is a brief expression for a a a . . . .. (m times) . Accordingly m must be a positive integer. In the follow- ing we shall find a meaning for negative, fractional, and 2ero exponents. 114. Zero exponents. The law a m +a n =a m ~ n has been shown to hold for positive integral values of m and n, with the understanding that m is greater than n. If we assume the law to hold also for m = n, we have ?L = a 2 - 2 = a; - = a 3 ~ s = a a 2 a 3 However, by dividing numerator and denominator of each fraction by the common factors a, we have a 2 <* ' a 3 # 1 Thus we may think of a as Me result obtained by divid- ing a power by itself, and we may assign to it the value 1. EXPONENTS. RADICALS. IRRATIONALS 109 In general, if we assume the law a m -i-a n = a m ~ n to hold for m = n } we have CL m j ^-~ = a m - m = a a m a m By reducing the fraction - to the simplest form we a have ^ = i a m ~ Hence we may assign to the symbol a the value 1, i.e., o=l It should be added that a must not be zero. For, m TO is 0, and ^ = ^ has no meaning. EXERCISES Give the value of each of the following expressions: 10; z; (-15); (x'+y); (a-6+c). 115. Negative exponent. Assuming the law to hold also if m< n, we have for m = 3 and n = 5 : " a 3- 5 = a - 2 a 5 a 3 Since by reducing -j to the simplest form, we have a 3 _ & > fi a a a 2 we may de^ne ~ 2 to mean 2 . Similarly, or 1 --, a~ 3 = - 3J a ~ 4 = - 4 - tt a Ui In general, a m = 110 THIRD-YEAR MATHEMATICS EXERCISES Find the value of each of the following expressions: 1. 2 5 2 4. 8 2~ 4 7. 3- 2 2. 6(a-6) 5. 5 3 - 5~ 2 8. .125" 1 ^ -' - Change the following to identical expressions free from nega- tive exponents and simplify: 15. \3zx-V 10. a 8 a~ 2 11. (-2x- 4 )(-Sx~ 1 ) 12. -4a4-a~ 1(J (z 2 ?/- 1 )- 2 13. (2a)~ 2 6 2 (a 2 6~ 2 )- 2a -2 b 17. (a+a-^ia-a- 1 ) 14, 3a 3 6~ 4 18. (rc-far" 1 ) 2 Solve the following equations for x, assuming a^O, a?* I: 19. a*- 2 = a 3 21. (o-*)*-W(o-*)i-* 20. a 2aj + 5 =a 7 -* 22. 4*+* = 8 2 X + 2 116. Fractional exponent. Assuming the law a m .a n = a m+n to hold for fractional exponents, we have a* a* = a; a* a* a$= a; a* a* a* a* = a; etc. Since l^a . Va = a; i^a . #"a l^a = a; Va l/a V a Va = a, etc., we may define a* to repre- sent the same number as V a. Similarly, a^ = f / a; a^ = Va; etc. i In general, a m =Va EXPONENTS. RADICALS. IRRATIONALS 111 / 1\ TO 111 Show that \a n ) = a n a n a n . . . . (m factors) rrrrr' " " -(wi terms) = a =a 1 1 Show that (a m ) n = [a a a. . . . (m factors)]" 111 = a n a n a n . . . . (m factors) 1 / l\m m .'. (a")=UV =a n , or V^=(V~a) m =a n Thus a fractional exponent indicates a root of a power or a power of a root, the numerator indicating the power and the denominator the root. EXERCISES Find the value of each of the following expressions: 1. 4* 5. (-125)* /27 \-l 2. 27* 6. 49-* ' ^ 8 ' 3. (-8)* ,27,-* 9. (-25)* 4. (64)* 7# V64/ 10. (32ar- 5 & 10 )* 117. Summary of the laws of exponents and of the meaning of fractional, negative, and zero exponent. The laws and definitions given in 107 to 116 are as follows: I. a m >a n =a m+n V. {a m ) n =a mn VI. a = l, a^O a VII. <*" m =^ III. {a-b.c) m =a m b m c m i VIII. a n =Va IV ' \b) = r IX. a* = (v / i) m =V / J l 112 THIRD- YEAR MATHEMATICS These laws have been shown to hold for all rational values of m and n. However, we shall find that, by enlar- ging our conception of powers, quite clear and definite meanings can be given to powers with irrational exponents, 146, and even to powers with imaginary exponents. MISCELLANEOUS EXERCISES 118. Change the following to identical expressions free from negative or zero exponents. Simplify as far as possible: 2. \x V m W Viyi 3. (VS)- 12 . ^5^ n n 1 a a~ 4. \x *> ) m ~ n 13. a- 3 +b- 3 5. (a&- 2 c 3 )! a 4 (m-fn\ X * )' 14. n-i a- AJ ra- h b n 6. -j- 15. x- 1 +2x- 2 +3x- 3 * =? (5SMP~* 9 ' V a?V ) 18 * U- 2 6/ ' \zb-*J Multiply as indicated: 19. (x*+i/*)(x* i^) 22. (ar^+i-V- l +ir , )(jr- 1 -r" 1 ] 20. (z- 4 +x- 2 +z)(z-2) 23. (a*+ai&*+6)(ai-&i) 21. (a;-l 2/~t) 3 24. (ma^+njr^+p) 2 EXPONENTS. RADICALS. IRRATIONALS 113 Divide as indicated : 25. (a?+a 2 +ai+a+ 7 6^T= ^42 3f xy*c 5#ri*=dy&xyc bdfcd 2 = lhdyf xyc 2 d 2 EXERCISES Multiply as indicated: 1. Vtf- Va 4. (1/2-1/3)1/5 2 6. (2v / 5-5)(3-v / 5) 3. 3>^5 /10 7V^5 6. (/6+ vlo) (/3 - -/5) 126. Multiplication of radicals of different orders. If the radicals to be multiplied do not have the same index, they should be changed to the same order before multiplying. For example, ^Vl = ^ . 5* = 4 - &m V&V&= ^2000 Similarly, 4 /sit/ Sf^y = 4:VxY 3v^V = 12v^V = 12xVxy b EXERCISES Multiply as indicated: 1. V&Vx 3. V2xf3x 2 5. f2*V\ 2. f&Vx* 4. fa^i/a; 6. ^9^ Vlbx Division of Radicals 127. Division by a monomial. As in multiplication, radicals are to be brought to the same index before divid- ing. Thus: V&+ Vx b = V&+&Vx4V&+&V&*-^yi x EXPONENTS. RADICALS. IRRATIONALS EXERCISES Divide as indicated : i/T2 Vxy %\ r x l -w 3. 7=- V x 6. 57= 2fx , Vl 2f 54 fs '71 4 - 71 6 - v^ l/9 119 MISCELLANEOUS EXERCISES 128. Simplify the following : 3 2 c~ 2 96 S 4 b 24a * 8c 4 1 ^ *" >3 4 6 3. **?($*()'* (Sheffield) 4. ffcr-S-V'iP; (\-x) + (l-i / x); 5. (^) -^3 27-i+(-243)=+( v -^rj (Chicago) 6. -x 2 (9-x 2 )~ i + > / 9 : ^ 2 H . 3 (Sheffield) __ _ V*-(!) 2 7 - VS5-VS+ A 17 ^ (SheffieId) 8. / a 3 - a*b - V a 6 2 - 6 s - V (a+b) (a? - b 2 ) (Chicago) 9. 3^+1/40+^?--^== (Sheffield) 10. 3>I+2"\_-4aJJ (Sheffield) 120 THIRD-YEAR MATHEMATICS 11. /2X^3; 2v / 20-i / 80+\/^ (Sheffield) 12. A^+/63+5/7; (4v / 7-8v / 2l+6^42)^-2v / 7 14. Determine, without extracting roots, which one of the following is the greatest : t^ 10, / 6 , / 17 . Rationalizing the Denominator 129. Rationalizing the denominator. The process of changing a fraction with irrational denominator to an equivalent fraction with rational denominator is called rationalizing the denominator. By means of this process it is possible to avoid dividing by a decimal fraction when the value of the fraction is required. 1 . For example, to find the value of 2 , ^* it would be neces- sary to approximate the square root of 3, to add the result to 2, and to divide the sum into 1. However, multiplying numerator and denominator by 2-i/3, we have 1 _ 1(2-/3) _2-v / 3_^ 2 Vz 2+/3 (2+/3)(2-/3) 4 -3 Hence the value of y may be found easily by subtract- ing /3 from 2. Moreover, by rationalizing the denominator it is possible to reduce the number of square roots required to find the value of a fraction. , 1/5-/2 (/5 -Z2) 2 7-2/10 For example, y^r 2 = {Vb ^ V - 2){Vl - V% T" EXPONENTS. RADICALS. IRRATIONALS 121 It is seen that the given fraction calls for the approxi- mation of two square roots and division by a decimal fraction. After rationalizing the denominator it is neces- sary to extract only one square root and to divide by 3. The number by which numerator and denominator are multiplied to make the denominator rational is called the rationalizing factor of the denominator. Radical expressions of the forms a-\-Vb and aVb, Vx-\-Vy and VxVy, are called conjugate radicals. EXERCISES Change the following fractions to equivalent fractions with rational denominator: VI z-Vl 1. -7= 7. 7= Vz 3+1/5 Vl = VlVz = Vib 3^5-4 VS 1/31/3" 3 2l/5+3 2 4 9 ^+2/5 1/2 2/2-3V5 _ 3+/18 31/3-1/7 3 * vz Va+b V2 i/3-l V2 2-V2 3+i/2- 14. Find the value of each of the fractions in exercises 6 to 10 to three significant figures. 122 THIRD-YEAR MATHEMATICS MISCELLANEOUS EXERCISES |130. Solve the following exercises: 1 a i * Vx+2aVx-2a x _ .. 1. Solve for x: , -== (Board) Vx+2a+Vx-2a 2a /i/3-V2\ 2 /V3+i/2\ 2 2. Snnphfy: (^^(__) (Board) 3. Simplify: P / 20+> / 12 (Board) 4. Solve: 1/5-1/3 1 1 , 1+3^2 3s 2 -2:r3 __ . - + T^7i = ~^^~ (Yale) 5. .bind the approximate value of 7= r=-. 7=. V2-V6 Vz-2 to three decimal places. (M.I.T.) 6. Rationalize and find correct to two decimal places: }-' ,- (Yale) 2+i/5-i/2 1/2+21/3 7. Simplify -7= ^= and compute the value of the fraction to two decimal places. (Yale) 8. Find the value of x from the equation 5x = i/3(l+2x) and express it as a fraction having a rational denominator. 9. Simplify: 2z 2 i / 9x 2 +81+27i / 4:r. 2 +36 Square Root of a Radical Expression 131. By squaring the binomial V a-\-Vb we have a+6+2V / a&. Therefore Va-\-Vb is the square root of the binomial (a+6)+2l^a6. Hence it is possible to find the square root of a binomial of the form x+al/y if it can be changed to the form a+b+2Vab. EXPONENTS. RADICALS. IRRATIONALS 123 The following examples illustrate the process: 1. Find the square root of 84-1/48. 8+/48 = 8+2/l2 = 8+2/o^ = 6+2+2/(T~2 .'. V / 8+/48=/6+/2 2. Find the square root of 38+3/32. 38+3i/32 = 38+i/9 32=38+1/9 -8-4 = 38+2/72 = 38+2v / 36^2 = 36+2+2i / 72 .\ V38+3/ 32 = 6+ 1/2 EXEKCISES Find the square root of each of exercises 1 to 9: 1.3-2/2 4.7+4i/3 7.3-1/5 2. 6-2/8 5. 11-3/8 8. 7+/I8 3. 11-4/7 6. 14+6/5 9. 11-6/2 10. In finding the sine of 15 by two different methods, we obtain the results |(/6 /2) and |/2 /3, respectively. Show that these results have the same value. Irrational Equations 132. Irrational equations in the form of quadratics. By changing the form some irrational equations may be solved like quadratics. For example, the equation 7x 2 -5x+l-8/7x 2 -5x+l=-15 may be changed to a 2 -8a+15 = 0, where a= /7x 2 5z+l Show that ai = 5, a 2 = 3 /. /7x 2 -5x+l = 5, /7x 2 -5x+l = 3 By squaring both sides of these equations quadratic equa- tions are found which may be solved for x. 124 THIRD-YEAR MATHEMATICS EXERCISES Solve the following equations: 1. a; 2 -5x+2v / x 2 -5x-2=10 Subtract 2 from both sides of the equation. 2. 6y 2 -3y-2=V2y 2 -y Notice that 6y 2 -3y = S(2y 2 -y) 3. x 2 -3x+4+Vz 2 -3a;+15=19 4. 2/ ! -|/ -l = 5. 4x-^-3x-^-l = 6. z-i-5:r-f+4 = 7. 3jr l +20fr i -32*0 8. (a+2)*-(a+2)i-2 = 9. f 7^6+4 = 4 i 7 7^6 133. Irrational equations solved by reducing them to rational equations. The following examples show the method of solving irrational equations which can be reduced to rational equations. 1. Solve i/z+13- t/x+6 = 1, and check. Adding Vx~+ to both sides, Vx+\Z = l+l/z+6 Squaring, 3+13 - 1 +2Vx~+ +x +6 .'. 3 = l/^+6 Squaring again, 9 = x +6 x = 3 2. Solve t / x+i'2x+1-v / 5x+5 = Isolating V 5x+5, Vx-\-V2x+l = V5x+5 Squaring, x +2'V2x J +x +2x + 1 = 5x +5 2V2x 2 +x=2x+4: Dividing by 2, V2x 2 +x = x+2 Squaring, 2x 2 +x = x 2 +4x +4 .*. xi = 4, x 2 =-l EXPONENTS. RADICALS. IRRATIONALS 125 The value x = 1 does not satisfy the original equation, but it satisfies the third equation. Hence it was brought in by the process of squaring the second equation. It is said to be an extraneous root of the original equation. Example 2 shows that the results obtained by solving an irrational equation are not always roots. Hence it is necessary to check all results in the original equation. EXERCISES Solve the following equations and check the results: 1. y+2Vy^l-4: = (Sheffield) 2. V x~+i+V 2x^1 = 6 (Sheffield) 3. i / 7x+l-> / 3x+10=l (Board) 4. V x+2Q-V~x~=i = Z (Board) 6. V2x+-Vx~=i=Vx~+l (Princeton) 6. V a -x+VoT+x = V 2a+2b 7. Vz+5+ v / 2x+8=v / 7x+21 (Princeton) 8. Vlx-b+ VAx- l = V 7x-4+V / 4x^2 (Harvard) Vbx-+Vb=x 2/x+l Vbx-- v / 5-x~2i / x-l Apply the process of addition and subtraction. 2x-2 10. V8x-7 =V2x+3 V2x+3 Clear of fractions. 11. V3+x+Vx= / J ^3+s J12. ^4=2V^P2-1 13. ^L_4 = 2^^i Reduce the first fraction to the simplest form. H* x ~ b ^x-Vb , nWT a-x , x-6 / r Vx+Vb 6 V a-x Vx-b 126 THIRD-YEAR MATHEMATICS J16. Reduce V (x-4) 2 +2/ 2 + V (z+4) 2 +?/ 2 = 10 to an equation free from radicals and as compact as possible. (Board) J 17. Simplify the following expression as far as possible: Yx^+axt-tfx-a?- V^-3a^+Sa 2 x-a 9 -a^4x-4a Assume that both xa and x+a are positive. (Harvard) Trigonometric Equations 134. Some trigonometric equations reduce to irrational equations. For example, tan 0+sec = 3. Since sec = i/l+tan 2 6, we have i/l+tan 2 <9 = 3-tan0 Squaring both sides, 1+tan 2 = 9-6 tan 0+tan 2 .-. 6 tan = 8 4 tan 0= The value of may be found from a table of tangents. EXEKCISES Solve the following equations: 1. 2sin0 = l+cos0 2. sin 0+cos 0= V2 Summary 135. The chapter has taught the meaning of the fol- lowing terms : base index of a root exponent order of a radical power rationalizing a number radical irrational equations similar radicals zero exponent conjugate radicals negative exponent radicand fractional exponent EXPONENTS. RADICALS. IRRATIONALS 127 136. The following problems review the essentials of the chapter: 1. Give the meaning of each of the following: I. a m . a n =a m+n VI. a=l II. m VII. a m III. IV. V. (a . b c) m =a m b m c n fa\ m a m \bj b m (a m ) n =a mn VIll. IX. a n = Va 2. Explain how to solve the following: 1. To change an expression containing negative or zero exponents to an identical expression free from negative or zero exponents. 2. To remove a factor from a radicand. 3. To reduce a fractional radicand to the integral form. 4. To reduce the order of a radical. 5. To add and subtract radicals. 6. To multiply radicals of the same order, or of different orders. 7. To divide by a radical. 8. To rationalize the denominator of a fraction. 9. To find the square root of a binomial of the form x+avy. 10. To solve irrational equations. 11. To solve trigonometric equations leading to irrational equations. CHAPTER VII _41 .RL Fig. 58 LOGARITHMS. SLIDE RULE Labor-saving Devices 137. Precision of measurement. When we compare a line-segment with a known segment such as an inch or a centimeter, we are measuring the line-segment. To measure AB, Fig. 58, we may lay off the distance AB on squared paper, as A'B' . Using 2 cm. as a unit A B we find the measure of A'B' to be 1.76. The 6 being estimated, the number 1 . 76 does not mean that the length of AB is exactly 1 . 76, but rather that it is between 1 . 755 and 1 . 765. The result, 1 . 76, is said to be expressed to three significant figures, the precision being indicated by the number of figures. Three-figure accuracy may be obtained with ordinary instruments. In surveying, four-figure accuracy is usually sufficient, but with skill and good instruments five-figure accuracy is possible. 138. Abridged multiplication. In adding, subtract- ing, multiplying, or dividing two numbers obtained by measurement it is useless to express the result to greater accuracy than that of the less accurate of the original numbers. Much labor may be avoided by omitting the meaningless figures in the product or quotient. The 128 LOGARITHMS. SLIDE RULE 129 following example illustrates the process of abbreviating the multiplication of two numbers which are known only approximately : Find the product of 2.4301 by 7.8043, to five significant figures. The work may be arranged as follows: 2.4301X7.8043 9 1944 17010 72903 7204 08 7 18965 22943 . A study of the complete process of multiplying the two num- bers brings out the following facts: Since in 7.8043 the last figure, 3, is uncertain, the first partial product, 72903, is uncer- tain. This may be indicated by a line drawn under each figure. Similarly, the 4 in the second partial product, 97204, is uncertain, etc. Evidently the final product, 18.96522943, is accurate only to five significant figures, the last figure, 5, being uncertain. Hence we may omit in all partial products the part to the right of the vertical line. A further simplification is obtained by writing the partial products in the reverse order, i.e., multiply 2.4301 by 7, then 2.430 by 8, 2.4 by 4, and 2 by 3. The work may now be arranged in the following form: 2.4301X7.8043 17.0107 1.9440 96 6 18.9649 130 THIRD-YEAR MATHEMATICS * EXERCISES 1. Find by abridged multiplication the following products : 12.13X119.4; 14. 625 X .32814; .1342X2.16 2. A cubic centimeter of mercury weighs 13.596 g., approxi- mately. What is the weight of 7 . 43 cubic centimeters ? 3. How many square inches are contained in a square meter, if a meter is approximately 39 . 37 inches ? 139. Abridged division. The process is illustrated on the following example : Divide 6 . 384 by 1 . 231. Placing the divisor to the right 6 . 384 1 1 . 231 = 5 . 189 of the dividend and marking the 6 155 uncertain figures by a line, we find Z that 1231 is contained in 6384 five 229 times, leaving the remainder 229. 123 We now cut off the last figure in the divisor and find that 123 is _ contained once in 229, leaving the _ remainder 106. The process of j q cutting off a figure from the divisor g is kept up until the whole divisor is used. Hence the quotient is 1 5 . 189, the 9 being uncertain. EXERCISES k 1. Find the following quotients: 63.4^-26.8; 86.423-5-18.25 2. The lunar month has 29.531 days. How many lunar months are there in a year which is equal to 365 . 24 days ? 140. Use of logarithms. By the use of logarithms the operations of multiplication and division may be reduced to addition and subtraction. When only one multiplication or division is to be made, it can be performed quickly by using the abridged pro- LOGARITHMS. SLIDE RULE 131 cesses given in 138 and 139. The use of logarithms is especially valuable where a series of operations is involved. The meaning of logarithms and the theory of computa- tion by logarithms will be discussed fully in this chapter, 145 to 160. 141. Use of the slide rule. Products, quotients, powers, and roots of numbers may be found mechanically by an instrument called the slide rule. A knowledge of logarithms is necessary to understand the principles on which the slide rule is constructed. A full discussion of these principles and of the use of the slide rule is found in 163 to 166. 142. Use of tables. The student is familiar with tables of roots and powers. They may be used to save time and to avoid unnecessary labor. Logarithms 143. Table of exponents. The table, Fig. 59, gives the first 25 powers of 2. By means of this table it is 2 = 2! 1,024 = 2 10 262, 144 = 2 18 4 = 2 2 2,048 = 2 n 524,288 = 2 19 8 = 2 3 4,096 = 2 12 - 1,048,576 = 2 20 16 = 2* 8,192 = 2 13 2,097,152 = 2 21 32 = 2 5 16,384 = 2 14 4,194,304 = 2 22 64 = 2* 32,768 = 2 15 8,388,608 = 2 23 128 = 2 7 65,536 = 2 16 16,777,216 = 2 24 256 = 2 8 131,072 = 2 17 33,554,432 = 2 25 512 = 2 9 Fig. 59 possible to reduce multiplication and division of numbers to addition and subtraction respectively. This follows from the theorem a m a n = a m+n . 132 THIRD-YEAR MATHEMATICS For example, let it be required to find the product 512X16,384. The table gives : 512 X 16,384 = 2 9 X 2 14 = 2 23 = 8,388,608. Thus by adding the exponents 9 and 14 we are able to locate the required product. Similarly, to find the quotient 524,288 + 8,192, we find from the table that 524,288 -f- 8, 192 = 2 19 -v-2 13 = 2 6 = 64. EXERCISES Find the value of each of the following: 1. 16X512 4. 4,194,304+131,072 2. 2,048X128 5. 8,192-256 3. 65,536X256 6. 131,072-16,384 ? 128X16X16,384 * 256X32X17024 Evidently a more complete table of exponents would be very useful in performing multiplications and divisions. The following examples show how the table may be used to find powers and roots: 1. Find the square of 2,048. From the table 2,048 = 2". Therefore 2,048 2 = (2 11 ) 2 = 2 22 = 4,194,304. 2. Find the square root of 1,048,576. From the table 1,048,576 = 2 20 . Therefore 1^1,048,576 = Vlfi = (2 20 ) * = 2 10 = 1,024. The exponents in the table, Fig. 59, are called the logarithms of the left members of the corresponding equations. Thus, if 2 is used as a base, the logarithm JOHN NAPIER JOHN NAPIER, BARON OF MERCHISTON JOHN NAPIER, a wealthy Scotch baron, made *J political and religious controversy the main business of his life, but his pet amusement was the study of mathematics and science. He was born in 1550 and died in 1617. The stupen- dous labors in calculating of some of his contem- poraries impressed him with the desirability of devising some way of shortening multiplications and divisions. Rheticus, with forty helpers, had spent years calculating the trigonometrical tables published in 1596 and 1613; Vieta seems to have enjoyed calculations requiring many days of hard labor; Ludolph von Ceulen (1539-1610) gave most of his life to calculating ir to 35 decimal places; and Cataldi (1548-1626) gave years of hard labor to numerical calculating. Napier devised a set of rods, known as "Napier's bones," containing sets of products in convenient form for facilitating multiplying. His virgulae, another invention, were to aid in the extraction of square and cube roots. He dis- covered certain trigonometrical formulas, known as Napier's analogies, and stated his "rule of cir- cular parts," a mnemonic aid to recalling the laws of right spherical triangles. His chief service to science was his invention of logarithms, which, after suitable modification by Briggs (1561-1631), became the powerful aid to calculation that we employ today. His great work on logarithms, entitled Rabdologia, was published in 1617. [See Ball, pp. 235-36; also Encyclopaedia Britannica.] LOGARITHMS. SLIDE RULE 133 of 128 is 7. This is expressed briefly in symbols by the equation log 2 128 = 7, read the logarithm of 128 to the base 2 is 7. Express by means of equations the logarithm, to the base 2, of the following numbers: 16, 256, 2,048, 16,384. 144. .Using 3 as base we have the following table of exponents : 3 = 3* 243 = 3 5 19,683 = 3 9 9 = 3 2 729 = 3 6 59,049 = 3 10 27 = 3 3 2,187 = 3 7 177, 147 = 3 11 81 = 3 4 6,561^=3 8 531,441 = 3 12 EXERCISE Express by means of equations the logarithms, to the base 3, of the following numbers: 243, 2,187, 19,683. 145. Logarithm.* The logarithm of a number, N, to the base, a, is the exponent to which a must be raised to give a power that is equal to N. Thus, if a x = N, then logo N = x. These two equations are equivalent. * Tables of logarithms were first published by John Napier, a Scotch baron, in 1614. Jost Biirgi (1552-1632) had calculated and used extensively tables of logarithms before 1617, but did not publish his tables until 1620. Henry Briggs (1561-1631) introduced the modern idea of logarithms to the base 10, and it was mainly through his influence that logarithms rapidly came into use all over Europe. Kepler introduced them in Germany about 1629, Cavalieri in Italy in 1624, and Edmund Wingate in France in 1626. Briggs also introduced the method of long division now commonly used in arithmetic. See Ball, Short Accourit of the History of Mathematics, pp. 235-37, and Tropfke, Geschichte der Elementar-Mathematik, Band II, S. 145-55. 134 THIRD-YEAR MATHEMATICS r\ EXERCISES 1. Using 10 as base find the logarithms of 10, 100, 1,000, 10,000. Express each result in the form of an equation. 2. Find the logarithm of 1 to the base 1, 2, 3..., a. Express each result in the form of an equation. 4*+^ * fc 3. Find the logarithm to the base 2 of 1, \, \, h T V- ^ + 4. Find the logarithm of a number using the same number as J* . base. Common Logarithms 146. Common logarithms. Logarithms to the base 10 are called common logarithms, or Briggs's logarithms. The base 10 is generally not written, it being understood that 10 is the base unless another base is indicated. Thus, log x means logi z. The table of exponents, 150, contains only numbers that are exact powers of 10. Hence the values of the logarithms of these numbers can be given exactly. The logarithm of a number which is not an exact power of 10 is written as a decimal fraction. Thus the logarithm of 56.23, to five decimal places, is 1.74997, since 56.23 = 10 1 - 74997 , approximately. 147. Characteristic. Mantissa. The integral part of a logarithm is the characteristic, the fractional part the mantissa of the logarithm. 148. Graphical representation of the logarithmic func- tion. We may use the exponential equation 10 y = x to find corresponding values of x and y, satisfying the equation y = log x. The table below gives some of these values. y 1 1 1. i 4 3 4 i 8 X 1/10=3.16 > / l0= 1/3716 = 1.78 1.78 3 = 5.62 1.33 y 1 i 1 4 l ~8 3 "~ 4 X 10 ttot 316 rW=- 562 .749 .178 LOGARITHMS. SLIDE RULE 135 Plotting these pairs of values we obtain the graph of the equation y = log x, Fig. 60. By means of this graph o -.1 .1 .3 -a -.5 1 i i % Fig. 60 it is possible to find the approximate value of the logarithm of a given number. 136 THIRD-YEAR MATHEMATICS From the graph find the logarithm of each of the following numbers : ,1.5,4,5,7.4,8.8. A study of the graph shows the following: 1. Log x is negative for x / 352 420 lo A r = log 49+1 log 352-log 86,420 log 49 = log 352 = adding, log 86,420 = subtracting, dividing by 3 log N N = 23.40X.8625 .00459X6.3804 8 0il 5. f 92; ^183 9. J 0.6712 5.327 n l/25.7\ 2 (-2582) 2 X (.05805) 6 * \|V286/ 2587X(-316) Find the numerical value 3 . 0436 by logarithms, then prefix the \ 3 . 187 proper sign. 11. log 4 64-log 3 9+log 2 1 (Yale) * When a logarithm is to be subtracted from a smaller one, 10 is both added to and subtracted from the minuend. For example, the form of the logarithm 2.34778 is changed to 12.34778-10. 146 THIRD-YEAR MATHEMATICS 13. (v / 278.2X2.578) 3 f .00231X^76.19 ^3.416X^25.9_^Q46 (Board) % 4 $15. A number N has 17 significant figures to the left of the decimal point. What is the characteristic of log Nf of log (log N) ? How long can this process of finding successive logarithms be kept up ? (Harvard.) X 16. Find by logarithms the first three figures of the num- ber 2 61 1. How many figures will this number contain? (Harvard.) 17. Given log 2 = 0.30103, log 3 = 0.47712. Find log 12; log*; log I; log ^6. Exponential Equations 162. Exponential equations. Equations in which the unknown occurs in the exponents are exponential equa- tions. The following example illustrates the method of solving exponential equations by logarithms: Solve the equation 5* =354 Taking the logarithm of both members, log 5*=* log 354, or 2 log 5 = log 354 _ log 354 _ 2. 5490 *~Tol5~ "0.6990' EXERCISES Solve the following equations: 1. 3^ = 226 4. (3.142)^ = 2.718 2. 2< = 437 6. 3 12 ~ 2 * = 243 3. 10*/ = 2. 71828 6. 7*+ 3 =5 LOGARITHMS. SLIDE RULE 147 The Slide Rule 163. Description of the slide rule. The slide rule is an instrument for determining mechanically products, quotients, powers, and roots. It consists of two pieces of rule, Fig. 65, capable of sliding by each other. Taking as unit the length A-B on the rule, Fig. 64, we may mark off, beginning from one end, the logarithms 1 Fig. 64 of numbers from 1 to 10, or from 10 to 1,000, or from 100 to 1,000. Thus, Al = log 1=0. A6 = log 6= .78 A2 = \og 2 = .30 A7 = log 7= .85 A3 = log 3= .48 AS = \og 8= .90 44 = log 4= .60 A9 = log 9= .95 A5 = log 5= .70 A10 = log 10 = 1.00 - In general, the logarithm of a number is the distance from A to that number. 7 8 9 1 Mv tVt Fig. 65 The arrangement of the slide rule, Fig. 65, makes it possible to find the sum or difference of logarithms even more rapidly than with the ordinary table of logarithms. For example, to find the sum of two logarithms, as log 2+ log 3, place scale B, Fig. 65, in such a way that the 148 THIRD-YEAR MATHEMATICS division marked 1, on scale B, falls directly under the division marked 2, on scale A. Then division 3, on scale B, falls on division 6, on scale A, and the distance A 6, or log 6, is equal to log 2+ log 3. It is evident that this process is practically the same as that of finding the product 2X3 from a table of logarithms, which is as follows : Let N = 2 X 3. Required to find N. From the table, log 2 = 0. 3010 and log 3 = 0.4771 adding, log N = 0.7781 From the table, N = 6. To find the difference between two logarithms, as log 6 log 3, place division 3, on scale B, directly below division 6, on scale A. Then division 1, on scale B, falls directly below division 2, on scale A. Hence the distance A 2, or log 2, is equal to log 6 log 3. Compare this process with that of finding the quotient 6 by logarithms. The two preceding examples show how scales A and B may be used to multiply and divide numbers. 164. The Mannheim slide rule. The Mannheim rule, Figs. 65 and 66, has four scales, denoted A, B, C, and D. l: ' : ''i Fig. 66 Scales C and D are laid off to a scale twice as large as that of A and B. Hence the logarithm of a number on scales C or D is represented by a segment twice as large as the LOGARITHMS. SLIDE RULE 149 segment representing the logarithm of the same number on scale A. For example, log 2 on scale D = 2 log 2, or log 4, on scale A; log 3 on scale D = log 9 on scale A, etc. It follows that a number on scale A is the square of the number vertically below on scale D, and that a number on scale D is the square root of the number vertically above on scale A. Therefore scales A and D may be used to find the squares and the square roots of numbers. The Mannheim rule gives results to three significant figures which is sufficiently accurate for ordinary use. Fig. 67 If greater accuracy is required, Thacher's slide rule, Fig. 67, is used. This rule gives results to four or five significant figures. The distance from 1 to 2 on scales C and D, on the Mannheim rule, is divided into 10 parts, and each of these is again divided into 10 parts. These subdivisions make it possible to read off between 1 and 2 numbers containing from 2 to 4 digits, as 1.2, 1.34, 1.526, the 6 in the last number being estimated by the eye. Since the mantissas are the same for all numbers having the same digits in the same order, we can give the left index the value 10, 100, or 1,000. Thus, if the value of the initial 1 be 100, the divisions between 1 and 2 will be 101, 102 . . . . 110, 111 .... 120 .... 199; the division between 2 and 3 will be 200, 210, 220 .... , etc. 150 THIRD-YEAR MATHEMATICS On account of these subdivisions scales C and D, when used to multiply and divide, give more accurate results than scales A and B. 165. The use of the slide rule. The following examples illustrate the use of the rule : 1. Multiplication. To find the product 4X3. The directions are given in Fig. 68 : The index l of scale C is put over one factor on scale D. The product is then found on scale D under the other factor on scale C. 2. Division. To divide 18 by 3. Put the divisor on C over the divi- dend on D. Find the quotient on D under 1 on C, Fig. 69. 3. The product of several factors, several factors the runner, r, is used. Follow the directions in Fig. 70 to find the product 8X6X5X2. C I Put r D over 4 under 3 find 12 = 4X3 Fig. Put 3 over 18 under 1 find 6 = 18^3 Fig. 69 To find the product of c Put 1 1 r to 6 11 to r r to 5 1 to r 1 under 2 D over 8 I 1 find 480 = 8X6X5X2 Fig. 70 4. Reduction of fractions. To reduce - Follow the directions given in Fig. 71. Put 42 26X16.8X35X18 42X15X91X1.2 over 26 r to 168 15 to r r to 35 91 to r r to 1 12 to r below 18 find 4 = result Fig. 71 5. Squares. Find the value of 12 2 . To find the square of a number place the runner on the num- ber on scale D, Fig. 72. A I Find 144= 12 2 The square is found directly above, on scale A. * The right index 1 is used here, 1) Put r on 12 Fig. 72 LOGARITHMS. SLIDE RULE 151 6. Square root. To find the square root of a number proceed as follows: Place the runner on the given number on scale A. The square root of the number is directly below, on scale D. 166. Trigonometrical computations. On the reverse side of the slide three scales are found. Scales S and T are the scales of angles. Scale A gives the sines of the angles in scale S, and scale D the tangents of the angles in scale T. The third scale gives the logarithms of the num- bers on scale D. By means of these scales it is possible to find such products as a sin x, or a tan x. The preceding rules exemplify most of the important applications of the slide rule. There are various makes of rules, and makers generally furnish with each rule a pamphlet giving complete instructions as to its use. Summary 167. The chapter has taught the meaning of the fol- lowing terms : precision of measurement table logarithms abridged multiplication common logarithms abridged division characteristic logarithm mantissa slide rule exponential equation 168. The following problems review the essential parts of the chapter: 1. Explain the processes of abridged multiplication and division. 2. Discuss the uses of logarithms and the slide rule as labor- saving devices in numerical calculations. 3. Give a discussion of the graph of the logarithmic function. 4. State the rule for determining the characteristic of a logarithm. 152 THIRD-YEAR MATHEMATICS 6. Explain (1) how to find the logarithm of a number by means of the tables; (2) how to find the number corresponding to a given logarithm. 6. State and prove the theorems regarding the properties of logarithms used in finding products, quotients, powers, and roots. 7. Explain the use of logarithms in the solution of expo- nential equations. CHAPTER VIII LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS. SOLUTION OF TRIANGLES Use of the Table of Logarithmic Functions 169. Logarithms of trigonometric functions. In chap- ter vii logarithms were used to calculate expressions in- volving products, quotients, powers, and roots. When logarithms are to be used to find the value of an expression involving trigonometric functions, the values of the func- tions could be looked up in a table of trigonometric func- tions and the logarithms of these values could then be found in a table of logarithms of numbers. To save labor the logarithms of the sines, cosines, tangents, and cotan- gents of angles between and 90 are given in a special table. The logarithms of secants and cosecants are rarely used and may be obtained from the logarithms of the cosines and sines, respectively. 170. Arrangement of the table. Since the values of the sine, cosine, and tangent of angles between and 45, and of the cotangent of angles between 45 and 90 are less than 1, their logarithms will have negative character- istics. To avoid negative characteristics the form 9-10, 8-10, 7-10, etc., is used in place of 1, 2, 3, etc. The 10 is, however, omitted from the table. Hence, to have the true value of the logarithm, 10 must be sub- tracted from the logarithm found in the first, second, and fourth columns. When the angle is less than 45, the number of degrees is indicated at the top of the page and the number of 153 154 THIRD-YEAR MATHEMATICS minutes is given in the left-hand column. When the angle is more than 45 and less than 90, the number of degrees is indicated at the bottom of the page and the right-hand column gives the number of minutes. 171. To find the value of the logarithm of a function of a given angle. The following examples illustrate the method. 1. Find the value of log tan 5250'12". The mantissa of log tan 5250' = 12026 The mantissa of log tan 5251' = 12052 .-. The tabular difference for 60" = 26 The difference for 12" = jf X26 = 5.2 This difference may be obtained quickly by means of the' table of proportional parts as follows: Changing 12" to minutes, 12"= (^x) =.2'. This means that the required number is in the second line of the table headed 26. .'. log tan 5250 , 12" = 0.12026+5.2 = 0.12031 2. Find the value of log cot 4825'38". log cot 4825' = 9 . 94808-10 log cot 4826 , = 9. 94783-10 Since the tabular difference is equal to 25, and since ) = . 63', we find in the sixth line of the table headed 25 the number 15 . 0. This means that . 6 of 25 is 15. Similarly we find that .03 of 25 is .75. Hence, .63 of 25 is 15.7, or 16 units of the fifth-decimal place. Since the cosine-function decreases as the angle increases, we must subtract 16 from the logarithm of cot 48 25' to get the logarithm of 4825'38r .'. log cot 4825'38" = 9.94808-10-16 = 9.94792-10 LOGARITHMS. SOLUTION OF TRIANGLES 155 EXERCISES Find the value of the following logarithms: 1. log sin 7123'41" 5. log tan 2725 , 10" 2. log cos 4115'35" 6. log sin 4157'36" 3. log tan 3947'36" 7. log tan 3736.'5 4. log sin 6558'24" 8. log tan 2313.'3 172. To find the angle corresponding to a given loga- rithmic trigonometric function. The following examples illustrate the method. 1. Given log sin A = 9.98357-10. Find A. We find that the mantissa lies between the mantissa of log sin 7420' and log sin 7421', that the tabular difference is 3, and that the difference between the mantissa of the given loga- rithm and that of log sin 7420' is 1. In the table of proportional parts headed 3 we find . 9 nearest in value to 1. Hence we may write 1= .9+.1. In the first column and in the same line with . 9 we find 3. Similarly the number nearest to . 1 in the table of propor- tional parts is .09 and the corresponding number in the first column is .03. .-. A = 7420.'33 = 4720'21" 2. Given log cos A = 9. 85981 -10. Find A. The table shows that the mantissa lies between the mantissas of log cos 4336' and log cos 4337: The tabular difference is 12. The difference between the mantissa of the given logarithm and that of log cos 4336' is 3. Hence in the table of proportional parts headed 12 in the second column we look for the number nearest to 3. This is either 2 . 4 or 3 . 6. Let 3 = 2.4+. 6. The corresponding numbers in the first columns are 2 and .05 .'. A = 4336.'25 156 THIRD-YEAR MATHEMATICS EXERCISES Find the value of A in each of the following equations: 1. log sin A = 9. 97527 -10 4. log tan A = 0.25936 2. log sin A = 8. 73997- 10 6. log cos A = 9. 94749- 10 3. log cot A = 9 . 40146 - 10 6. log sin A = 9 . 42443 - 10 Use of Logarithms in the Solution of Right Triangles 173. Solution of triangles. To solve a triangle is to find the values of some of the sides and angles by means of the given sides and angles. In the course of the second year right triangles were solved by use of the natural values of the trigonometric functions. We are now able to carry on by logarithms all multiplications and divisions involved in the solution. 174. Formulas. The relations between the sides and angles of a right triangle, Fig. 73, are expressed in the following formulas: B a = c sin A b = c sin B a c cos B b = c cos A a =b tan A b = a tan B , / a =b cot B b = a cot A c 2 = a 2 +b 2 Fig. 73 The equation c 2 = a 2 -\-b 2 is usually taken in the form a = i/c 2 -6 2 = l/(c+6)(c-6) The area of the right triangle is given by the formulas ao 2 c 2 S=2=2V / (c+6)(c-6) =2^TB = 2 sin B C0S B The preceding formulas are all adapted to logarithmic computation. LOGARITHMS. SOLUTION OF TRIANGLES 157 Taking the logarithm of both sides of the equa- tions they take the forms: log a = log c + log sin A; log & = log c-flog sin B, etc. To determine an angle, the tangent or cotangent should be used because these functions change more rapidly than do the sine- and cosine-functions. To determine a side, it is best to use the sine- or the cosine-function of the given angle. 175. Arrangement of the solution of a right triangle. The following example illustrates the plan to be followed in solving the right triangle: Case I. Given the sides of the right angle. To find the angles and the hypotenuse. Let a = 418 and 6 = 325, Fig. 74. (a) Draw a figure marking the given and required parts. (6) The formulas to be used in the solu- tion are: tan = -; c = - 5 ; A=90-. a sin5 The equation a = V (c+b)(c b) is to be used as a check. (c) Make a detailed outline of the computation to be made, thus: log 6= log 6= Check: log(c+&) = log a= log sin B = log (cb) = log tan B = log c = B= c = A=90-B = (d) Carry out the computation log a 2 =2 log = loga = Compare this with log a, found above. according to the plan in (c). Case II. Given the hypotenuse, c, and one of the sides, b, of the right angle. The formulas to be used are: sin B = cos A=-; a c' w tan B' a = V / (c+&)(c-6). 158 THIRD-YEAR MATHEMATICS Case III. Given one angle, B, and one of the sides, 6. Use the formulas A=90-B; a = K=; 0=-^-^: tan B' sm B' a = V(c+b)(c-b). Case IV. Given one angle, B, and the hypotenuse, c. Use the formulas A =90 B; b = c sin B; a = c cos B; a = V(c+b)(c-b). EXERCISES By means of logarithms solve the following right triangles: 1. c = 25, a = 22 6. a=194. 5, 6 = 233.5 2. c = 35 . 145, A = 2524'30" 7. b = 547 . 5, B = 3215 , 24 ,/ 3. a = 316.5, c = 521.2 8. c = 672.4, = 3516'25" 4. = 239', 6 = 75.48 9. a= 3.414,6 = 2875 5. c = 369.27, a = 235.64 10. a = 617.57, c = 729.59 Solve the following problems: 11. In order to determine the width of a river, a surveyor measured a distance of 100 ft. between two points A and B on one bank. A tree stood at a point C on the opposite bank. The angle ABC was found to be 6340' and the angle BAC to be 5535? Calculate the width of the river. (Yale.) 12. The base of a certain triangle is 3,248 ft., and the base angles are 4615'[ = 46?25] and 10037 , [ = 100?62]. Find the altitude. Sketch the figure (roughly) to scale, and see whether your result is reasonable. (Harvard.) 13. A and B are two points on opposite banks of a river 1,000 ft. apart, and P is the top of the mast of a ship directly between them. The angle of elevation of P from A is 14?33 (1420') and from B the angle of elevation is 8?17 (810'). How high is the mast ? (Harvard.) 14. The shadow of a tower standing on a horizontal plane is observed to be 100 ft. longer when the sun's altitude is 30 than LOGARITHMS. SOLUTION OF TRIANGLES 159 when the altitude is 45? What is the height of the tower ? Do not use tables, but express the result in terms of radicals. (Yale.) 15. A circle of radius 5 subtends an angle of 20 at a point A, and M and N are the points of contact of tangents drawn from A. Find the perpendicular distance from M to AN. (Harvard.) 16. The value of the smallest division on the outer rim of a graduated circle is 30'[ = 0?50], and the distance between the successive graduations, measured along a chord, is 0.02 inch. What is the radius of the circle ? (Harvard.) 17. Each of two ships A and B, 415 yd. apart, measures the horizontal angle subtended by a cliff and the other ship; the angles are 48 17' and 90 respectively. If the angle of eleva- tion of the cliff from A is 1524' what is the height of the cliff ? (Board.) 18. At the top of an observation tower which is 200 ft. high and whose base is at sea-level the angles of depression of two ships are observed to be 3032' and 1840 .' At the bottom of the tower the angle subtended by the line joining the two ships is found to be 90? What is the distance between the ships to the nearest foot ? (Board.) 19. A man who is walking on a horizontal plane toward a tower observes that at a certain point the elevation of the top of the tower is 10 and after going 50 yd. nearer to the tower the elevation is 15? Find the height of the tower. (Princeton.) 20. The diameter of the moon is 2,164 mi. long. Find the distance from the earth to the moon if its apparent diameter subtends an angle 31 '1. 176. Isosceles triangle. The perpendicular from the vertex to the base divides the isosceles triangle, Fig. 75, into two congruent right triangles. Since a right triangle is determined by two parts it follows that two independent parts must be given to solve the isosceles triangle. 160 THIRD-YEAR MATHEMATICS The following equations are used in the solution : +^=90 ^ a = 2b' sin ^ = 26' cos h= W + l){ b '~t) = l tan B=h ' sin B . ah Area = -y 177. Regular polygon. Lines drawn from the center of a regular polygon to the vertices divide the polygon into congruent isosceles triangles. Denoting the side by a, the radius of the inscribed or circumscribed circle by r, and the number of sides by n, we have - = r sin -= , or a = 2r sin for the inscribed polygon 2 2n n and 7% = r tan -p, or a = 2r tan for the circumscribed 2 2n n polygon. The area in both cases is one-half the perimeter multi- plied by the apothem. Relations between the Sides and Angles of Oblique Triangles 178. By means of certain relations between the sides and angles of oblique triangles it will be possible to compute from certain given parts the remaining parts of a triangle. These relations are stated in the form of three laws, called the law of sines, the law of cosines, and the law of tangents. LOGARITHMS. SOLUTION OF TRIANGLES 161 179. The law of sines. 1. Let h be the length of the perpendicular from C to A B in triangle ABC, Fig. 76. Show that sin A = T and h = b sin A . Show that sin B = - and h = a sin B. a ,\ a sin B = 6 sin A. It follows that - t = - ^ . sin A sin B By drawing a perpendicular from A to BC we obtain in a similar way Fig. 76 sin B a sin C b sin A sin. B sin C This equation is known as the law of sines. It may be expressed in words as follows: The sides of a triangle are proportional to the sines of the opposite angles. In the obtuse triangle ABC, Fig. 77, sin A= T and h = b sin A o sin x = sin (180 5) =sinB = - and/i a b sin A = a sin B a b a sin B and sin A sin B It will be seen in 182, 187, 189 how the law of sines is used in the solution of triangles. Fig. 77 180. Diameter of the circumscribed circle. The con- r = = Ti has an interesting geo- A sin B sin C to stant ratio sin 162 THIRD-YEAR MATHEMATICS metrical meaning. If a circle is circumscribed about AABC, Fig. 78, it follows that ZA= ZD. .' . sin A = sin D = -.,d denot- ing the diameter. d=- 7. sin A Thus the constant ratio of the side of a triangle to the sine of the opposite angle is equal to the diameter of the circumscribed circle. Fig. 78 181. The law of cosines. 1. Let Z A, Fig. 79, be acute. Then a 2 = b 2 +c 2 -2cb' (The square of the side opposite the acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides and the projection of the other upon it.) Since b' = b cos A, it follows that a 2 = b 2 +c 2 -2bc cos A. This means that the square of a side of a triangle is equal to the sum of the squares of the other two sides diminished by twice the product of these two sides and the cosine of the included angle. This theorem is the law of cosines. 2. If Z.A is obtuse, Fig. 80, fig. 80 a 2 = b 2 +c 2 +2cb' Since b' = b cos x = b cos (180 - B) = - b cos B, it follows that a 2 = b 2 +c 2 -2bc cos A. LOGARITHMS. SOLUTION OF TRIANGLES 163 Thus the same equation holds for acute and obtuse angles A . Similarly, we find b 2 = c 2 +a 2 -2ca cos B c 2 = a 2 +fc 2 -2a&cosC Show that the theorem of Pythagoras is a special case of the law of cosines. The following simple device makes it unnecessary to memorize each of these three equations. Imagine the letters a, b, c and A, B, C placed on a circle, Fig. 81. Following the direction indicated by the arrows we pass from A to B, then to C, and again to A. By changing, in this order, the letters in the first equation above, we deduce the second equation. Similarly the third equation may be deduced from the second. One formula is said to be obtained from the other by cyclic substitution. Fig. 81 182. The laws of sines and cosines are sufficient to solve oblique triangles. For, if two angles and one side are known, the equation A+B+C = 1&0 determines the third angle and the law of sines the other two sides. If two sides, a and b, and the angle, A, opposite one of them are known, the third side, c, is found by solving the equation a 2 = 6 2 +c 2 26c cos A for c. The other angles are then found by means of the law of sines. If two sides and the included angle are known, the law of cosines gives the third side and the law of sines the other angles. If three sides are known, the law of cosines gives the angles. 164 THIRD-YEAR MATHEMATICS However, the law of cosines is not adapted to the use of logarithms because it involves terms and not factors. The computation by means of the cosine law without logarithms is likely to be tedious for numbers containing 3 or 4 figures. Hence we shall now obtain formulas that are adapted to logarithmic computation. 183. The law of tangents. Let ABC, Fig. 82, be any triangle. Fig. 82 With C as center and the shorter of the sides passing through C as radius, draw a circle cutting CB and AB in D and E, respectively. Extend BC meeting the circle at D' Draw CE, AD, and AT>' Then ZD'CA=A+B. .'. D'DA = (A+B) Why? A=ZCEA=ZECB+B .'. ZECB = A-B Why? .-. ZDAE = (A-B) ZD'AD = 90 Applying the law of sines to AADB, DB sin DAB a-b = sinJ(A-) AB~ sin ADB' 0Y c "sin [180-J(A+)] = sinf(A-fl) "sin|(A+B) LOGARITHMS. SOLUTION OF TRIANGLES 165 a b sin %(AB) ,. HenCe ' -T- = S inl(A+B) (1) Applying the law of sines to AAD'B, D'B^ sinD'AB AB sin AD'B ' q+b ^ sin [D'AD+%(A-B)] or c sin AD'B = sin[90+|(A-)] sin [90- J(A+)] = cos|(A-) cos(A+#) q+6_cos ^(A ) ' * ~~cT~ cos i(A+B) (2) Dividing equation (2) by equation (1), q+fr cosf(A-) c cos|(A+J5) a +6 cos(A-) sin J (A +5) a-b sin %(A-B) ' a-b cos %(A+B) sin%(A-B) * sin $(A+B) a-b tanl(A-B) This is called the law of tangents. The formula may also be written in the form b + a Janl(B+A) b-a tan^B-A) which is to be used if b>a. By cyclic substitution two similar formulas are ob- tained, involving b and c and c and q, respectively. 166 THIRD-YEAR MATHEMATICS Since A+B ISO C_ ftAO C 2 "2 2~ yU 2' tan(A+5)=cot it follows that C 2 .'. By substitution, IX, cot o a+6 2 a-b tan|(A-J8) Solving for tan (A B), a form of the tangent law to be used when a, b, and C are known. (j 184. Mollweide's equations. By substituting 90 -x A-\-B for 5 in equations (1) and (2), 183, a -b _ sml(A-B) a +b ^ cosl(A-B) ~c~ cos \C ' * sin-C These formulas are called Mollweide's equations.* They are especially useful for checking. * Tropfke (Band II, S. 241-42) says these equations were pub- lished in 1808 by the astronomer Mollweide (1774-1825), and on account of their high applicability rapidly found wide acceptance under his name. The naming is, however, false to history, for at least the second one was known a hundred years earlier. Newton proved it in substance in his Arithmetica universalis of 1707. Both equations were derived as independent trigonometric theorems by F. W. De Oppel in his Analysis triangulorum of 1746. LOGARITHMS. SOLUTION OF TRIANGLES 167 185. Tangents of half the angles of a triangle. Radius of the inscribed circle. Let be the point of intersection of the bisectors of the angles of a triangle and let r be the length of the radius of the inscribed circle. Let x, y, and z denote the lengths of the tangents from A, B, and C, respectively. Then the perimeter = 2x+2y+2z. Denoting the perimeter by 2s and dividing by 2, s=x+y+z Since, a= y-\-z Similarly, s a = x s-b = y; From A A OD, tan =- 2 x tan c = z Fig. 83 Similarly, tan -= = 2 s-b . C tan -= = 2 s-c From plane geometry it is known that the area F of A ABC is given by the formulas F = Vs(s-a)(s-b)(s-c) Before Mollweide's rediscovery they are found also ia Thomas Simpson's Trigonometrie (1765, 2d ed.) and in Mauduit's Principles of Astronomy of 1765. Oppel derived the equations from the law of tangents, Simpson gave a geometrical proof, and Mauduit was content with merely applying Napier's Analogies to plane triangles. Mollweide derived the equations from the law of sines. His real service was to draw effective attention to the great usefulness of these equations in astronomy. 168 THIRD-YEAR MATHEMATICS and F = 2 (a+b+c)=rs _T/(s-a)(8-6)(s-c) Hence, _'i(s-a)(s-b)(s-c) H Solution of Oblique Triangles 186. An oblique triangle can be constructed if three of the six parts are known, at least one of these being a side. Hence in solving oblique triangles we shall consider the following four cases. 187. Case I. Given one side and two angles. The following example illustrates the method of solution: Given. Formulas: A = 4938'30" = 7021'15" 6 = 229.38 C= 180- (A + B) 6 sin A a = Required: C, a, and c c = sin B b sin C sin B Solution: 180 = 17959W A+ = 11959'45' .*. C = 60 15" log 6 = 2.36055 log sin A =9. 88198 -10 Adding, 12.24253-10 log sin = 9.97396-10 Subtracting, log a = 2. 26857 .'.a =185. 59 Fig. 84 log 6 = 2.36055 log sin (7 = 9.93757-10 Adding, 12.29812-10 log sin 5 = 9.97396-10 Subtracting, log c = 2. 32416 .\ c = 210.94 LOGARITHMS. SOLUTION OF TRIANGLES 169 Check: -A = 2042'45" log c = 2. 32416 (-A) = 1021'22" \ogsm(B-A) =9.25473-10 iC=30 7" Adding, 11.57889-10 6-a = 43.79 i og cos JC = 9. 93752-10 log(6-a) = 1.64137 Subtracting, log (b-a) = 1.64137 Since five-place tables are used, the results are computed only to 5 significant figures, the last figure being uncertain, and angles are taken only to the nearest second. Hence fractions of a second are discarded. 188. Cologarithm. The principle log -^ = logl log N may be used to avoid the subtractions in the solution above. Since log 1 = 0, we have log t? = log N 1 = (10 log N) 10. The logarithm of -^ is called the cologarithm of N. When more than one addition and sub- traction is involved, the use of the cologarithm has a real advantage, as it is very easy to subtract mentally a loga- rithm from 10. If cologarithms are used, the solution in 187 is arranged as follows: log b = 2 . 36055 log b = 2 . 36055 log sin A =9.88198-10 log sin C = 9. 93757 -10 colog sin B = 0.02604 colog sin B = . 02604 adding, log a = 2 . 26857 adding, log c = 2. 32416 EXERCISES Solve the following triangles and check the results: 1. a = 29. 73, A = 5236', J5 = 6740' 2. a=788, C=7212'35", = 5543'18" 3. c = 3795, A = 1853'22", = 8112'5" 4. 6 = 37, A = 11536'24", = 2718'10" 5. c = 913.45, A = 6456'18", = 4729'11" 6. c = 327.85, A = 4031'42", = 11052'54" 170 THIRD-YEAR MATHEMATICS 189. Case II. Given two sides and the angle opposite one of them. It is known from c * geometry that it is not always possible to con- struct a triangle with these given parts. 1. We will consider first the case where the angle A is obtuse. Then the side opposite A is the greatest side of the triangle, and one and only one triangle can be con- structed satisfying the given conditions. 2. If Zi is a right angle, one triangle can be con- structed. The solution of the right triangle has been discussed in 175. 3. If LA is acute, various possibilities may arise: (1) If ah and b, the circle will meet A B in two points, F and F', but only AADF satisfies the conditions of the problem, Fig. 89. Since the length of the perpendicular, h, may be expressed trigonometrically in terms of two of the given parts by means of the equation h = b sin A, the preceding discussion may be summarized briefly as follows : 1. A > 90 ; then a > b; one solution : an obtuse triangle. 2. A =90; one solution: a right triangle 3. A <90; and if a a > b sin A ; two solutions if a ^ 6; one solution Ca,se II is called the ambiguous case. EXERCISES State, without solving, how many solutions are possible if the given parts are as follows: 1. A = 5042', a = 204, b = 204 2. A = 2010.'3, a =57, 6 = 42 3. A = 7418 , 13",a=20, 6 = 75 4. A = 326 / , a = 802, 6 = 785 6. A = 45, a= 108, 6=152.71 6. A = 7717.'6, a = 210, 6=196 172 THIRD-YEAR MATHEMATICS Fig. 90 Solve the following triangles: 7. a = 140. 5,6 = 170. 6,A = 40 Discussion: log 6 = 2.23198 log sin A =9. 80807-10 * log 6 sin A = 1.03005 log a =2. 14768 .'. a>b sin A and there are two solutions, AABC and AB'C, Fig. 90. n 6 sin A c = 180 o_ (A+jB) ^smC sin A log a =2. 14768 log sin C=9.99989-10 colog sin A =0.19193 log c= 2. 33940 c = 218.49 log a =2. 14768 log sin C' = 9. 29239-10 colog sin A =0.19193 log AB' = 1.63200 c'=A5' = 42.855 log c = 2. 33940 log sin i(B-A) =8.99348-10 colog cos C = 0.14562 log (6 -a) =1.47850 log c' = 1.63200 log sin i(B'~ A) =9.84447-10 colog cos \C'= 0.00212 r ui triuiua. am a ' v ~' 6- Solution: log 6 = log sin A = colog a - csin (BA) cos | C =2.23198 =9.80807-10 =7.85232-10 log sin B i .'. '=ZA'C = =9.89237-10 =5118.'4 = 12841.'6 B+A: C=ZACB-- B'+A-- ;. C'=Z.ACB'-- = 9118.'4 = 8841.'6 = 16841.'6 = 1118.'4 Check: B-A=ll18.'4 \{B-A)= 539.'2 JC = 4420.'8 6-a = 30.1 log (6 -a) = 1.47857 '-A=8841.'6 J(B'-A)=4420.'8 C' = 539.'2 8. a=491.2, c = 385.7, 9. o = 629, c = 462, 10. a = 723, c = 483, log (6 -a) = 1.47859 C = 4615' A = 4610' A = 140 o ll , LOGARITHMS. SOLUTION OF TRIANGLES 173 11. a = 342.6, 6 = 745.9, A=4335.'6 12. a = 345.46, 6 = 531.75, .A = 2647'32" 1 13. In the triangle ABC, A = 3721', a = 93, 6 = 85, find the angle B. (Sheffield.) 14. A road OA is 9 mi. long and makes an angle of 31 16' [ = 3127] with a straight beach OX. From the point A two straight roads, AB and AB', each 6 mi. long, run to the beach. Find the distance along the beach from to the nearer of the points B and B' . (Harvard.) 190. Case III. Given two sides and the included angle. The following example illustrates the method : Given: = 3733'40", c = 95,721, a = 25,463 Required: A, C, and 6. c+a Formulas: The equation tan J(C A)= cot hB deter- mines J(C A). The equation i(C+A)=90-} determines (C+A). C = |(C+A)+J(C-A) 6 = Solution: A = i(C+A)-(C-A) csinB c-a_sin %(CA) sin C cos hB c- a = 70,258 c+a= 121,184 =i846'50" i(C+A)=7113'10" log (c- a) = 4. 84670 colog (c+a) = 4. 91656-10 log cot %B = 0.46847 log c=4. 98100 log sin = 9.78505-10 colog sin C= 0.12110 log tan %(P-A) = 0.23173 i(C-A)=5936'30" i(C+A) = 7113'10" C=13049'40' A = ll36'40" log 6 = 4.88715 6=77117 Check: log 6=4.88715 log sin J(C-A) = 9. 93580- 10 colog cos |= 0.02376 log (c-o) = 4. 84671 174 THIRD-YEAR MATHEMATICS EXERCISES Solve the following triangles and check: 1. a =748, 6 = 375, C=6335'30" 2. a = 486, 6 = 347, C=5136' 3. a= 34.645, 6 = 22.531, C = 4331' 4. a= 145.9, 6 = 39.90, C = 9211'18" 5. a = 540, 6 = 420, C=526' 6. a = 469. 71, 6 = 264.37, C = 9657'48" 7. a =103.21, 6 = 152.37, C=1418'54" 8. a= 167.38, 6 = 152.37, C=15020'6" Solve the following problems: 9. The diagonals of a parallelogram are 83.66 and 92.84 and one of the angles of their intersection is 84 . 28. Find the sides and angles. 10. Two trains start from the same station at the same time, one going north at 40 mi. per hour, the other going 10 south of east at 30 mi. per hour. How far apart will the trains be at the end of three quarters of an hour ? (Harvard.) 11. An aeroplane is observed at the same instant from two stations on a level plane, 5,280 ft. apart. At the first station the horizontal angle between the aeroplane and the other station is 1837'[ = 18?62], and the angle of elevation of the aeroplane is 3741'[ = 37?68]. At the second station the horizontal angle between the first station and the aeroplane is 6416'[ = 64?27]. Find the height of the aeroplane. By the horizontal angle between two points, A and B, each viewed from a point C, is meant the angle between the vertical plane through C and A and the vertical plane through C and B. (Harvard.) 12. The two diagonals of a parallelogram are 122 and 44, and they form an angle of 4728'[=47?47]. Find the lengths of the sides and the angles of the parallelogram. (Harvard.) LOGARITHMS. SOLUTION OF TRIANGLES 175 13. In a square ABCD a circular arc BD is described, with A as center and AB as radius. If AB = 5 ft., find the distance from the point C to one of the points of trisection of the arc BD. (Harvard.) 191. Case IV. Given the three sides. The following example illus- trates the method: Given: a=116.26, 6 = 172.36, c= 149.54, Fig. 91. Required: A, B, and C Formulas: s=i(.a+b+c), r =^(fr^Ki=Wz) tan ^A = sa r , tan ^B = :r ! ^ , tan \C r sc A+B+C = 180 Solution: 2s = 438. 16 s = 219.08 s-a=102.82 s-b= 46.72 s-c= 69.54 log(s-a) = 2.01208 log (s-6)= 1.66950 log (s-c)= 1.84223 cologs= 7.65940 10 Check: 2s = 438. 16 logr 2 = 3.18321 log r= 11. 59160- 10 log r=ll. 59160-10 log (s-a)= 2.01208 logtan|A= 9.57952-10 |A = 2047'42" A = 6135'24" log r= 11. 59160- 10 log(s-6)= 1.66950 logtanf = 9.92210-10 iB = 3953'18" = 7956'36" log r= 11. 59160- 10 log (s-c) = 1.84223 logtanJC= 9.74937-10 C = 2918'54" C=5837'58" Check: A+B+C =m59'5S" 176 THIRD-YEAR MATHEMATICS EXERCISES Solve the following triangles: 1. -^ . JC 1 cos* SU1 2 \ 2 Similarly, from the equation cos 2a = 2 cos 2 a 1 we derive cos ~ -V 2 \ 2 x x By dividing sin 5 by cos 5 , we have x 1 cos a t ll+cosa SL- EXERCISES Prove that the following statements are identities: . . 2 tan x 1. sin 2x = 1+ tan 2 re sin 2x = 2 sin re cos re sin re 2 tan re cos x 2 sin re cos re 2 sin re cos re 1+ tan 2 re sin 2 re cos 2 re + sin 2 re 1 cos 2 re 192 THIRD-YEAR MATHEMATICS 2. (sin +cos ) =l+sinA sin 2a 3. = tana 1+cos 2a 4. tan A+cot A = 2 esc 2A _ tan z-ftan y 6. - - = tan x tan y cot x+cot y . sin 2x 6. cot x 1 cos 2x x l-tan 2 2 7. = cos x 1+tan* | 8. 1+tan A tan = sec A MISCELLANEOUS EXERCISES 201. Verify the following statements: cos x + sin x 1. tan 2x+sec 2x = 2. 2 sin z+sin 2x cos x sin x 2 sin 3 x 1 cos x 3. (seca+tana) 2 = ^ +sma 1 sin a 4. sin (^7r+a) sin (^7r a)=sina 5. cos 3z = 4 cos 3 x 3 cos z 6. 1-ftan a tan - = sec a 7. tan z = tan -+tan - sec x 8 cos (a 45) _ cos 2a ' cos (a+45)~l-sin2a FUNCTIONS OF SEVERAL ANGLES 193 Solve the following problems : 9. Given tan x f and x in the second quadrant. Find sin 2x. (Sheffield.) 10. tan z = ^ and x is in the third quadrant, sec y = y and y is in the second quadrant. Find cos (x 2y); cot 2x; sin \y. (Board.) 11. If sin z=- r, find tan -. (Harvard.) m 2 +n 2 2 12. If tan ^ = y, find the values of sin x and cos x in terms of y. (Harvard.) 13. Prove for a right triangle that the cosine of the difference between the acute angles is equal to twice the product of the two legs, divided by the square of the hypotenuse. (Harvard.) 2d x 14. If x = tan _1 -, find the value of sin - in terms of a \a z & (consider only values of x between and 90, and values of a between and 1). (Harvard.) 16. Solve the equation tan- 1 x+tan" 1 2x = tan~ 1 3. (Sheffield.) Let a = tan-i x, /3= tan -1 2x. Then tan (a+jS) =3, etc. 16. Prove arc tan ^-+arc tan \ = 45? Trigonometric Equations 202. Solve the following equations for values of x between and 360: 1. cos 2x-h3 sin x 2 Show that 1-2 sin 2 x +3 sin x = 2. Solve for sin x. Then find x. 2. cos 2z-f-cos x = 3. cos x cos 2z-f-2 cos 3 x = Solve by factoring. 194 THIRD-YEAR MATHEMATICS 4. tan (^H-zj+tan t-. x ) =4 Expand each term. 5. sin z+sin 2x=\ 6. cos 2x+ sin a; = 4 sin 2 $7. Solve the equation 3 sec 2 x 7 tan 2 z = tan x. Obtain all solutions for x between and 180 and give the answers to the nearest degree. (Yale.) $8. Solve the equation sin 2z+J = sin x-f-cos x. Obtain all solutions for x between and 180 and give the answers in degrees. (Yale.) $9. Find all the values of x between and 360 which satisfy the equation 4 cos 2x+3 cos x=l. (Harvard.) $10. Find all the values of x between and 360 which satisfy the equation 6 cos 2z+6 sin 2 a: = 5+sin x, and verify your results. (Harvard.) Jll. Find all the values of x between and 360 which satisfy the equation 3 cos 2rc+sin x (3 sin x+5) = 5. (Harvard.) 1 12. Find all the values of x between and 360 for which 2 sin 2x = cos x. (Harvard.) $13. The sum of the tangents of the acute angles of a right triangle is equal to 4. Find the values of these angles. (Harvard.) $14. Solve the equation cos 50+ cos 36=^2 cos 40. (Princeton.) FUNCTIONS OF SEVERAL ANGLES 195 Summary 203. The following formulas have been proved: 1. sin (a+p) = sin a cos p+cos a sin p 2. cos (a+P) =cos a cos p sin a sin p 3. sin (a P) =sin a cos p cos a sin p 4. cos (a p) =cos a cos p+sin a sin p 5. sin a+sin p=2 sin \ (a+p) cos|(a p) 6. sina-sin p=2cos *(a+p) sin (a-p) 7. cosa+cos p=2cos *(a+P) cos^(a-p) 8. cosa-cos p = -2 sin^(a+p) sin^(a-P) tan a+tan p 9. tan (a+p) 10. tan (a-p) = 1 tan a tan p tan a tan p 1+tan a tan p 11. sin 2a = 2 sin a cos a 12. cos 2a = cos 2 a sin 2 a =2 cos 2 a-1 = 1-2 sin 2 a 2 tan a ._ a fl+cos a 13. tan 2a =z i = 15. cos l-tan 2 a 2 \-4 . . a ll cosa am a a 14. sm^ = ^ 2 16. ten g-*^ cos a 1+cos a 204. The chapter has shown how to solve trigo- nometric equations. Explain your method of solving such equations. 205. Explain how to prove trigonometric identities. CHAPTER X BINOMIAL THEOREM. ARITHMETICAL AND GEOMETRICAL PROGRESSIONS Binomial Theorem 206. The binomial theorem enables us to state by in- spection the expansion of a power of a binomial. By actually multiplying, the following identities are obtained: (a+&) 2 = a 2 +2a&+6 2 (a+by = a i +3a'>b+3ab 2 +b 3 (a+&) 4 = a 4 +4a 3 &+6a 2 6 2 +4a& 3 +& 4 (a+b) 5 =a 5 +5a 4 b+10a?b 2 +10a 2 b*+5ab*+V> y etc. A study of these identities brings out the following facts: 1. The number of terms in the right member is one greater than the exponent of the binomial in the left member. 2. The exponent of a in the first term of the expansion is the same as the exponent of the binomial and decreases by one in each succeeding term, being in the last term. 3. The exponent of b increases by 1 from term to term, being in the first term and the same as the exponent of the binomial in the last term. Omitting the coefficients, these three facts give the following expansion of any binomial, as {a+b), raised to any positive integral power, n: (a+b) n = a n +( )a n ~ 1 b-\-( ) a n ~ 2 b 2 +( )a-*&+..., 196 BINOMIAL THEOREM. PROGRESSIONS 197 The coefficients may be determined by the following simple device: Arrange the coefficients of (a+b), (a-f-6) 1 , (a+6) 2 , etc., as in the form given in Fig. 94. Notice that each coefficient in this arrangement is equal to the sum of the coefficients which 1 are nearest to the right and left of it 12 1 13 3 1 in the line above. 14 6 41 Fig. 94 is known as Pascal's tri- 1 5 10 10 5 1 angle.* Fig. 94 Moreover, after the second term any coefficient may also be determined by means of the coefficient of the term just preceding, according to the following rule: Multiply the coefficient of the preceding term by the exponent of a in that term and divide the product by the number of that term. Thus in (a+6) 5 the coefficient of the fifth term is : = 5; the coefficient of the third term is -^- = 10, etc. 4 2 207. The binomial theorem. According to the rules given in 206 the expansion of (a+b) n takes the following form: (a+b)"=a"+na"- 1 b+ n ^~ 1) a-*b* + n(n-lKn-2) an _ 3bS+ This is known as the binomial theorem for positive integral powers. The theorem is assumed without proof. The proof is usually given in a course in advanced algebra. * See First-Year Mathematics, pp. 200-201. 198 THIRD-YEAR MATHEMATICS 208. The factorial notation. The products 1 -2, 1 -.2.3, 1*2* 3*4 ,...., l*2 a 3*4 . . . r, are called factorial 2, factorial 3, factorial 4, .... , factorial r. They are usually denoted briefly by the symbols: 2!, 3!, 4!, r! or by |2, |3, |4, , \r. Thus, (a+6) w = a"+na- 1 6+^^a"-- 2 6 2 . n(n l)(n 2) , . + 3^ -a n ~ 3 V+ .... EXERCISES Expand the following powers to four terms: 1. (2x-Zyy In this example a=2x, b=(3y), n=7. 3 Hence Vx-Zy)-* = (2xy+7(2x)*( -Sy) + 7 -(2x)H -32/) 2 + ^W(-3y)'+ = 2V-7-2-3x 6 y4-21-2 5 -3 2 x 6 i/2-35-2 4 3 3 x 4 ^H-.... = 128x 7 -1344^2/+6048x 5 2/ 2 -15120xV+. . ' (; + ?)' Herea=- , 6=-|-,n=4 ( 2 ,4r= 4 ^ 3 ( 3 f) 4f ?)W) 2 4 2 +'(? r _2* 4-2 3 -3y 2 6-2 2 -3V 4-2-3Y 3V y* -1 " 4y + 4 2 y 2 + 4?y + 4 4 BINOMIAL THEOREM. PROGRESSIONS 199 -") Let a = T^> b= bV a '2a ba* a\ 4 . Then(|- 6 / a )' = (|)V4(|)V^) + V 8 w>* 1)5 2> 4. (x-i/) 8 6. (2a+4c) 4 6. (3a 2 -6 2 ) 7 7. (3a 2 6 4 - 8 * (4*-^ * K)* 10. (Vz+vV) 5 11. (2+1/3) 4 12. (^-^Y \ y x / 2 4 a 4 4-2 3 a 3 a*6 6-2 2 aW 6 b 6 + 6 4 16a 4 _ 32a 3 v / a 24a 3 6 8 6 5 + 62 13. Op-^Y \ Vb z a J 14. (z*+?/ ) 4 15.. (a- 3 -6- 3 ) 4 16. (3a6" 3 -a- 3 6) 6 (S-f)' " ('-*)' 20. (4a f -a*6*) 6 209. The rth term of (a+b) n . The number of the term and the numerator of the coefficient of the same term may be arranged in the following table : Number of Term Numerator 3 4 ....... 5 6 n(n 1) n( l)(n 2) n(n-l)(n-2)(w-3) n(w-l)(n-2)(w-3)(n-4) 200 THIRD-YEAR MATHEMATICS From a study of this table show that the numerator of the coefficient of the rth term is n(w-l)(n-2)(n-3). . . . (n-r+2). Similarly, find that the denominator of the coefficient of the rth term is 1-2-3-4- .... (r-1). Show that the exponent of a in the rth term is n r+1. Show that the exponent of b in the rth term is r 1. Thus the rth term in the expansion of (a+6) n is given by the formula _ n(n-l)(n-2) .... (n-r+2) _, lr 1l2._3- (r-1) a * EXERCISES In the expansions of the following binomials find the term called for in each case: in (*r*>y find the fourth term. Here a = 2x, b= Fr , n = 6, r = 4 20.23-x 3 20 ^(M) 2 3 x* \8 find the fifth term. 3. In (^x+fty) 7 find the sixth term. 4. In (%x^y) 10 find the fourth term. 5. Write the last three terms of the expansion of (4a 3 a*s*) 8 . (Yale.) BINOMIAL THEOREM. PROGRESSIONS 201 MISCELLANEOUS EXERCISES J210. The following problems are taken from college- entrance examination papers: 1. Expand and simplify ( ^--77= ) . (Smith.) \y* xv 6/ 2. Write the last three terms of the expansion of (4a* a*x*) 8 . (Yale.) 3. Prove that (a+b) 7 -a 7 -b 7 = 7ab(a+b)(a 2 +ab+b 2 ) 2 . (Harvard.) / 1 \ 10 4. Find the fifth term of (^ 4 +fi) an d reduce to the simplest form. (Dartmouth.) ^ ' 6. In the expansion of ( 2x+ J the ratio of the fourth term to the fifth is 2:1. Find x. (Princeton.) / x ^yV 6. Write the sixth term of ( ^ ) . (Pennsyl- vania.) 7. Find and simplify the twenty-third term in the expansion of ( - !) 23 - (ComeU - ) 8. If the middle term of ( Sx r= ) is equal to the fourth / ._ 1 V V 2Vx) 4 - term of ( 2 / x+^= J , find the value of x. (M.I.T.) 9. Write the first term of (x* x~*) s which in its simplest form has a negative exponent. (Board.) 10. Find the coefficient of x 4 in ( 2x 2 J . 11. Show ttfat the coefficient of the middle term of (l-fz) 16 is equal to the sum of the coefficients of the eighth and ninth terms of (1-fx) 15 . (Princeton.) 202 THIRD-YEAR MATHEMATICS 12. In the expansion of (2xSx~ 1 ) 8 find that term which does not contain x. (Princeton.) 13. Write the sixth term in the expansion of 10 W 64aW , . / o (Yale0 81m 2 n 8 \ m" 14. Find the third and fifth terms in the expansion of (1- /). (Sheffield.) (2a; 2 3 \ 8 J find the coefficient of x 4 . 16. In the expansion of ( a+- ) write the term which does not contain a. 17. Find the middle term in the expansion of ( -+- ) . \b a) Arithmetical Progression 211. Arithmetical progression. A succession of num- bers formed according to a definite law is called a series. Thus the expression 2+4+6+8.... is a series, the law of formation being that any term in the series is obtained by adding the fixed number, 2, to the preceding term. An arithmetical progression is a succession of numbers in which each term after the first may be found by adding a constant number to the preceding term. EXERCISES 1. Show that the following are examples of arithmetical progressions: 3, 5, 7, 9 ; 84, 74, 64 ; the logarithms, to the base 3, of the numbers 3, 9, 27, 81 2. Show that the equation ab = bc expresses the fact that three numbers a, b, and c are in arithmetical progression. 212. Elements. In general, the form of 'an arith- metical progression is a, (a+d), (a+2d), (a+3d),.... BINOMIAL THEOREM. PROGRESSIONS 203 The first term is denoted by a, the constant difference by d, the number of terms considered by w, the nth term by I, and the sum of n terms by s. The numbers a, d, n, I, and s are the elements of the arithmetical progression. 213. Relations between the elements. The table, Fig. 95, shows one of the relations between the elements of an arithmetical progression. Number of Term Term Second a+d Third a+2d Fourth a+Sd Fifth a+4d Tenth a+9d Fifteenth a+Ud nth a + (n-l)d .-. l = a+(n-l)d Fig. 95 Another relation may be obtained as follows : s = a+(a+d) + (a+2d)+ .... a+(n-l)d Similarly, 8 = l + (l-d) + (l-2d)+ .... l-(n-l)d Adding, 2s=(a+Z) + (a+Z) + (a+Z)+ .... +(a+l) Combining terms, 2s = n(a+l) s=^(a+l) The two relations just established enable us to find the values of two of the elements if the other three are known. 204 THIRD-YEAR MATHEMATICS EXERCISES Solve the following problems: 1. Find the tenth term of the series 7+10+13 Let a = 7, d = S, n = 10. Substitute these values in the equation l = a + (n l)d, and find the required value of I. 2. Find the seventh term and the sum of seven terms of the series 2+4+6 3. If a=7, d= 2, and w = 8, find s and I. 4. If Z=30, n = 9, and s = 162, find a and d. 5. The fourth term of an arithmetical progression is 11 and the fourteenth term is 39. Find the common difference. 6. The sum of the second and twentieth terms of an A.P. is 10, and their product is 23|~|. What is the sum of 16 terms ? (Pennsylvania.) 7. Given Z = 23, d=2, s = 143. Find a and n. By substituting for I, d, and s the given values, 23 = -'>+(J-)+(g-S)+- tfw 2. The sum of three numbers in geometrical progression is 70. If the first be multiplied by 4, the second by 5, and the third by 4, the resulting numbers will be in arithmetical progression. Find the three numbers. (Board.) 3. If r , -7 , and r are in arithmetical progression, o a o o c show that a, b, and c are in geometrical progression. (Yale.) 4. An arithmetical progression and a geometrical progres- sion have the same first term, 3, equal third terms, and the difference of the second terms is 6. Determine the progressions. 5. The sum of 9 terms of an arithmetical progression is 46; the sum of the first 5 terms is 25. Find the common difference. (Vassar.) 6. The difference between two numbers is 48. Their arith- metical mean exceeds the geometrical mean by 18. Find the numbers. (Smith.) 212 THIRD-YEAR MATHEMATICS Infinite Geometrical Series 220. Infinite geometrical series. We have learned how to find the sum of n terms of a geometrical series, n being a finite number. If the number of terms in the series a+ar-\-ar 2 + is unlimited, it is called an infinite geometrical series. If the sequence of the partial sums, a, a-\-ar, a-\-ar-\-ar 2 , a-{-ar-\-ar 2 -{-ar 3 , etc., approaches a definite finite number, S, as a limit, we say S is the sum of the infinite series a+ar +ar 2 -|-ar 3 + For example, to find the sum of A ' b Ep c the series l+|+J+|+.... we t"utK-~-i may represent the partial sums i 1-&* \- l A\% 1 graphically, Fig. 98. Fig. 98 Let AB = 1 =BC. Then AD = 1+|. Bisecting the remainder, DC, Bisecting the second remainder, EC, AF=i+Hi+i As the process of increasing the number of terms con- tinues, the partial sum increases, approaching the num- ber 2 in such a way that it can be made to differ from 2 by less than any assigned number, however small. Hence 2 is said to be the sum of the infinite series The series is a convergent series. Another example of a convergent infinite geometrical series is found in a recurring decimal fraction. The number . 333 .... is really a brief way of writing TU+TTnr+TUinT+TiFiFinr a geometrical series in which a = t 3 q- and r = T 1 Tr . BINOMIAL THEOREM. PROGRESSIONS 213 Show that the sum of this series is J. The two preceding examples are particular cases of an infinite geometrical series whose ratio r is numerically less than 1. If in the series a-\-ar+ar 2 +ar s -\- . . we take r numerically equal to 1, we have either a+a+a+a+ , or aa+aa-J- . . . . In the first case the sum increases without bound as the number of terms increases indefinitely. In the second case the partial sums a, a a, a a + a, etc., have alter- nately the value a or 0. Hence in either case the series has no definite sum. The question as to the sum of an infinite geometrical series may be considered by making a study of the formula for the sum of n terms, 1 r n a ar n a ar n s n = a 1 r 1 r 1 r 1 r a ar n 1. If r>l, then r n increases without bound as n increases indefinitely. This is expressed symbolically by the statement nm v ) = oo which is read, "The limit of r n , as n increases indefinitely, is infinite." Hence s n also increases numerically without bound and the series has no sum. 2. If r * r= A, we have the sequence . 1, .01, .001, .0001,.... lim / a ar n \ = a , n^co\l r 1 r) 1 r' Hence the infinite geometrical series a+ar-\-ar 2 -\- . . . . has a sum, s, given by the formula, s= lim( Sn ) = , if r< i. n->00 1-r EXERCISES Find the sum of each of the following infinite series: 1.3+1+1+1+ Let a = 3, r Thens 1 3" a 3 1-r 1 2' 3 2. 10+5+2J+1J.. 3 . _ 3+1 -l + l... 4. -2- 1 -!. 4 32 5. 4 + 2 + 1 +.. 3^3^3^ 6.6-4 +3 .... (Princeton.) 9. i-M- 1 .. 2^4 8 8. 1.35+0.045+0.0015+.. (Harvard.) Also find the sum of the positive terms. 10. In a geometrical progression the sum to infinity is 64 times the sum to 6 terms. What is the common ratio ? (Prince- ton.) 11. The first term of a geometrical progression is 225 and the fourth term is 14J-. Find the series and the sum to infinity. BINOMIAL THEOREM. PROGRESSIONS 215 Find the limiting value of each of the following repeating decimals: 12. .1666.... 14. 1.2121.... Leta = ^, ri. 15. .83333.... Find s and add . 1 to the result. 16. . 234234 13. .363636.... 17. .23737 221. Historical note. The arithmetical and geometrical progressions are among the oldest topics of all mathematics. Problems leading to both kinds of progression are found in the oldest extant historical document, the papyrus of Ahmes. The forms of progression called for by the problems mentioned in this manuscript are very far from the simplest, indicating that, for thousands of years before the Christian era, Egyptian scholars had studied these progressions and by 2000 to 1700 B.C. they had attained to an advanced stage of knowledge of them. The Babylonians made use of both forms of progression in recording the phases of the moon, and the Greeks were zealous students of the progressions. The theory was very greatly advanced by the Pythagoreans in connection with their work in figurate numbers, which was also a favorite subject of the school of Plato. Archimedes was even well acquainted with the laws of summation of the progressions. Heron made extended prac- tical use of the laws. Hypsicles and Nicomachus both studied and taught the topics very fully. The Hindus never advanced beyond the attainment of the Greeks. They solved a few problems requiring a knowledge of the progressions, and regarded the study as belonging to arithmetic. The Arabs advanced considerably beyond the Hindus. They were in possession of the completed theory of these topics, if we may judge from the work of Leonardo of Pisa, who, in his liber abaci of 1202 a.d., brought together what was known by the Greeks and Arabs and made it available for European scholars. Leonardo even made summing of these series one of the nine fundamental processes of arithmetic, thus 216 THIRD-YEAR MATHEMATICS putting what he called progressio on a par with additio, subtract, multiplication etc. The progressions formed a basis for the ancient Greek method P ^ J * 7I ( 4, -' iIL.2. u 1/ V , J 1(5'0J r J V r <;4 i '(1-1 * ( 4) 4- * b (0 # Jvi-)- Fig. 102 222 THIRD-YEAR MATHEMATICS 3. Graph these values and draw the curve through the points thus obtained. The curve in Fig. 102 is an ellipse. Equation (2) readily shows: 1. That the curve is symmetric with respect to the axes. 2. That the intercepts on the axes are = a and =*= b. The lengths a and b are the semiaxes of the ellipse. If a = b, the ellipse reduces to a circle EXERCISES 1. Graph the equation 4x 2 +?/ 2 = 20. 2. Graph the equation x 2 +4?/ 2 =16. 229. The Hyperbola. If in the equation ax 2 +q/ 2 +/=0, c is negative and a positive, the equation may be changed to the form a 2 b 2 For a = 5 and 6 = 4, this reduces to 25 16 The equation may be graphed as follows: 1. Solving for y, 2/=== b(.8)l/x 2 "^25. EQUATIONS IN TWO UNKNOWNS 223 2. The corresponding values of x and y are computed and tabulated as in Fig. 103. X y *(-8) 5 6 2.6 7 3.9 8 5. 9 6. ; ( . 8) y 25, imaginary .... i ' (-9, ^,6) N L / (- 3T V r V" -finvp) X=k&61> jFi 5.2.6) C T (- 5. w >.( >J JT 1 ? i k .v> j v> s (-' ' -3 I >y . p) (-9,- 5) y Fig. 103 3. The curve, Fig. 103, is the graph of the equation 16 The curve is called a hyperbola. 25 16 EXERCISES 1. Graph the equation z 2 -?/ 2 -4 = 0. 2. Graph the equation 9 4 230. The graph of an equation of the form xy=c is also a hyperbola. This equation was discussed in 24. 224 THIRD-YEAR MATHEMATICS If c = 8, the graph xy = 8, Fig. 104, is easily obtained by means of the values in the table, Fig. 104. . x y 1 +8 2 +4 3 +2.7 4 +2 5 + 1.6 6 + 13 7 +11 8 +1 -1 -8 -2 -4 -3 -2.7 -4 -2 -5 -1.6 -6 -1.3 -7 -1.1 -8 -1 1 Tfl -| \ i V \ \ V s s N s - ( S \ \ \ \ Fig. 104 EXERCISES 1. Graph the equation 2. Graph the equation xi/ = 24. xy = 5. 231. Two straight lines. If the general quadratic equation is the product of two linear factors, the graph consists of two straight lines, as illustrated in the following example : z 2 +2:n/+2/ 2 +2z+2i/-3 = 0. By grouping terms, (*+2/) 2 +2(z+2/)-3 = /. (x+H-3)(z+2/-l) = 0. The graph consists of two parallel straight lines, Fig. 105. Fig. 105 sij: x ^ + _Vc S,_ V s^_ s^ N, O- ^- -X s Sr _L $>. S^. ^U S^ 4g S^. s*r s^ %- s^ S S,. s s ^ - EQUATIONS IN TWO UNKNOWNS 225 Solution of Simultaneous Quadratics 232. To solve a system of quadratic equations in two unknowns, as r y-x 2 = 2 x y 2 = 5 one of the unknowns may be eliminated by substitution. Since y = 2-\-x 2 , the second equation is changed to z-(2+x 2 ) 2 = 5, or x4:4:X 2 x 4 = 5, .'. x*+4:X 2 -x+9 = 0. This is an equation of the fourth degree. So far the student knows no general method of solving an equation of the fourth degree and therefore he will find it impossible to solve some systems of simultaneous quadratics. A similar situation was found in the study of factoring polynomials. Not being able to work out a general method by which any polynomial may be factored, we made a study of certain typical forms of polynomials. Methods were then worked out for factoring such special forms. Similarly we shall now study only certain cases of simultaneous quadratics and find the proper methods of solution. 233. Case I. The form of the equations is such that either x or y may be eliminated by addition or subtraction. The following example illustrates case I: EXERCISES Solve the system of equations f x 2 + 2/ 2 = 25 \4z 2 +9?/ 2 =144 226 THIRD-YEAR MATHEMATICS 1. Graphical solution. The graph of the first equation is a circle, 226. The graph of the second equation is an ellipse, 228. The two graphs intersect in four points, Fig. 106, the co-ordinates of which must satisfy both equations. Hence there are four solutions: m s i) Fig. 106 (4.1, 2.9), (4.1, -2.9), (-4.1, 2.9), and (-4.1, -2.9). 2. Algebraic solution. Multiplying the first equation of the system by 4 and leaving the second equation as it is, we have and By subtracting, 4x 2 +4?/ 2 =100 4z 2 +9?/ 2 =144 5y 2 = 44 .\ yi=vX8, 2/2= -VX8. Substituting the value of y\ in the first of the given equations, x 2 +8.8 = 25 /. z 2 =16.2 /. 3==*^ 16.2. Hence we have the two following solutions: (/lO, v / 878) and (-VTjO, /O). Similarly, by substituting ?/ 2 = ^8.8 in the first of the given equations, we have the solutions: (VTO, -/O) and (-/16^2, -/O). Verify the four solutions by means of the graph, Fig. 106. EQUATIONS IN TWO UNKNOWNS 227 2. Solve the following systems : 5x 2 -2?/ = 30 2z 2 +?/ 2 = 57 f2x 2 +2/ 2 -33 = \z 2 +2i/ 2 -54 = x 2 +y*=lQ 4z 2 -9i/ 2 = 36 z 2 x?/_8 3 ~4 ~6 15. 6. 7. J8. x 2 +y 2 =l x 2 y 2 =l 4x 2 -9?/ 2 = 36 z 2 -f-4?/ 2 = 4 (9x 2 +4?/ 2 = 36 \9x 2 +162/ = 33 x 2_j_ 2/ 2 = 25 -2_ 3j/ = 21 234. Case II. One equation can be resolved into two linear factors. The following example illustrates case II: Solve the following system of equations: |4?/ 2 -3z 2 +a;?/+15x-202/ = \ z 2 + i/ 2 = 25 1. Graphical solution. By factoring, the first equation is changed to the form (4y-Sx)(x+y-5)=0. Hence the graph consists of two straight lines, Fig. 107. The graph of x 2 +y 2 = 25 is a circle, 226. .'. the co-ordinates of the four points of intersection give the solutions of the system. 2. Algebraic solution. According to the graph, Fig. 107, two of the solu- tions of the system are obtained from the points of intersection, A and B, of the circle with the straight line whose equation is x-{-y 5 = 0. The other two solutions are given by the points of intersection, C and D, of the circle with the straight line whose equation is 4y-&*-0. c. \> \U? r t ^SL &* Nl. 1& \ \'*y i \y T jt tlS ^ s "">->-^ -^"^ Fig. 107 228 THIRD-YEAR MATHEMATICS This suggests that the solutions of the original system be obtained by solving the systems: (x +y -5 = (4?/-3x = \ x 2 +y 2 = 25 and \ x 2 +y 2 = 25 Both systems should be solved by eliminating one of the unknowns by substitution. Solving the first system, we have: x = 5 y. Substituting, (5-?/) 2 +?/ 2 = 25, .'. 2y 2 -10y = 0, ' 2/i = 0, 2/2 = 5. The corresponding values of x are found by substituting these values of y into the equation x = 5 y. This gives the solutions: (5, 0) and (0, 5). Find the remaining solutions. EXERCISES Solve the following systems: , (x 2 5xy # \x 2 -y 2 = , (x 2 -y 2 = " \x 2 +?/ 2 = 8 5xy+6y 2 = . fx 2 =y 2 27 + \Sx 2 +5y 2 = 32 . Ux 2 -9y 2 = 1 ' \4x 2 +9?/ 2 -i = x*+xy = . (x 2 -3xy = x 2 -xy+y 2 = 27 * * \5x 2 +3?/ 2 -9 = x 2 +2:n/+?/ 2 +z+?/-30 = a:?/ = 15 ft jm 2 -{-n 2 5m 5n= 4 Multiply the second equation by 2 and add to the first equation. EQUATIONS IN TWO UNKNOWNS 229 235. Case III. One equation is of the form xy = c. The following example illustrates this case : Solve the system (x 2 +y 2 *=5 \xy=2 1. Graphical solution. The graph of the first equation is a circle whose radius is 2 . 2, approximately, Fig. 108. The graph of the second equation is a hyperbola. The two curves intersect in points A, B, C, and D, which determine the solutions. 2. Algebraic solution. Multiply the second equation of the given system by 2 and add the resulting equation to the first. This gives the equation Sj % \ ^ ' s. . 7 \ s* * \ N\ / SvJs \ ^ 1 \ \ \ 3 1 r 7 >\ \ q5& j \ -4 f * \ ^Y\ X -i \ j s \ \ Fig. 108 x 2 +2xy+y 2 = 9, in which the left number is a perfect square. Extracting the square root of both members, we obtain the two linear equations x+y = +S. and x+y=-S. The graphs of these two equations must pass through the four points of intersection of the hyperbola and circle, Fig. 108. Therefore two of the four required points may be' located by graphing the straight line x -\-y = 3 and the simpler of the given equations, xy = 2. The other two points may be determined by the graphs of x+y= 3 and xy = 2. 230 THIRD-YEAR MATHEMATICS This suggests that, instead of solving the original system, we may solve the two systems, (x+y = S Substituting for x its equal 3-2/, (3-2/)2/ = 2, /. ^-32/4-2 = 0. From this we find the solutions: (1, 2) and (2, 1). and z+?/=-3 xy = 2 Substituting for x its equal -3-2/, (~3-2/)2/ = 2, 2/ 2 +32/+2 = 0. Show that the solutions are: (-2, -1) and (-1, -2). Often, when neither of the given equations is of the form xy = c, we can obtain linear equations by addition or subtraction, and extraction of the square root or factoring. Thus, in exercise 2 below, the two given equations may be added. The resulting equation may be then divided by 2. The two linear equations are obtained by extracting the square root of both members of the last equation. In exercise 3 the two given equations may be added. By factoring both members of the resulting equation, we obtain two linear equations. EXERCISES Solve the following systems: fz 2 +2/ 2 \xy-Q = 13 2. >{ (x 2 -5xy+y 2 =-2 \z 2 +9a:2/+2/ 2 = 34 x 2 +xy+y 2 = 7 x+xy+y = 5 r 2 -f-rs = 5 rs-f-s 2 = 4 5. 7. 8. z 2 +2/ 2 +*+2/=18 xy = Q (x 2 +xy = 6 * W+2/ 2 =10 (t 2 +u 2 = 5 \tu-\-t+u = 5 2 (r 2 -H \rs = ! EQUATIONS IN TWO UNKNOWNS 231 236. Case IV. All terms containing the unknowns are of the second degree. If in an equation all terms contain- ing the unknowns are of the same degree the equation is homogeneous with respect to these terms. The following example illustrates case IV: Solve the following system of equations: % (x 2 +xy+y 2 = 7 \x 2 -xy+y 2 =l9 Multiplying the first equation by 19, the second by 7, 19x 2 +19xy+19y 2 =19-7 and 7x 2 - 7xij+ 7y 2 =19>7. Subtracting one of these equations from the other, we have 12x 2 +2Gxy+ 12y 2 =0 Dividing by 2, Qx 2 +I3xy+ 6y 2 = 0. Factoring, (2x+3y)(Zx+2y) = 0. .'. we can replace the given system by the following: (2x+3y = , (3x+2y = \x 2 +xy+y 2 = 7 \x 2 +xy+y 2 = 7 Solve these two systems. EXERCISES Solve the following systems: (x 2 +3xy-y 2 = 3 ' \2x 2 +5xy+y 2 = 8 (4a 2 -2xy = y 2 -l6 ' \5x 2 -7xy+36 = (x 2 +xy = 75 ' \y 2 +x 2 =125 (x 2 +3xy = 7 ' \x 2 -xy+y 2 = 3 (x 2 +2xy+2y 2 =10 * [3x 2 -xy-y 2 = 51 (x 2 -xy-\-y 2 = 21 ' \y 2 -2xy=-15 Solution of Equations of Degree Higher than the Second 237. One equation is divisible by the other. Some systems of quadratic equations and of equations of higher degree may be solved by dividing one equation by the other, as in the following example: 232 THIRD-YEAR MATHEMATICS Solve the system of equations: \x+y = 3 (1) (2) Since x+y is a non-zero constant, we may divide the first equation by the second without losing a solution of the given system. This gives y x*-xy+y* = S. (3) Graphical solution. Solving equation (1) for y, y=f9-x\ By means of the table of cube roots verify the correspond- ing values of x and y, as given in table (1), Fig. 109. X y X y 2.1 *1.7 1 2 1 2, -1 2 1 2 1, 1 3 -2.6 3 Imaginary 4 -3.8 -1 -1, -2 -1 2.1 -2 -1, -1 -2 2.6 -3 Imaginary -3 3.3 3.8 -4 (1) (3) \ \ \ \ < 3 ^ L* < ^tt * V \ - _ j ! '/ -i Q --1 J A .i \ , A \ k ^ \ i S - 4 > s Fig. 109 Graph equations (1) and (2) and give the solution of the given system. Solving equation (3), '.Vl2-Sx 2 EQUATIONS IN TWO UNKNOWNS 233 By means of this equation obtain the values in table (3), Fig. 109. It is seen that the graph of equation (3) passes through the points of intersection of the graphs of equations (1) and (2). Hence, by solving the system fx 2 -^+2/ 2 = 3 1 x+y =3, we are able to find the required solutions of the given system. Give the complete algebraic solution. EXERCISES Solve the following systems: ,'a 2 -6 2 = 3 6 fs 2 -* 2 = 228 a-b = l \st-t 2 = 42 ,'a 3 +& 3 = 18 6 jV 3 -s 3 = 56 a+6 = 6 \r 2 +rs+s 2 = 28 r x 3 +2/ 3 ==2 7 7 (a 4 +a 2 & 2 +6 4 = 91 \x+y = S V+a&+& 2 =13 J a 2 6+a& 2 = 126 fa 2 +a&+26 2 = 74 \a+6 = 9 \2a 2 +2a&+6 2 = 73 Solution of Irrational and Fractional Equations 238. Introduction of a new variable. Some equa- tions in which the unknowns appear in combinations may be simplified by using a new symbol in place of these combinations. Thus the equations (x+2/) 2 +3(z-h/) = -2, x+y+V^+y^, and^+^8 may be written respectively a 2 +3a=-2, a 2 +a = 6, and x 2 -f?/ 2 = 8. This device may be used in the exercises below 234 THIRD-YEAR MATHEMATICS EXERCISES Solve the following systems: f: a+b+V a+b = ' 2 -B 2 =10 Denote V a+b by x. Solve the first equation for x and obtain two linear equations in a and b. [1 1 1 X y 3 1 l l X 2 y 2 4 D ,l 1 Put - = a, -i 6. 3 * {v*+fi=6 Let fx = a, Vy = b. 4. 6. 1 X2/ I/ 2 1+1 = 4 r+s r 2 s 2 r 2_|_ s 2_ r 4 s 4 = fa6+5a+56 = 23 \4a6+3a+36 = 2a6(a+6) MISCELLANEOUS EXERCISES 239. Solve the following problems : 1. The perimeter of a rectangle is 22 inches. If the cube of its length is added to the cube of its width the result is 407. Find the area of the rectangle. 2. The difference of the cubes of two consecutive numbers is 817. Find the numbers. (Chicago.) 3. The sum of two numbers multiplied by the greater is 126 and their difference multiplied by the less is 20. Find the numbers. (Princeton.) 4. The area of a rhombus is 24 sq. in. and the sum of its diagonals is 14 inches. Find the length of one side. (Harvard.) |5. The sum of the volumes of two cubes is 559 cu. in. and the sum of their lengths is 13 inches. What is the height of each cube ? EQUATIONS IN TWO UNKNOWNS 235 6. At his usual rate a man can row 15 mi. downstream in 5 hr. less time than it takes him to return. Could he double his rate, his time downstream would be 2 hr. less than his time upstream. What is his usual rate in still water and what is the rate of the current ? (Board.) J7. The diagonal of a rectangle is 13 ft. long. If each side were longer by 2 ft., the area would be increased by 38 square feet. Find the length of the sides. 8. Two men, A and B, start at the same time from a certain point and walk east and south respectively. At the end of 5 hr. A has walked 5 mi. farther than B, and they are 25 mi. apart. Find the rate of each. 9. If a number of two digits is divided by the sum of the digits, the quotient is 2 and the remainder is 2. If it is multi- plied by the sum of the digits, the product is 112. Find the number. (Board.) 10. Three men, A, B, and C, can do a piece of work together in 1 hr. and 20 minutes. To do the work alone C would take twice as long as A and 2 hr. longer than B. How long would it take each to do the work alone ? (Board.) 11. Find two numbers such that their sum, difference, and the sum of their squares are in the ratio 5:3:51. (Yale.) 12. The sum of the ages of a father and his son is 100 years, and one-tenth of the product of the numbers of years in their ages minus 180 equals the number of years in the father's age. What is the age of each ? 13. Solve the equations l = a+(n-l)d 2s = n(a+l) taking I and n as the two unknown numbers. Find I and n when 113 a = -, d = -,s=--. (Princeton.) 14. Two men, A and B, dig a trench in 20 days. It would take A alone 9 days longer to dig it than it would B. How long would it take A and B each working alone ? (Yale.) 236 THIRD-YEAR MATHEMATICS 15. Two automobiles run 336 miles. The winning car wins by 4 hr. by going 2 mi. an hour faster than the other. What was the winner's time and speed ? (Sheffield.) 16. Two men work on a job and each receives 36 dollars. One of them, however, has worked 2 days less than the other and is paid 20 cents more a day. Find his daily wages and the number of days he worked. (Sheffield.) 17. An audience of 360 persons is seated in rows each con- taining the same number of people. They might have been seated in four rows less if each row contained 3 more people. How many rows were there ? (Board.) 18. Find the sides of a rectangle whose area is unchanged if its length is increased by 4 ft. and its breadth decreased by 3 ft., but which loses one-third of its area if the length is increased by 16 ft. and the breadth decreased by 10 feet. (M.I.T.) Solve the following systems: (x*-Sxy-y> = 9 ' \2x 2 +2xy+3y* = 7 (Board) ( s *+st+s-t=-2 \2s 2 -st-P = (x+y = 5-xy 21 - L+2/=- (Yale) I xy 22 (x>+xy+y*=133 ' \xVxy+y = l 23. (dxy = l |36:c 2 -r- 180xy+S6y 2 = - 35 (Sheffield) EQUATIONS IN TWO UNKNOWNS 237 (x 2 -4xy+4if-x+2y-G = ' \x 2 +12xy+9y 2 +2x+Zy-12 = 27. x+- = l y 2/+- = 4 (Harvard) a; {; 2 +st+P=l3s '*-H-*7 (Chicago) ;5xY-2 = 3^ ^+5?/=l (Princeton) 30. J^-^ = 2 \(i / a;-> / y)(i/^)=30 (Yale) 81 (2x+-Sx~y=10 ' {Sy-2Vxy=-l (xy = 80 32. jl_l = l [x y 5 33 fzy+28x?/-480=Q 34 Jz 2 +:n/+ AVB'C, etc. VB AB VB BC w , 9 Why? A'B" BC A'B'~B'C. VB' AB VB' B'C Why? AREAS OF SURFACES 249 Similarly show that BC CD 7 , etc. etc. B'C CD' Prove that ZABC= LA' B'C '; LBCD= ZB'C'D' \ .. ABCDE co A'B'C'D'E'. Why ? 3. To prove that Proof: ABCDE VO 2 A'B'C'D'E' VO' 2 ' ABCDE A'B'C'D'E' AB 2 AJB' 2 ' Why? AB 2 VB 2 VO 2 A'B ,Z VB' 2 VO' 2 ' ABCDE A'B'C'D'E' VO 2 VO' 2 ' Why? Why? 261. Theorem: A section of a cone made by a plane passing through the vertex is a triangle. Given the cone V-AB cut by plane P, Fig. 131. To prove that the section is a triangle. Proof: Plane P intersects the base AC B in the straight line, CD. Draw the straight lines VC and VD. Then VC and VD are elements. Why? Therefore VC and VD lie in the conical surface. Why? But they also lie in plane P. Why ? Hence the straight lines VC and VD are the inter- sections of P with the conical surface, and the section is a triangle. Why? Fig. 131 250 THIRD-YEAR MATHEMATICS 262. Theorem: A section of a circular cone made by a plane parallel to the base is a circle. Given the circu- lar cone V-A'B', Fig. 132, and plane Q || plane P. To prove that the section ACDB is a circle. Proof: Take any two points, C and D, on A B. Draw the elements VCC and VDD*. Draw VO' intersecting the plane of AB in 0. Draw CO, DO, CO', D'O'. Prove that A VOC oo A VO'C; &VOD AVO'D'. VO = CO VO' CO'' VO DO VO' DV CO^ = DO_ Why? CO' DV y Since C'O'^D'O', it follows that CO = DO. .'. the section ACDB is a circle. Why ? EXERCISES 1. The area of the base of a pyramid is 1 10 square feet. The area of the section of the pyramid parallel to the base and 5 ft. from it is 80 square feet. Find the altitude to two decimal places. 2. The base of a pyramid is 50 sq. in. and the altitude 6 inches. How far from the vertex must a plane be passed that the area of the section may be half as large as the area of the base ? Then Why? Why? AREAS OF SURFACES 251 EXERCISES 1. Show that the axis of a right circular cone passes through the center of every section parallel to the base. 2. Prove that the radius of the section of a circular cone made by a plane parallel to the base, and the radius of the base are proportional to the distances from the vertex to the cutting plane and to the plane of the base. 263. Parallelopipeds. A prism whose bases are paral- lelograms is a parallelopiped, Figs. 133 to 136. A paral- Parallelopiped Fig. 133 M ^ \ ^ ^ \ \LA Right Parallelopiped Rectangular Parallelopiped Fig. 134 ' Fig. 135 Cube Fig. 136 lelopiped whose lateral edges are perpendicular to the bases is a right parallelopiped, Fig. 134. A right parallel- opiped whose bases are rectangles is a rectangular paral- lelopiped, Fig. 135. A parallelopiped all of whose faces are squares is a cube, Fig. 136. EXERCISES 1. State the difference between a right parallelopiped and a rectangular parallelopiped. 2. Show that the faces of a rectangular parallelopiped are all rectangles. 3. Prove that the opposite faces of a parallelopiped are parallel. Use 545. 4. Prove that a section of a parallelopiped made by a plane cutting four parallel edges is a parallelogram. 5. Prove that the diagonals of a cube are equal. 252 THIRD-YEAR MATHEMATICS 6. Find the diagonal of a cube whose edge is 2; 3.4; e. 7. Prove that the square of a diagonal of a rectangular parallelopiped is equal to the sum of the squares of three edges meeting in the same vertex. 8. Find the diagonal of a rectangular parallelopiped whose edges are 6, 8, and 10 respectively. 9. Prove that the diagonals of a rectangular parallelopiped are equal and bisect each other. 10. Find the length of the diagonal of a rectangular paral- lelopiped whose edges from any vertex are 4, 6, and 8. 11. Find the edge of a cube whose diagonal is 12 inches. 264. Truncated prism. A portion of a prism included between the plane of the base and the plane of a section not parallel to the base is a truncated prism, Fig. 137. 265. Frustum of a pyramid. Altitude. The portion of a pyramid included be- tween the plane of the base and the plane of a section parallel to the base is a frustum of a pyramid, Fig. 138. The perpen- dicular, 00', intercepted between the planes of the bases is called the altitude of the frustum. EXERCISES 1. Show that the lateral faces of a frustum of a pyramid are trapezoids. 2. Show that the lateral faces of a frustum of a regular pyramid are congruent trapezoids. 3. Prove that a plane bisecting the altitude and parallel to the plane of the bases of a frustum of a pyramid forms a section whose perimeter is half the sum of the perimeters of the bases. Fig. 137 Fig. 138 AREAS OF SURFACES 253 Fig. 141 266. Slant height of a frustum. The altitude, AB, Fig. 139, of a lateral face of a frustum of a regular pyramid is the slant height of the frustum. 267. Frustum of a cone. The portion of a cone included between the plane of the base and the plane of a section parallel to the base is a frustum of a cone, Fig. 140. 268. Sections of a cone. In the discussion of the plane sec- tions of a right circular cone the following cases may be considered : Let P be a plane perpen- dicular to plane AVB, Fig. 141. 1. If P, Fig. 141, passes through the vertex V and an element VD, it cuts the surface in two intersecting straight lines, as DD' and CC. 2. If P does not pass through the vertex V, and if it is per- pendicular to the axis VC, the sec- tion is a circle, Fig. 142. 3. If P, Fig. 143, is not per- pendicular to the axis, but meets both of the elements VA and VB, the section is an ellipse. 4. If P, Fig. 144, is parallel to one of the elements, the section is a parabola. Fig. 143 Fig. 144 254 THIRD-YEAR MATHEMATICS 5. If plane P, Fig. 145, meets some of the elements produced, the section is a hyperbola. Fig. 145 Thus we have the following sections of a cone: C U) D (2) (8) U) Two intersecting Circle Ellipse Parabola straight lines Hyperbola Fig. 146 It was seen in chapter XI that these curves represent graphically quadratic equations in two unknowns. A very extensive study of these sections is made in analytic geometry. AREAS OF SURFACES 255 Areas 269. Theorem: The lateral area of a prism is equal to the perimeter of a right section multiplied by the lateral edge. In symbols this may be expressed by the equation L=p e, L denoting the lateral area, p the perimeter of the right section, and e the length of a lateral edge. Given the prism AD'; the right section FK, Fig. 147. To prove that L = p e. Proof: Show that the lateral edges are equal. Show that the lateral faces are parallelograms. Show that the sides of the section FK are the altitudes of these parallelo- grams. Hence AB' = FG - BW=FG e, BC'=GR CC 7 =GH -e, etc. Adding, AB'+BC'+ etc. = (FG+GH+eic.)e, or L=p e. EXERCISES 1. Prove that the lateral area of a right prism is equal to the perimeter of the base by the altitude. 2. Find the lateral area of a prism whose lateral edge is 18 cm. and whose right section has a perimeter equal to 29 centimeters. 3. Find L (1) if e = 2.75, p = 5.26 (2)ife = 5i,p = 10f . (3) if 6 = 12.14.^ = 25^ 256 THIRD-YEAR MATHEMATICS 4. Find p (1) if L = 20.26, e = 12.48 (2) if L=19f, e= 6.92 6. Find the lateral area of a column having the form of a regular hexagonal right prism, if one side of the base is 5 ft., and if the altitude is 8 feet. 6. Find the total surface of a cube whose edge is 2. 6 centi- meters. 7. Find the total area of a right triangular prism, if the base is an equilateral triangle whose side a =2. 7 in. and if the altitude h = 8 . 4 inches. 8. How many square inches of copper lining will be required to line the sides and base of a tank 9 in. high, 9f in. wide, and 20 in. long ? 9. How many square feet of lead will be required to line a rectangular cistern 9 J ft. long, 7 ft. wide, and 5 ft. deep ? 270. Theorem: The lateral area of a regular pyramid is equal to one-half the product of the slant height by the perim- eter of the base. In symbols L = \s-p, L denoting the lateral area, s the slant height, and p the perimeter of the base. Given the regular pyramid V-ABCDE, Fig. 148; the slant height VK. To prove that L = ^s p. Fig. 148 Proof: Show that L = Js AB + Js BC + Js CD + etc. AREAS OF SURFACES 257 271. Theorem: The lateral area of the frustum of a regular pyramid is equal to one-half the product of the sum of the perimeters of the bases by the slant height, or in symbols, =i(A+A). Prove. EXERCISES 1. The slant height of a regular triangular pyramid is 8 feet. The side of the base is 3 feet. Find the lateral area. 2. The altitude of a regular pyramid is 5 feet. The base is a regular hexagon whose side is 6 feet. Find the lateral area. 3. The altitude of a regular pyramid is 8 feet. The base is a square whose area is 25 square feet. Find the lateral area. 4. The base of a regular pyramid is a square whose side is 6. The slant height makes an angle of 45 with the plane of the base. Find the lateral area. 5. The base of a regular pyramid is a square whose area is 900. The altitude is 12. Find the lateral area. 6. The sides of the bases of a frustum of a regular hexagonal pyramid are 6 and 14 respectively. The slant height is 20. Find the lateral area and total area. 7. Find the cost of painting a church spire at the rate of 20 cents per square yard. The altitude of the spire is 80 ft. and a side of its hexagonal base is 10 feet. ^_ ^ 27rr Lateral Surface of Cylinder 272. Lateral area of a right cylinder and of a right cone. The lateral area of a right cylinder and of a right cone may F IG 150 be found by rolling the lateral surface along a plane. The lateral surface of a right cylinder is found to be a rectangle, Fig. 150, whose width 258 THIRD-YEAR MATHEMATICS is equal to the altitude of the cylinder and whose length is equal to the length of the circle forming the base of the cylinder. Hence, L = 2irrh, where L is the lateral area, r the radius of the base, and h the altitude of the cylinder. Similarly the lateral surface of a right cone is found to be a sector of a circle, Fig. 151, whose arc equals the length of the circle forming the base of the cone, and whose radius is equal to the slant height of the cone. Hence L = ttts. 2 TT r Fig. 151 EXERCISES 1. Roll an oblique cylinder along a plane and make a drawing of the lateral surface. 2. Make a drawing of the lateral surface of an oblique cone. 3. The extreme length of a clothes boiler is 24 in., the width is 11 J in., and the depth 12| inches. The ends of the boiler are semicircular. Allowance has to be made for locking as follows: \\ in. on the width of the side piece and 1 in. on the length; \ in. all around the bottom piece. How much tin is required to make the boiler ? 273. Lateral area of a frustum of a right cone. Show that the lateral surface of a frustum of a right cone, Fig. 152, is the difference of the lateral surfaces of two right cones. Fig. 152 AREAS OF SURFACES 259 Hence the lateral area is given by the formula L = 7r(s2^2 $it*i). Since si and s 2 are not parts of the frustum, this formula will be changed to a different form, as follows: * = i 2 Why? n r 2 .' . S2n Sir 2 = Why ? . * . L = 7T (2^2 + S 2 n Sif2 Sifi) = 7r[s 2 (r 2 + ri) si (r 2 -f n) ] = 7r(r 2 +ri)(s 2 --Si) .*. I = ir(ri+r 2 )s This may be written : L=4(2Trri+2irr 2 )s. Hence the lateral area of a frustum of a right circular cone is equal to one-half the product of the slant height and the sum of the perimeters of the bases. EXERCISES 1. Show that the total area of a cylinder of revolution is given by the formula T = 2irr(h+r), h being the altitude and r the radius of the base. 2. Show that the total area of a cone of revolution is given by the formula T=irr(s+r), s being the slant height and r the radius of the base. State in words the law expressed by this formula. 3. Show that the lateral area of a frustum of a cone of revolu- tion is equal to the slant height multiplied by the length of a 260 THIRD-YEAR MATHEMATICS circle obtained by cutting the frustum by a plane at equal distances from the bases. Show that r = f(ri+r 2 ), Fig. 153. Hence ri+r 2 = 2r. Substituting this in the equation L = 7r(ri+r 2 )s, it follows that =2irr.s. Fig. 153 4. How much metal is required to construct a galvanized iron pail which is 9 in. in diameter at the top, 8j in. at the bottom, and 11 in. in the slant height, allowing lj in. on the width, 1 in. on the length of the side piece for locking, and 1 in. on the diameter of the bottom piece ? 6. The lateral area of a frustum of a right circular cone is 607r square feet. If the radii of the bases are 4 ft. and 6 ft. re- spectively, find the slant height. 274. Similar cylinders. Two right circular cylinders are similar if they are generated by revolving two similar rectangles about corresponding sides, Fig. 154. Fig. 154 Fig. 155 275. Similar cones. Two right circular cones are similar if they are generated by revolving two similar right triangles about corresponding sides, Fig. 155. AREAS OF SURFACES 261 276. Theorem: The lateral areas, or the total areas, of similar right circular cylinders, or cones, are 'proportional to the squares of the altitudes, or to the squares of the radii of the bases. Proof: Denoting the radii by r and r', Fig. 154, the altitude by h and h' , the lateral areas by L and U, and the total areas by T and T r , show that L _ 2-irrh _ rh _r h _ r 2 _h 2 V~2Trr , h , ~7h'~r ,y "h , ~r T2 ~V 2 ' T 2irr(h+r) r(h+r) = r h+r V~2>irr'(h'+r') ~ r f {h' +r')~ r ,X h' +r' ' Since 77 = -,, it follows that tj- , = -, = t} h r h+r r h By substitution, r r' 2 h' 2 ' The proof for the cones, Fig. 155, is similar and is left to the student. EXERCISES 1. Show that the lateral areas, or total areas, of two similar right cones are to each other as the squares of the slant heights. 2. How many square feet of surface are there in a tank formed by a cylinder and cone of the dimensions and shape shown in Fig. 156? Fig. 156 262 THIRD-YEAR MATHEMATICS 3. How many square feet of material, not allowing for waste, have been used in the construction of a silo, Fig. 157, the diameter of whose base is 16 ft., whose total height is 24 ft., and the height of whose roof is 8 feet ? GENERAL EXERCISES J277. Solve the following problems: 1. Find the lateral surface and the total surface of a cylinder of revolution if ft = 8. 5 in. and r = 5.3 inches. 2. Find the lateral surface of a quadrangular right pyramid, the side of whose base is 7 cm. and whose altitude is 6.8 centimeters. 3. Find the lateral surface and the total surface of a right cone whose radius is 4.2 and whose alti- tude is 5.7. 4. Find the lateral surface of a Fig. 157 frustum of a pyramid whose altitude is 10 and whose bases are squares with sides equal to 4 and 6 respectively. 5. The altitude of a right prism is 40. The base is a right triangle having the sides of the right angle equal to 36 and 43 respectively. Find the lateral and total area. 6. The base of a right prism is a regular hexagon whose side is 6. The altitude of the prism is 10. Find the lateral and total area. 7. The curved surface of a cylindrical column made of granite is to be polished. What will be the expense at the rate of 60 cents per square foot if the diameter of the base is 4 . 5 ft. and the column is 24 ft. high ? AREAS OF SURFACES 263 8. The great pyramid of Cheops is about 460 ft. high. The base is a square whose side is 746 ft. long. What is the lateral area? 9. A steeple is of the form of a regular hexagonal pyramid. The perimeter of the base is 60 feet. The slant height is 48 feet. How many square feet must be allowed for slating the steeple ? 10. At 26 cents a square yard what will be the cost of paint- ing a gas tank of the form of a right circular cylinder if the height is 72 ft. and the diameter of the base 45 feet ? 11. A funnel is 8 in. in diameter at the widest end, lj in. at the spout, and 1 in. at the smaller end of the spout. The slant height of the funnel is 6 in. and that of the spout is 4 inches. Allowing for locking J in. on the length and width of each part, find the amount of tin needed to make the funnel. 12. How far from the vertex of a right circular cone must a plane be passed parallel to the base and so that the lateral area of the small cone cut off shall be .equivalent to the lateral area plus one base of a right circular cylinder ? The altitude of the cone is 12 in., the radius of the base 8 inches. The altitude of the cylinder is 4 in. and the radius of its base is 2 inches. How far from the vertex of the cone must the plane be passed ? 13. A windmill water-supply tank, Fig. 158, is 8 ft. in diameter and 12 ft. high. The roof is 9 ft. in diameter and 3 ft. high. How much mate- Fig. 159 rial was used in its construction ? 14. The width and length of a tent, Fig. 159, are 12 ft. and 18 ft. respectively. The height of the pole is 8 ft. and the height Fig. 158 264 THIRD-YEAR MATHEMATICS of the wall 3| feet. How much material was used in making the tent? Surfaces of Revolution 278. Surface of revolution. If a line segment, AB, Figs. 160-164, revolves about a straight line, CD, in the same plane as an axis, every point of the segment describes a circle whose plane is perpendicular to the axis. Why ? The surface generated by the segment is a surface of revolution. According to the position of the segment with reference to the axis, the surface of revolution of the segment is a lateral surface of a right cone, Fig. 160, a Fig. 163 frustum of a right cone, Fig. 161, a right cylinder, Fig. 162, a surface of a circle, Fig. 163, or a circular ring, Fig. 164. AREAS OF SURFACES 265 279. Theorem: If half of a regular polygon having an even number of sides is revolved about a diagonal joining two opposite vertices, the area of the surface thus generated is equal to the product of the diagonal by the length of the circle inscribed in the polygon. C } Proof: The surface, Fig. 165, is composed of cones, frus- tum of cones, and cylinders. Hence the area may be found by adding the areas of these cones, frustums, and cylinders. It will be shown that one formula may be used to find the lateral surface of each. 1. The area of the surface generated by AB, Fig. 166, is given by Fig. 165 L^wBB XAB, 272. Bisect AB at M. Draw MM'AF. Show that MM' = \BB'. Then L l = 2irMM'XAB. Draw MOAB. AOMM'oAABB'. . AB AB' "MO MM'' .\MM'XAB = MOxAB'. /.L 1 = 2ttMOxAB 7 . F Fig. 166 Why? Why? Why? Why? Why? 266 THIRD-YEAR MATHEMATICS 2. The area of the surface generated by BC, Fig. 167, is given by L 2 = 7r(CC'+')C, 273. . Bisect BC at M . Draw MM'AF. Show that CC'+BB' = 2MM'. Then Draw Then U = 2ttMM , XBC. BB"CC. ACBB"i*AOMM\ Why? CB BB" B'C * * MO MM' MM' ' MM'XCB = M0XB'C. :.U = 2irM0xB , C c M 3. The area of the surface generated D by CD, Fig. 168, is given by L z = 2irDD'XCD. Bisect CD at M. Draw MO. Then Ls = 2tM0XCD. Similarly the area of the remaining part of the surface is found. Thus, Adding, Fig. 165, L l = 2wM0xAB' U = 2TvM0XB r C' U = 2ivM0xC , D , i etc. L = 2TMd(AB'+B'C'+. . ..E'F), or L = 2ttMOXAF AREAS OF SURFACES 267 280. Area of the surface of a sphere. To find the area of the surface of a sphere inscribe in a semicircle half a regular polygon as ABODE, Fig. 169. The area of the surface generated by the polygon ABCDE is 2tMOxAE. Let the number of sides of the polygon be increased indefinitely. Then the polygon approaches the circle as a limit. The area of the surface generated by the polygon approaches as a limit the area of the surface of the sphere generated by the semicircle. MO .approaches the radius r as a limit. Hence 2wMO approaches 2wr, and approaches 2irrXAE, which is equal to 27rrX2r = 47rr 2 . Hence the area of the surface generated by ABCDE approaches 47rr 2 as a limit. Thus the preceding discussion leads to the following theorem: The area of the surface of a sphere is equal to the product of the diameter by the length of a great circle, or Fig. 169 2ttMOXAE S = 4irr 2 . 281. Zone. A portion of a spherical surface included between two parallel planes is a zone, Fig. 170. The distance between the planes is the altitude of the zone. The sections made by the planes are the bases of the zone. If one of the planes is tangent to the spherical surface, the zone is said to have one base. Fig. 170 268 THIRD-YEAR MATHEMATICS EXERCISES 1. Show that the area of a zone is equal to the product of the altitude by the length of a great circle, or Z = 2-n-rh. Determine the area of a zone whose altitude is 12 if the radius of the sphere is 15. 2. The areas of two spherical surfaces are to each other as the squares of the radii. Prove. 3. How many square feet should be allowed for polishing a hemispherical dome whose diameter is if feet ? 4. The earth is approximately a sphere of diameter equal to 7,920 miles. How large is its surface ? 6. Find the area of the north temperate zone, assuming its altitude to be about 1,800 miles. 6. Show that the surface of a sphere is equal to the lateral surface of the circumscribed cylinder. 7. Find the ratio of the area of the surface of the moon to that of the earth, assuming the diameter of the moon to be 2, 162 miles. 8. Two parallel planes, equidistant from the center of a sphere of radius r, cut from the sphere a zone whose area is \ the area of the curved surface of the cylinder having the same bases as the zone. Find the distance of the planes from the center of the sphere. (Harvard.) 9. How far in one direction can a man see from the top of Mount Etna ? The required distance is the geometric mean between the height of the mountain, 3,300 m., and the sum of the height and the diameter of the earth, 6,374 kilometers. 10. Show that if a man ascended in a balloon to a height equal to the earth's radius he would see one-quarter of the earth's surface. (Harvard.) 11. The lateral area of a cone of revolution and the area of a sphere are each equal to 49 square feet. If the radius of the sphere equals the radius of the base of the cone, find the altitude of the cone. AREAS OF SURFACES 269 12. The eight vertices of a cube all lie on a sphere. Prove that every diagonal of the cube is a diameter of the sphere. If one edge of the cube is a, find the area of the zone of one base cut off by the plane of one face of the cube. (Harvard.) 13. If the temperate zones were between the 30 and 60 parallels of latitude, what proportion of the earth's surface would they comprise? Give the details of the computation. (Board.) 14. Two parallel planes on the same side of the center of a sphere of radius r bound a zone. The area of this zone is one- fourth that of the sphere. The area of the circle cut by the plane nearer to the center is double that cut by the farther. Find the distance from the center of the sphere to the nearer plane. (Harvard.) 282. The chapter has taught the meaning of the fol- lowing terms: polyedron face, edge, vertex, surface of a polyedron tetraedron, hexaedron octaedron, dodecaedron, icosaedron pyramidal and conical sur- face directrix, generatrix triangular, quadrangular, pentagonal pyramid regular pyramid slant height circular, right circular cone cone of revolution cylindrical surface, cylinder right cylinder, oblique cylinder cylinder of revolution prismatic surface, prism right and oblique prism triangular, quadrangular, etc., prism section, right section parallelopiped truncated prism frustum of a pyramid frustum of a cone sections of a right circular cone circle, ellipse, parabola, hyperbola similar cylinders, similar cones surface of revolution zone 270 x THIRD-YEAR MATHEMATICS Summary 283. The truth of the following theorems has been established : 1. The equation J-\-v = e+2 expresses the relation between the number of faces, vertices, and edges of a convex polyedron. 2. The lateral edges of a regular pyramid are equal. 3. The lateral faces of a regular pyramid are congruent isosceles triangles. 4. The lateral edges of a prism are equal. 5. The lateral faces of a prism are parallelograms. 6. The sections of a prism made by parallel planes are congruent. 7. The right sections of a prism are congruent. 8. A section of a prism parallel to the base is congruent to the base. 9. The section of a cylinder made by a plane passing through an element is a parallelogram. 10. The sections of a cylinder made by parallel planes cutting all elements are congruent. 11. The sections of a cylinder parallel to the bases are congruent to the base. 12. A section of a cone made by a plane passing through the vertex is a triangle. 13. A section of a circular cone made by a plane parallel to the base is a circle. 14. If a pyramid is cut by a plane parallel to the base, the edges and altitude are divided proportionally; the sec- tion is a polygon similar to the base; the areas of the section and the base are proportional to the squares of the distances from the vertex. AREAS OF SURFACES 271 15. The lateral areas, or the total areas of similar circular cylinders, or cones, are proportional to the squares of the altitudes, or to the squares of the radii of the bases. 16. The plane sections of a right circular cone are two intersecting straight lines, a circle, a parabola, an ellipse, and a hyperbola. 284. The following is a summary of the formulas of this chapter: I. Lateral area: 1. Of a prism, L=pXe 2. Of a right prism, L=p-h 3. Of a regular pyramid, L = \sXp 4. Of a frustum of a pyramid, L = \{p,+p,)s 5. Of a right cylinder, L = 2-rrrh 6. Of a right cone, L = irr 5 7. Of a frustum of a right circular cone, L=Tr(ri+r 2 )s II. The area of a surface generated by revolving half of a regular polygon about a diagonal joining two directly opposite vertices, L = 2ttMOxAF III. The area of the surface of a sphere, S=4-rrr 2 IV. The area of a zone, Z=2irrh CHAPTER XIII VOLUMES Volume of a Rectangular Parallelopiped 285. Volume. To measure the space bounded by the surface of a solid, a cube is used whose edges are the unit of length. This cube is said to be the unit of volume, and the number of times it is contained in the solid is the volume of. the solid. 286. Volume of a rectangular parallelopiped. Let a, b, and c be the lengths of three concurrent edges, Fig. 171. By drawing planes parallel to the base, the parallelopiped, p, may be divided into a equal layers, (Z), Fig. 172. Each layer, Z, ////// / x / / / / / y ' / / / / / X rfi^V y i i c y ////// > ////// s '///// s / r Fig. 171 Fig. 172 may be divided into b equal strips, (s), and each strip, s, into c equal cubes, (u). 272 VOLUMES 273 Hence the volume of a strip s is c, the volume of a layer I is bXc, and the volume of the parallelopiped p is aXbXc. So far we have assumed that a, 6, and c are com- mensurable. Let us suppose a, b, and c to be incommen- surable, e.g., a = V6m; b = VlOm; c = Vlbm. Then a=2.4494. . . . m,6=3.1623 .... m,c=3.8730 m. In this case the volume of the parallelopiped may be deter- mined to any desired degree of accuracy, as follows : 1. Taking a = 2. 4, 6 = 3.1, and c = 3.8 and using 1 decimeter as the unit of length, we have 7 = 24X31X38 cubic decimeters = 28,272 cubic decimeters = 28 . 272 cubic meters. 2. Similarly, for a = 2 . 44, b = 3 . 16, and c = 3 . 87, using a centimeter as unit, 7 = 244X316X387 cubic centi- meters =29,993,456 cubic centimeters = 29. 839248 cubic meters. 3. For a = 2.449, 6 = 3.162, and c = 3.873, using a millimeter as unit of length, 7 = 29,991,497,274 cubic millimeters = 29. 99 14 .... cubic meters. 4. For a = 2.4494, 6 = 3.1623, and c = 3.8730 we find 7 = 29,999,241,802,260 cubic one-tenth millimeters = 29.9992 .... cubic meters. By taking a, 6, and c to a still greater number of decimal places, we may obtain an approximate value of 7 differing from the actual value by a number less than any assigned quantity. Assuming the formula V = a 6 c to hold for incom- mensurable values of a, 6, and c, we find F = V / 6Xl / 10XV / 15 cubic meters = ^6X10X15 cubic meters = 30 cubic meters. 274 THIRD-YEAR MATHEMATICS However, this is the value approached by the sequence 7 = 28.272, 29.8392, 29.9914, 29.9992, etc. Thus, whether a, b, and c have commensurable or incommensurable values, the preceding discussion shows that the volume of a rectangular parallelopiped is equal to the product of the three dimensions. EXERCISES Prove the following: 1. The volume of a rectangular parallelopiped is equal to the product of the base by the altitude. 2. The volume of a cube is equal to the cube of an edge. 3. The volumes of two cubes are to each other as the cubes of the edges. 4. Two rectangular parallelopipeds are to each other as the products of the three dimensions. 5. Two rectangular parallelopipeds having equal altitudes (bases) are to each other as the bases (altitudes). 6. Two rectangular parallelopipeds having two (one) dimen- sions equal are to each other as the third (product of the other two) dimension. 7. Show that the volume of a cube varies directly as the cube of the edge. If the edge of a given cube is doubled, trebled, etc., how does the volume of the new cube compare with that of the given cube ? 8. How many dimensions are needed to determine the volume of a rectangular parallelopiped ? The area of one face ? 9. A room is 12 . 5 ft. long, 12 ft. wide, and 11 ft. high. Find how many cubic feet of air it contains. 10. How many bricks will be needed to build a wall 20X4X2 ft., making no allowance for mortar? Assume the size of a brick to be 9 X3 X4 inches. VOLUMES 275 Comparison of Volumes 287. Theorem: The plane passed through two diago- nally opposite edges of a right parallelopiped divides the parallelopiped into two equal triangular right prisms. Given the right parallelopiped AG, Fig. 173, plane ACGE passed through AE and CG. To prove that prism ABC-F 2 prism CDA-H. Proof: AABC&ACDA. Why? Imagine ABC-F placed on CDA-H making AABC coincide with AC DA. Then BF will coincide with DH, AE with GC, and GC with AE. Why ? Hence AEFG will coincide with AGHE. Why ? .*. Prisms ABC-F and CDA-H coincide throughout and are congruent. 288. Theorem: An oblique prism is equal to a right prism whose base is equal to a right section of the oblique prism, and whose altitude is equal to a lateral edge of the oblique prism. Given the oblique prism AD', Fig. 174, and the right prism FT, FI being a right section of prism AD'; FF' = AA'. To prove that AD' = Fr. Proof: Imagine the truncated prism AI to be placed on the truncated prism AT making AD coincide with A'D'. Show that FA, GB, HC, etc., coin- cide respectively with F' A' , G'B', H'C, etc. py 276 THIRD-YEAR MATHEMATICS Show that AG, BH, etc., coincide respectively with AW, B'H', etc. Hence AI = A'I r , since they can be made to coincide. But AI'mAI ' .'. AD' = Ff (Equals subtracted from equals give equals.) Exercises 1. The diagonal of a cube is l$Vz. Find the volume. 2. Find the surface and volume of a cube whose diagonal is 24 inches. 3. How many gallons of water are contained in a tank whose shape is that of a rectangular parallelopiped whose dimensions are 12, 20, and 10.4 feet respectively? A gallon contains 231 cubic inches. 4. Given a sphere whose diameter is 10 inches. Find the volume and the surface of the inscribed cube. (Sheffield.) Volume of a Prism 289. Theorem : The volume of a right triangular prism is equal to the product of the base by the altitude. Given the right triangular prism ABC-F, Fig. 175. j^\ To prove that ABC-F=ABCXCF. A Proof: Draw FGDE and H Fig. 175 CHAB. Show that FG and CH are both perpendicular to plane AE. Pass a plane through FG and CH. Draw AK \\ EC, CK \\ HA. Draw KI \\ HG, meeting plane DGF in /. VOLUMES 277 Then AHCK-F is a rectangular parallelopiped. AHC-F = iAHCK-F (287). .-. AHC-F = \AHCKXCF (286). .*. AHC-F = AHCXCF. Why? Similarly, prove that hbc-f=hbcxcf: Adding, ABC-F=ABCXCF. 290. Theorem: The volume of a right parallelopiped is equal to the product of the base by the altitude. Divide the parallelopiped into two equal right triangular prisms ( 287). Then find the sum of the two triangular prisms. 291. Theorem: The volume of an oblique parallel- opiped is equal to the product of the base by the altitude. Given the oblique parallelopiped AG, Fig. 176, whose base is ABCD and h h' g g' whose altitude is h. To prove that AG=ABCDXh. Proof: Con- A ^ b ~b' struct the right yig. 176 section A' H' '. On A'H' as base construct the right parallelopiped A'G', having its edge A'B' equal to AB. Then AG = A'G'. Why? But A'G' = A'H'XA'B' = (hXA'D')XA'B' = hXA'D'XAB = hX(A'D'XAB) = hXABCD. Briefly, this result may be expressed by the equation V=h>b. 278 THIRD-YEAR MATHEMATICS 292. Theorem: The plane passed through two diago- nally opposite edges of any parallelopiped divides the parallel- opiped into two equal triangular prisms. Draw the right section IJKL, Fig. 177. Show that the triangular prism ABC-F = the right prism having the base UK and the altitude equal to BF. Show that CDA-H = the right prism having the base KLI and the altitude equal to BF. Show that these two right prisms are equal to each other ( 289). .-. ABC-F = CDA-H. rr s * IF Dl '^9* 1 / V Fig. 177 293. Theorem: The volume of any triangular prism is equal to the product of the base by the altitude. Construct the parallelogram p IG -^g ABCD, Fig. 178. Construct the parallelopiped ABDC-B'. Then ABC-B' = ^ABDC-B' ( 292). ABDC-B' = ABDCXh (291). ABC-B' = %ABDCXh = ABCXh. 294. Theorem: The volume of any prism is equal to the product of the base by the altitude. By drawing planes through AA', Fig. 179, and all the other lateral edges, the prism may be divided into triangular prisms, having the same altitude, h, and bases 6i, 62, b 3 , etc VOLUMES 279 Then ABC-A' = hh, ACD-A' = b 2 h, APE - A' = b z h, etc. Adding, ABODE- A' = (6i+6 2 +6 3 + etc.)h. This result may be expressed briefly by means of the equation y=b-h EXERCISES 1. Prisms having equal bases and altitudes are equal. Prove. 2. The volumes of two prisms having equal bases are to each other as the altitudes. Prove. 3. How many cubic yards of earth must be removed to build a trench 200 ft. long and 10 ft. deep, 4 ft. wide at the bottom and 6 ft. at the top ? 4. What will be the cost, at 42 cents a cubic yard, to dig a ditch 10 rd. long, 4 ft. deep, 7 ft. wide at the top, and 4^ ft. wide at the bottom ? 5. Find the volume of a right triangular prism, 8 in. high, whose base is an equilateral triangle with sides of 2 inches. 6. A triangular prism is 10 ft. high. The base is a right triangle whose sides are 3, 4, and 5 ft. respectively. Find the volume. 7. The volume of a triangular prism is 250. The base is an equilateral triangle whose side is 7. Find the altitude of the prism. 8. Find the volume of a triangular prism whose height is 30 in. and the sides of whose base are 12 in., 10 in., and 10 inches. 9. Find the volume of a prism whose base is a rhombus, one of whose sides is 40 in., and whose shorter diagonal is 48 inches. The height of the prism is 60 inches. 10. A regular hexagonal prism has the area of one base 12 and its total area 276. Find the volume of the prism. (Yale.) 280 THIRD-YEAR MATHEMATICS 295. Inscribed prism, pyramid, and frustum of a pyramid. A prism, a pyramid, and a frustum of a pyra- mid are said to be inscribed if the lateral edges are ele- ments of a cylinder, a cone, or a frustum of a cone, respectively, and if the bases of the former are inscribed in the bases of the latter, Fig. 180. It will be assumed that Fig. 180 the volume of an inscribed prism, pyramid, or frustum of a pyramid is less than the volume of the cylinder, cone, and frustum of a cone respectively. 296. Tangent plane. If a plane contains one, and only one, element of a cylinder, a cone, or a frustum of a cone, but does not intersect the surface, it is a tangent plane, Fig. 181. EXERCISES 1. Prove that a plane passing through a tangent to the base of a circular cone and the element drawn through the point of contact is tangent to the cone. Is this theorem necessarily true when the cone is not circular ? (Harvard.) 2. The intersection of two planes tangent to a circular cylinder is parallel to the elements of the cylinder. (Yale.) VOLUMES 281 297. Circumscribed prism, pyramid, and frustum of a pyramid. A prism, a pyramid, and a frustum of a pyra- mid are said to be circumscribed if the lateral faces are tangent to the lateral surface of a cylinder, a cone, and a frustum of a cone, respectively, and if the bases of the former are circumscribed about the bases of the latter, Fig. 182. It will be assumed that the volume of a circum- /C yi I ' ' iii j iJ IV r Fig. 182 scribed prism, pyramid, or frustum of a pyramid is greater than the volume of the cylinder, cone, or frustum of a cone respectively. Volume of a Cylinder 298. Theorem: The volume of a circular cylinder is equal to the product of the base by the altitude. Given the cylinder AC, Fig. 183, whose altitude is h and whose base is b. To prove that the volume, v = b-h. Proof (indirect method) : Assume that v^b'h. Then v > bh , or v B. Inscribe in the cylinder a prism whose base, B' , is greater than B, i.e., such that b>B'>B. Fig. 183 282 THIRD- YEAR MATHEMATICS Then B , h>Bh. Thus B'h, the volume of the inscribed prism, is greater than Bh, the volume of the cylinder. This is impossible, and v is not less ^ ^\ than b'h. . A. _^# Secondly, assume v>bh, or v = Bh, J! II where b2 ( 260) ABC CW M'N'P'Q'R' = W' 2 MNPQR sf 2 OE ,2 = ST' 2 OE 2 OT 2 A'B'Ci ^M'N'P'Q'R' * ABC = MNPQR .\ A'B'C = M'N'P'Q'R' Why? Why? Why? VOLUMES 285 301. Theorem: If two triangular pyramids have equal bases and altitudes, their volumes are equal. Fig. 186 Discussion: Place the bases of the pyramids, Fig. 186, in the same plane and divide the altitude h into equal parts, x. Through the points of division draw planes parallel to the plane of the bases, cutting the pyramids into sections equal in pairs ( 300). Using each section as lower base, construct prisms whose altitudes are equal to x and whose lateral edges are parallel to AV and A'V respectively. In this way the prisms P, Q, R, and S have been constructed in pyramid V-ABC. Imagine a similar set of prisms constructed in V'-A'B'C. Using each section as upper base, construct prisms whose altitudes are equal to x and whose lateral edges are parallel to AV and A'V respectively. This gives prisms P', Q', and R' in the pyramid V'-A'B'C Imagine a similar set of prisms constructed in V-ABC. Show that P = P', Q = Q', R = R'. Denote P+Q+R+S by X and P'+Q'+R' by F. ThenZ-F = >S. 286 THIRD-YEAR MATHEMATICS Denoting the volumes of V-ABC and V'-A'B'C by V and V, respectively, we have X>V>Y and X>V>Y. .'. the difference between V and V is less than the difference between X and Y, i.e., V-V'V, and denote the difference by d, i.e., V-V' = d. We have seen that by increasing the number of divi- sions in the altitude h we can make S less than any assigned quantity, therefore less than d. Hence, V-V'B. FlG 192 Then \B'h>\Bh. This means that the volume of the inscribed pyramid is greater than the volume of the cone. This is impossible, and v is not less than \bh. 2. Similarly we may show that v is not greater than \bh. 3. Hence v = A> h. 292 THIRD-YEAR MATHEMATICS EXERCISE A right triangle is revolved about one leg. Show that the volume of the cone thus generated is equal to the product of the area of the triangle and the circumference of the circle traced by the point of intersection of the medians. (Harvard.) Volume of a Frustum of a Cone 307. The formula giving the volume of the frustum of a cone is obtained in the same way as the formula for the volume of the frustum of a pyramid. Hence, v = \h(bi+b 2 +V bib 2 ) - 308. To find the volume of a frustum of a cone of revo- lution let bi = 7rn 2 and b 2 =irr 2 2 . Then Vb& 2 = V-n-r^irr^ = irnr 2 . Substituting these values in 307, v=lirh(ri 2 +r 2 *+rir2). EXERCISES 1. The bases of a frustum of a pyramid are regular hexagons whose sides are 8 in. and 4 in. respectively. The altitude of the frustum is 3 feet. Find the volume. 2. The radii of the bases of a frustum of a cone of revolution are 4 in. and 5 inches. The frustum is 12 in. high. Find the volume. 3. A conical heap of grain is 4 ft. high and has a circular base whose radius is 5 feet. How high must a bin be whose base is 4 ft. square to contain the grain ? 4. Find the number of bushels of wheat contained in conical heap thrown into a corner of a bin, the highest point of the heap being 4 ft. and the radius of the circular base being 6 . 5 feet ? A bushel contains 2,150 cubic inches. 5. A cone is 12 in. high and the area of the base is 15 square inches. Find the volume. VOLUMES 293 6. The height of the frustum of a cone is 6 in. and the radii of the bases are 4 in. and 8 in., respectively. Find the volume. 7. What must be the depth of a pail that is 18 in. aeross the top and 10 in. across the bottom in order that it may hold 5,280 cubic inches? (*-V.) (Yale.) 8. A pyramid is 6 in. high. The area of its base is 324 square inches. Find the volume of the frustum cut off by a plane 4 in. from the base. j, ig o 9. Find the volume of a grain tank, Fig. 193, 10 ft. high and 9 ft. in diameter, the height of the roof being 3 feet. Volume of a Sphere 309. Theorem: The volume of a sphere of radius r is Let ACB, Fig. 194, be a hemisphere, and let DF be a right cylinder whose circular base, DE, is equal to the i c 111 Q \\2 k G (, ^r?~~. r,~ . -^ \F K m\ 1 L H J -jf i p z / ^ f B d\ \e/ / Fig. 194 circle A B and whose altitude is equal to the radius, r, of the sphere. Suppose a cone H GF be cut from the cylinder leaving the solid, GDEFHG. Pass a plane parallel to plane Z and at a distance x from Z, and let KL and MNOP be the sections of ACB and GDEFH, respectively. 294 THIRD-YEAR MATHEMATICS Show that RL = Vr 2 - x 2 . au .. UO HU UO x Show that YF = Hf ,0V ~T = V :. OU=x. Show that the area of KL = -rr{r 2 x 2 ). Show that the area of the circular ring MNOP = 7rr 2 ttX 2 = ir{r 2 X 2 ) . .'. the hemisphere ACB is equal to the solid GDEFHG, 302. Since GDEFHG = (DF) -(H- GF) = ttt 2 r \-rrr 2 r fir 3 , it follows that the hemisphere ACB = \tti s '. .'. the volume of the sphere is given by the formula y = 3'irr 3 . EXERCISES 1. Find the weight of a cast-iron sphere 4 in. in diameter. Cast-iron weighs .26 lb. per cubic inch. 2. Find the volume of a sphere 4 in. in diameter. 3. Find the volume of metal in a spherical shell \ in. thick whose external diameter is 4 inches. 4. The area of a spherical surface is 6 square inches. Find the volume of the sphere. 6. A bar of metal of the form of a rectangular parallelopiped 12 X 8 X 4 in. is to be melted and cast into a spherical ball. What is the radius of the ball ? No allowance is to be made for waste. 6. Prove that the volumes of two spheres are to each other as the cubes of the radii. 7. Regarding the earth and the sun as spheres of radii 4,000 mi. and 860,000 mi., respectively, compare their volumes. VOLUMES 295 8. A rifle shell has the shape of a cylinder surmounted by a hemispherical cap. The total length of the shell is four times its diameter. Compare the surfaces and also the volumes of the cylindrical and the spherical portions. (Sheffield.) 9. Find the volume and surface of a sphere inscribed in a cube whose diagonal is 6V 7 3. (Yale.) 10. A sphere is inscribed in a cube. Find the ratio of the radius of the sphere to the edge of the cube. 11. What percentage of the volume of a sphere is contained in the inscribed cube ? (Harvard.) 12. A regular octaedron has an edge a. Find the volume of the inscribed sphere. (Harvard.) 13. In a semicircle of radius a is inscribed a right triangle one of whose acute angles is 30, the hypotenuse of the triangle being the diameter of the circle. The figure is revolved about the diameter as an axis. Find the ratio of the volumes generated by the triangle and the semicircle. (Yale.) 14. The inside of a glass is in the form of a cone whose verti- cal angle is 60, and whose base is 2 in. across. The glass is filled with water and the largest sphere that can be immersed is placed in the glass. How much water remains in the glass? (Yale.) 15. A hemisphere and a right circular cone have the same base, and the areas of their curved surfaces are equal. Find the ratio of their volumes. (Harvard.) Volume of a Spherical Segment 310. Spherical segment. The Xjg ^N portion of a sphere included between S~ ^ two parallel planes intersecting a ^ . Spherical Segment sphere is a spherical segment, of two bases Fig. 195. Fig. 195 296 THIRD-YEAR MATHEMATICS The perpendicular between the planes is the altitude, the sections of the sphere made by the planes are the bases of the segment. If one of the planes is tangent to the sphere the segment has only one base, Fig. 196. Spherical Segment of one base Fig. 196 311. Theorem: The volume of a spherical segment of one base is given by the formula V=l?i 2 Tr(3r-h), where r is the radius of the sphere and h the altitude of the segment. Proof: According to 302, 309, the segment KCL, Fig. 197, is equal to the solid MPFONG, which is the Fig. 197 difference between the cylinder M F and the frustum of a cone, GNOF. The radii of the bases of the frustum are TF = r and UO = x = r-h. .-. MF = 7rr 2 h and GNOF = lirh[r 2 +(r-h) 2 +r(r-h)] = %irh(3r 2 -3rh+h 2 ) .'. KCL = MF-GNOF = Trr 2 h-l7rh(3r 2 -3rh+h 2 ) = i*h(3r 2 -3r 2 +3rh-h 2 ) = \nh(3rh-h 2 ) :. V=lTTh 2 (Sr-h) VOLUMES 297 312. Theorem: The volume of a spherical segment of two bases is given by the formula V=l(Trrf+irr 2 *)+-~. Proof: The volume of a segment of two bases is equal to the difference of two segments having one base. Denoting the altitude of the segment of two bases by h and the altitudes of the segments of one base by hi and h 2 , respectively, we have h = hi h 2 . .-. v = ^7rhi 2 (Sr-hi)-^Trh 2 2 (Sr-h2) = irrhi 2 ^-rrhi 3 itrh 2 2 + ^vh 2 s = 7rr(hi 2 -h 2 2 )-^7r(hi z -h 2 s ) = irrihi-h 2 )(hi+h 2 )-^7r(hi-h 2 )(hi 2 +hih 2 +h 2 2 ) (hi-h 2 )[r(hi+h 2 )-\{hi 2 +hih 2 +h 2 2 )} = irh[rhi+rh - (h 2 - 2hih 2 +h 2 2 +3hh 2 )] = 7rh[rhi+rh 2 -^(h 2 +3hih 2 )] I h 2 = -rrh ( rhi +rh 2 hih 2 h 2 r 2 _...,,,_ _ x h n Show that = =- and that = r 2 2r h 2 n 2r hi } .-. 2rh 2 -h 2 2 = r 2 2 and 2rhi hi 2 = n 2 . Adding, 2rh+2rh 2 - (h 2 +h 2 2 ) = n 2 +r 2 2 n 2 +r 2 2 , h 2 +h 2 2 rhi-\-rh J n 2 +r 2 2 , h 2 +h 2 2 h 2 ,,\ .. ,.^__+_ ~-hih 2 ) t /ri ? fr 2 2 , h 2 -\-2hih 2 h 2 2hih 2 \ 777 2 K-J(rf+*i*)+^ 298 THIRD-YEAR MATHEMATICS 313. Spherical cone. A spherical cone, Fig. 198, is generated by revolving a circular sector, ABC, Fig. 199, about its bounding radius, BC, as an axis. 314. Theorem: The volume of a spherical cone is given by the formula. v=*pr 2 h. Proof: ABDC=ABD+ACD = 7rx*(r-h)+lTrh 2 (Sr-h) = ^[r 2 -(r-h) 2 ](r-h)+^7rh 2 (3r-h) = %Tr(2rh-h 2 )(r-h)+^h 2 (3r-h) = 7rh(2r 2 -hr-2rh+h 2 +3rh-h 2 ) = \irh-2r 2 .'. v = \-nr 2 h. 315. Spherical sector. The portion of a sphere generated by revolving a circular sector ABC, Fig. 200, about a diameter of its circle is a spherical sector, Fig. 201. 316. Theorem: The volume of a spherical sector is given by the formula v=\-nr 2 h. B Fig. 201 VOLUMES 299 Proof: A spherical sector is the difference between two spherical cones. Denote their volumes by vi and v 2 respectively. Then vi = %-n-r 2 hi V v 2 = 1 7rr 2 /l 2 v\ V2 = Trr 2 (hi hz) or v = Trr 2 /i. EXERCISES 1. The distance of a plane from the center of a sphere is one- third the radius of the sphere. Find the ratio of the volumes of the two solids into which the sphere is divided by this plane. (Harvard.) 2. In a certain sphere there are as many square feet in the surface as there are cubic feet in the volume. Find the radius and determine the. area of the segment of this spherical surface cut off by a plane perpendicular to the radius at its middle point. $3. How large a hole must be bored through a sphere 6 in. in diameter to remove one-half of the sphere ? The part cut from the sphere consists of a cylinder, C, and two spherical segments, S. 27rr 3 Show that 2S = 5 (2+ cos x 2 cos 2 z cos 3 x) o and that C = 2-irr 3 (cos x cos 3 x) . $4. The diameter of a sphere is 10 inches. If a cylindrical hole of 5 in. in diameter is bored through the sphere, what is the volume of the remaining solid? It is assumed that the center of the sphere lies on the axis of the cylinder. 5. The curved surface of a spherical segment of one base is 25V and the altitude is 3. Find the volume. 300 THIRD-YEAR MATHEMATICS Summary 317. The chapter has taught the meaning of the fol- lowing terms : unit of volume, volume inscribed prism, pyramid, and frustum of a pyramid circumscribed prism, pyramid, and frustum of a pyramid tangent plane spherical segment, cone, and sector 318. The following theorems have been studied : 1. The plane passed through two diagonally opposite edges of a right parallelopiped divides the parallelopiped into two equal triangular right prisms. 2. An oblique prism is equal to a right prism whose base is equal to a right section of the oblique prism and whose altitude is equal to the lateral edge of the oblique prism. 3. The plane passed through two diagonally opposite edges of any parallelopiped divides the parallelopiped into two equal triangular prisms. 4. Prisms having equal bases and altitudes are equal. 5. The volumes of two similar cylinders of revolution are to each other as the cubes of the altitudes, or as the cubes of the radii of the bases. 6. If two pyramids have equal bases and equal altitudes, sections made by planes parallel to the bases and at equal distances from the vertices are equal. 7. If two triangular pyramids have equal bases and altitudes, they are equal. 8. If two solids lie between two given parallel planes, having their bases in these planes, and if the sections made by any plane parallel to the given planes are equal, then the volumes of the solids are equal. VOLUMES 301 319. The following is a summary of the formulas in this chapter: Rectangular parallelopiped v = aXbXc v = bxh (6 = base) Cube v=e* Triangular right prism v = bxh Right parallelopiped v = bXh Oblique parallelopiped v=bXh Triangular prism v=bxh Prism v = bxh Cylinder v = bXh Cylinder of revolution v=tn 2 h Pyramid v = \bXh Frustum of a pyramid v=bi(bi+bi+V bib 2 ) Cone v=\hXb Frustum of a cone of revolution v^gvftfcH-itf+ivO Sphere i/= -Jirr 3 Spherical segment of one base v=*trh 2 (Zr-h) Spherical segment of two bases v=^(trri 2 +Tn 2 2 )+^w- 2 6 Spherical cone v=\trr 2 h 3 Spherical sector v=%r 2 h CHAPTER XIV POLYEDRAL ANGLES. TETRAEDRONS. SPHERICAL POLYGONS Polyedral Angles 320. Polyedral angle. If a line, AB, Fig. 202, moves with one endpoint fixed at A and always touching a con- vex polygon, CDEFG, whose plane does not contain A, it generates a convex polyedral angle. The fixed point A is the vertex, the bounding planes CAD, DAE, etc., are the faces, the lines AC, AD, etc., are the edges, A CAD, DAE, etc., are the face angles of the polyedral angle. 321. Triedral angle. A polyedral angle having three faces, as (1), Fig. 202, is a triedral angle. Point out several triedral angles in the classroom. 322. Theorem: The sum of two \d face angles of a triedral angle is greater than the third. Fig. 203 Given the triedral angle O-ABC, Fig. 203. To prove that IAOB+ Z BOO Z.AOC. 302 POLYEDRAL ANGLES 303 Proof: The theorem is easily proved for a triedral angle having equal face angles. Assume that the face angles are not equal and that ZAOC is the greatest face angle. In the plane AOC draw OD, making ZAOD = ZAOB. Lay off OD'=OB'. Pass a plane through B' and D', cutting the faces in lines A'B\ B'C\ and C'A', respectively. Prove that AA'OB' & AA'OD'. .'.A'B' = A'D'. Show that A'B'+B'OA'C. Subtracting, B'OD'C. .*. ZB'OO ZD'OC. .'. /.A'OB'+AB'OOAA'OD'+AD'OC, or ZAOB+ /.BOO ZAOC. EXERCISES 1. Show that the difference of two face angles of a triedral angle is less than the third. 2. Show that any face angle of a polyedral angle is less than the sum of the other face angles. 3. Show that the three planes bisecting a triedral angle intersect in a straight line. 323. Spherical polygon. Let the faces of the polyedral angle O-ABCD, Fig. 204, intersect the surface of a Fig. 204 304 THIRD-YEAR MATHEMATICS sphere whose center is in the great circle arcs AB, BC, CD, and DA. The figure ABCD on the surface of the sphere is a spherical polygon. Thus a spherical polygon is the section of a spherical surface made by a convex polyedral angle whose vertex is at the center of the sphere. To every polyedral angle at the center of the sphere corresponds a spherical polygon. A spherical polygon of three sides is a spherical triangle, Fig. 205.* The bounding arcs, AB, BC, etc., are the sides of the spherical polygon. The points of intersection of the sides are the vertices of the polygon. Fig. 205 EXERCISES 1. The sides of a spherical polygon are usually measured in degrees. Show that the sides of a spherical polygon have the same measure as the face angles of the corresponding polyedral angle at the center of the sphere, Fig. 206. * The properties of spherical triangles are applied in the solution of problems in astronomy, navigation, and geography. In fact, spherical geometry was first developed by astronomers. The follow- ing are some of the interesting applications: 1. To determine the position of an observer on the surface of the earth, i.e., his latitude and longitude. 2. To find the distance between two places and the bearing of each from the other when their latitudes and longitudes are known. 3. To determine the position of a star. 4. To determine the time of the day at a place on the surface of the earth. 5. To determine the course of a ship. POLYEDRAL ANGLES 305 2. Two sides of a spherical triangle are 88 and 70. What are the limits for the third side ? Exercise 2 indicates how some properties of spherical polygons may be inferred from a study of polyedral angles. , 3. Show that the sum of two sides of a spherical triangle is greater than the third side, Fig. 206. 4. The shortest line that can be drawn between two given points on the surface of a sphere is the minor arc of the great circle which passes through the two points. Prove. Proof: 1. Let A and B, Fig. 207, be the two given points and let ACB be the minor arc of a great circle joining A and B. Let C be any point onAB. With A and B as centers and radii equal to AC and BC, respectively, draw two small circles meeting at C. A Let D be any point on circle A, not point C, and draw the arcs of great circles AD and DB. Then AD+DB>AB, 323, exer- cise 3. ^ ^ But AD = AC , DB>CB. Thus D lies outside of circle B. .'. circles A and B are tangent to each other at C. 2. Let AEFB be any line joining A and B on the surface of the sphere and not passing through C. Then line AEFB meets circles A and B in two distinct points, E and F. Why? Whatever may be the form of AE, an equal line can be drawn from A to C; and whatever may be the form of BF, an equal line can be drawn from B to C. Why ? Hence it is possible to draw a line from A to B passing through C and equal to AE+FB. Since AE+FB Z ABC, 322. In the triedral angle C-BOD, ZBCO+ZOCD> A BCD, etc. Adding, Z ABO+ Z CBC+ Z BCO+ Z (XLD+etc, > ZABC+ZBCD+etc, i.e., the sum of the base angles of the triangles with vertex is greater than the sum of the base angles of the triangles with vertex 0'. But the sum of all angles of the triangles with vertex is equal to the sum of all angles of the triangles with vertex 0'. : . by subtracting unequals from equals we have the sum of the face angles at Oless than the sum of the angles about 0'. In symbols this may be stated as follows : ZAOB+ ZBOC+ ZCOD+etc, < ZAO'B+ZBO'C + ZCO , D+etc. Since ZAO f B+Z BO'C+ Z CO'D+etc. = 4 R.A. ,\ ZAOB+ ZBOC+ ZCOD+ etc. <4R.A. POLYEDRAL ANGLES 307 EXERCISE Prove that the sum of the sides of any convex spherical polygon is less than 360, Fig. 209. 325. Number of regular poly- edrons. In 245 five regular polyedrons were shown. The theorem in 324 may be used to prove that there are no other kinds of convex regular polyedrons. For the faces of a regular polyedron are all regular poly- f ig . 209 gons such as equilateral triangles, squares, etc., and the sum of the face angles of any polyedral angle of the polyedron must be less than 360. Why? Show that three, four, or five equilateral triangles, but not six or more, may be placed so as to form a polyedral angle. Hence no polyedron can be formed with six or more equilateral triangles at the vertex. The tetraedron has three equilateral triangles at one vertex, the octaedron has four, and the icosaedron has five. Show that three squares may be placed so as to form a polyedral angle, but not four or more. Hence no polye- dron can be formed with four or more squares at a vertex. The cube has three squares at one vertex. Show that three regular pentagons may be placed so as to form a polyedral angle, but not four or more. Hence the dodecaedron is the only polyedron whose faces are regular pentagons. Show that it is impossible to form a regular polyedron having six or more regular polygons at one vertex.* * The regular solids were studied so extensively by Plato and his school that they have received the name of " Platonic figures." 308 THIRD-YEAR MATHEMATICS Tetraedrons 326. Theorem : Two tetraedrons having a triedral angle of one equal to a triedral angle of the other are to each other as the products of the edges including the equal triedral angles. Given the tetraedrons T-ABC and T'-A'B'C, Fig. 210, with the triedral angle at T equal to the triedral Fig. 210 angle at T' and having the volumes equal to V and V respectively TAXTBXTC To prove that y , = rA , xrB , xrc , Proof: Place T'-A'B'C' on T-ABC, making triedral angle T' coincide with triedral angle T. Draw C'P' and CPLATB. V_ \XABTXCP V Then ABT v CP WaTB'TXC'P' A'B'T^C'P' 303. Since triangles ABT and A'B'T have one angle equal, ABT TAXTB A'B'T show that CP TA'XTB' TC TC or CP' By substitution, V^^ TAXTB V'~ V TAXTBXTC V TC TA'XTB'^TC" T'A'XT'B'XT'C POLYEDRAL ANGLES 309 327. Similar polyedrons. Two polyedrons are similar if their faces are similar each to each and similarly placed and if the corresponding polyedral angles are equal. 328. Theorem: Two similar tetraedrons are to each other as the cubes of the corresponding edges. V TAXTBXTC V TA'XT'B'XT'C" TA TB TC 326. Show that T'A' V V T'B' TA T'C TA / ^\ rni A / ^ TA TA' T'A'"T'A'"T'A' TA /3 EXERCISE A pyramid, the area of whose base is 36 sq. ft., contains -g 1 ^ of the volume of a similar pyramid whose altitude is 9 feet. Find the volume of each pyramid. 329. To construct a sphere through four given points not all in the same plane. Given the four points A, B, C, and D, Fig. 211, not all in the same plane. To construct a sphere passing through A, B, C, and D. Construction: Draw AB, AC, AD, BC, CD, and DB forming the tetra- edron A-BCD. Bisect CD at E. Draw plane FEG perpendicular to CD at E and inter- secting planes CAD and CBD in lines EF and EG respec- tively. Show that EF passes through the center F of the circle circumscribed about ACAD and that EG passes through the center G of the circle circumscribed about &CBD. 310 THIRD-YEAR MATHEMATICS Draw FI plane CAD and GH plane CBD. Since CD _L plane FEG, it follows that planes CAD and Cj5D are perpendicular to plane FEG. Why? Show that FI and (?# lie in plane FEG, 551. Show that FI and (x# are not parallel. Denoting the point of intersection of FI and GH by 0, show that is equidistant from A, B, C, and D. Therefore a sphere with as center and radius OB passes through A, B, C, and D. 330. To inscribe a sphere in a given tetraedron. Given the tetraedron A- BCD, Fig. 212. To construct a sphere tangent to all faces of A- BCD. Fig. 211 Construction: Draw plane BOD bisecting the diedral angle BD. Draw plane BOC bisecting diedral angle BC. These planes must meet in a line, as BO, since they have point B in common. POLYEDRAL ANGLES 311 Draw plane CO A bisecting the diedral angle AC. This plane will meet the line of intersection of planes BOD and BOC in point 0. Show that is equidistant from the four faces of the tetraedron A-BCD. Hence a sphere with as center and radius equal to the perpendicular from to one of the faces will be tangent to all faces. 1 331. To determine the diameter of a given material sphere. Fig. 213 Given a material sphere, 0, Fig. 213. To find its diameter. Construction: With P, a point on the surface of the sphere, as a pole, describe the circle ABC. Let A, B, and C be three points on this circle. Construct triangle A '.B'C'congruent to the triangle ABC. Circumscribe a circle about AA'B'C, and let D' be the center of this circle. Draw D"A" equal to the radius D'A'. Through D" draw a line P"Q" perpendicular to D" A" . From A" lay off A"P" equal to AP. At A" erect A"Q" perpendicular to A"P". Then P"Q" is the required diameter of the given sphere. The proof is left to the student. 312 THIRD-YEAR MATHEMATICS Spherical Angles 332. Spherical angle. Two intersecting curves, C and C, Fig. 214, are said to form an angle. The angle formed by two intersecting curves is the angle made by the tangents to the curves at the common point, as Z TOT'. Fig. 214 Fig. 215 The angle formed by two intersecting arcs of great circles is a spherical angle, as TOT', Fig. 215. The point of intersection, 0, is the vertex and the arcs OA and OB are the sides of the spherical angle. 333. Measure of a spherical angle. Draw AB, Fig. 215, an arc of a great circle with as a pole and terminated by the sides of the spherical angle AOB. Draw the radii CO, CA, and CB. Show that OC is perpendicular to OT and CA, .*. OTWCA. Similarly, OT'WCB. ... ZT0T'=ZACB. But Z.ACB has the same measure as arc AB. :. /.TOT', or spherical angle AOB, is measured by arc AB. POLYEDRAL ANGLES 313 This proves the following theorem: A spherical angle is measured by the arc of a great circle having the vertex as pole, and included between the sides, produced if necessary. EXERCISES 1. Prove that a spherical angle is equal to the diedral angle formed by the planes of the sides. 2. The angles formed by the sides of a spherical triangle are respectively equal to the diedral angles of the corresponding triedral angle. Prove. 334. Right spherical angle. When the planes of the sides of an angle are perpendicular to each other, a right spherical angle is formed. 335. Classification of spherical triangles. The terms isosceles, equilateral, and scalene have the same mean- ing for spherical triangles as for plane triangles. A spherical triangle is right, birectangular, or trirectangular, Fig. 216 according as it has one, two, or three right angles, Fig. 216. 314 THIRD-YEAR MATHEMATICS Polar Spherical Triangles 336. Polar triangles. Let AABC, Fig. 217, given triangle. Draw three great circle arcs as B'C, and CA', having as poles C, A, and B, respectively. Any two of these circle arcs, if far enough extended, have two points of intersection. Let C be that point of intersection of arcs A'C and B'C which is nearest to C, let B' be the point of intersection of arcs A'B' and CB' which is nearest to B, and let A' be the point of intersection of arcs B'A' and CA' which is nearest to A. polar triangle of AABC. be a A'B' Fig. 217 Then AA'B'C is the 337. Theorem : If a spherical triangle is the polar triangle of another, then the second is the polar triangle of the first. Given AABC, Fig. 218, and AA'B'C, the polar of AABC. To prove that AABC is the polar of AA'B'C. Proof: Since A is the pole of B'C, it follows that B' is a quadrant's distance from A. Since C is the pole of B'A', it follows that B' is also a' quad- rant's distance from C. .'. B' is the pole of AC, 563. Similarly, prove that A' is the pole of BC and C is the pole of A B. .'. AABC is the polar triangle of AA'B'C. Fig. 218 POLYEDRAL ANGLES 315 EXERCISES 1. Make a sketch of the polar triangle of a birectangular triangle and show that it is also birectangular. 2. Show that the polar triangle of a given trirectangular triangle is identical with the given triangle. 338. Theorem: In two polar spherical triangles each angle of the one is the supplement of that side of the other of which it is the pole. a! Fig. 219 Given the polar triangles A'B'C and ABC, Fig. 219, having the sides equal to a', b', c' and a, b, c respectively. To prove that A +a' = 180, B+b' = 180, C+c' = 180/ A'+a = 180, B'+b = 180, C'+c = 180. Proof: Let the sides of /.A, produced if necessary, intersect the side B'C in points D and E respectively. Since Z A is measured by arc DE, 333, A=DE. Since ITE = 90 = DC', B 7 E+DC / = 180. i^+(jD+ J E'C , ) = 180 o . Why? DE+ (B^E+E) = 180. Why ? DE+lfc' = 180. Why? .*. A+a' = 180. Why? Similarly, B+b' = 180, C+c' = 180, etc. 316 THIRD-YEAR MATHEMATICS EXERCISES 1. Find the sides of the polar triangle of a triangle whose angles are 75, 85, and 88 respectively. 2. The angles of a spherical triangle are 88, 125, and 96 respectively. Find the sides of the polar triangle. 339. Theorem: The sum of the angles of a spherical triangle is less than six and greater than two right angles. Given the spherical triangle ABC, Fig. 220. To prove that A + B+C<50; A + J5 + O180 . Proof: 1. Let AA'B'C be the polar triangle of AABC. Then A +a' = 180 B+b' = 180 C+c' = 180 Adding, A+B+C+a'+b'+c' = 540 ... A++C = 540-(a'+6'+c') or ^+J5+C<540 2. A+B+C+a'+b'+c' = 50 360>a'+6'+c' Adding, (See p. 307.) A+B+C+a'+b'+c'+ZQ0>540 o +a'+b'+c' .'. A+B+C>1$0 340. Spherical excess. The amount by which the sum of the angles of a spherical triangle exceeds 180 is called the spherical excess. POLYEDRAL ANGLES 317 EXERCISES 1. Show that the spherical excess of a triangle is equal to A o +B+C o -180. 2. The angles of a spherical triangle are 100, 65, and 190. Find the spherical excess. Symmetry and Congruence 341. Congruent polyedral angles. Two polyedral angles are congruent if they can be made to coincide. It follows that the face angles and diedral angles of one of two congruent polyedral angles are equal respec- tively to those of the other. However, it does not follow that two polyedral angles are congruent if the correspond- ing face angles and diedral angles are equal. For example, triedral angles O-ABC, O'-A'B'C, and 0"-A"B"C", Fig. 221, have the corresponding face angles and diedral angles equal. Fig. 221 O-ABC and O'-A'B'C are congruent, but O-ABC and 0"-A"B"C" are not congruent. 342. Symmetrical polyedral angles. Two polyedral angles are symmetrical if the face angles and diedral angles of one are equal respectively to the face angles and diedral angles of the other, but arranged in opposite order. 318 THIRD-YEAR MATHEMATICS 343. Congruent and symmetrical spherical polygons. If the sides and angles of one spherical polygon are equal respectively to those of another, the polygons are con- gruent, provided the parts are arranged in the same order, and symmetrical, provided the parts are arranged in opposite order. Thus, in Fig. 222, ABC^ AA'B'C and both triangles are symmetrical to AA"B"C". 344. Theorem: If two triedral angles have the three face angles of one equal respectively to the three face angles of the other, the corresponding diedral angles are equal. Given the triedral angles O-ABC and O'-A'B'C, Fig. 223, having AOB=A'0'B', ZBOC= /.B'O'C \ ZCOA= /.CO' A'. To prove that diedral angle AO diedral angle A'O r , BO = B'0' 1 and CO=CO'. POLYEDRAL ANGLES 319 Proof: Lay off OA=OB = OC = 0'A' =0'B' = 0'C. Draw AB, BC, CA, A'B', B'C, and C'A'. Prove that AAOB^AA'O'B'; ABOCz AB'O'C; ACOA ^ AC'O'A'. Prove that AABC^ AA'B'C. Take OD = 0'D'. Draw DE and DF perpendicular to AO and in faces A OB and AOC respectively. Similarly, draw D'W and D'F'. Prove that AEDA E'D'A'; AFDA sg AF'D'A'. Prove that AEAF 2 AE'A'F'. Prove that AEDF sg AE'D'F'. .'. ZEDF=ZE'D'F'. .'. Diedral /.AO = diedral Z A '0', having equal plane angles. Similarly, diedral Z B0 = diedral AB'O'. diedral ZC0 = diedral Z CO'. EXERCISES Prove the following: 1. Two triedral angles are congruent if the face angles of one are equal respectively to the face angles of the other, arranged in the same order. 2. Two triedral angles are symmetrical if the face angles of one are equal respectively to the face angles of the other, arranged in the reverse order. 3. If two spherical triangles on the same sphere or on equal spheres have three sides of one equal respectively to three sides of the other 1. They are congruent if the equal parts are arranged in the same order. 2. They are symmetrical if the equal parts are arranged in the reverse order. Show that the triangles being mutually equilateral are also mutu- ally equiangular, and therefore either congruent or symmetrical. 320 THIRD-YEAR MATHEMATICS 4. Two spherical triangles on the same or equal spheres are congruent 1. // two sides and the included angle of one are equal respec- tively to two sides and the included angle of the other, arranged in the same order. 2. 7/ two angles and the included side of one are equal respec- tively to two angles and the included side of the other, arranged in the same order. Show that AABC, Fig. 224, can be maMe to coincide with AA'B'C. a C >0 +0' Fig. 224 5. Two spherical triangles on the same or equal spheres are symmetrical 1 . 7/ two sides and the included angle of one are equal respec- tively to two sides and the included angle of the other, arranged in the reverse order. 2. If two angles and the included side of one are equal respec- tively to two angles and the included side of the other, arranged in the reverse order. Let AABC and A*B'C, Fig. 225, be the given triangles. Draw AA"B"C" symmetrical to AA'B'C POLYEDRAL ANGLES 321 Then A A'B'C and A"B"C" are mutually equilateral and mutually equiangular. Prove that AABC& AA"B"C". :. AABC is symmetrical to AA'B'C. 6. State and prove theorems on triedral angles correspond- ing to the theorems in exercises 3, 4, and 5. 7. 2/ two spherical triangles on the same or equal spheres are mutually equiangular, they are mutually equilateral and congruent if the equal parts are arranged in the same order. If the equal parts are arranged in the reverse order, they are symmetrical. Fig. 226 Given the spherical triangles ABC and A'B'C, Fig. 226, hav- ing ZA = ZA', ZB=ZB', ZC=ZC. To prove that a = a', b = b', c = c', and that AABC and A'B'C are either congruent or symmetrical. Proof: Construct the polar triangles DEF and D'E'F'. Show that ADEF and D'E'F' are mutually equilateral, 338. Show that A DEF and D'E'F' are mutually equiangular, exercise 3 . Show that AABC and A'B'C are mutu- ally equilateral, 338. ^C .'. ABC and A'B'C are either congruent or symmetrical, 343. 345. Theorem: The angles opposite the equal sides of an isosceles spherical triangle are equal. v Draw the arc of a great circle, CD, b Fig. 227, bisecting the side AB. Fig. 227 ;*0 322 THIRD-YEAR MATHEMATICS Then A A DC and BDC are mutually equilateral and therefore symmetrical, .*. ' ZA=ZB EXERCISES 1. State and prove the theorem on triedral angles cor- responding to 345. 2. Two symmetrical isosceles triangles are congruent. Prove. 346. Theorem: If two angles of a spherical triangle are equal, the sides opposite are equal. Construct the polar triangle of AABC, Fig. 228. Since ZA= ZB, it follows that B'C' = A'C', 338. .*. ZB'=Z A', 345. .*. AC = BC, 338. Fig. 228 347. Theorem: If two angles of a spherical triangle are unequal, the sides opposite are unequal and the greater side lies opposite the greater angle. Draw AD, Fig. 229, an arc of a great circle making ZDAB=ZDBA. Then or AC^ A, B, and C being the number Fig. 233 of degrees in the angles of AABC and S the number of spherical degrees in AABC. Proof: Extend the sides of AABC. The angles of AABC are the angles of lunes C'ACBC, B'CBAB', and A'BACA' respectively. Lune CACBC = AABC + AABC = 2(C), exercise 3, 352. Lune B'CBAB f = AABC+ ACB'A = 2(B) Lune A'BACA' = AABC+ACA'B = 2(A) S(AABC) + AABC'+ACB'A + ACA'B = 2(A+B+C) or (2 AABC) + AABC + AABC + AC B' A + AC A' B = 2(A+B+C) Show that ACA'B = ACAB' .\2(AABC) + AABC+AABC+ACB'A + AC'AB' = 2(A+B+C) .-. 2(AAC)+hemisphere = 2(A+J5+C) .'. 2(AABC)+%(720)=2(A+B+C) .'. AABC = A+B+C-i(720) or S=A+B+C-1S0. 326 THIRD-YEAR MATHEMATICS 354. Let ABODE, Fig. 234, be a spherical polygon of n sides. Divide ABODE into n-2 spherical triangles by drawing arcs of great circles, as AC and AD. Denoting by T h T 2 , T h etc., the areas of ABAC, CAD, DAE, etc., and by s h s 2 , S3, etc., the sums of the angles in triangles BAC, CAD, DAE, etc., it follows that Ti = si-180, T 2 = s 2 -180, T 3 = s 3 -180, etc. Fig. 234 Adding, P = [si+S2+s 3 +etc.-(rc-2)180], or P = s-(n-2)180, where P denotes the number of spherical degrees in the polygon and s the number of degrees in the sum of the angles. This may be stated as a theorem as follows : The area of a spherical polygon is equal to its spherical excess. EXERCISES 1. The angles of a spherical triangle are 90, 90, and 79. Find the area in spherical degrees. 2. Find the area of a spherical degree on the spheres having radius equal to 3 in. ; 14 in. ; a inches. 3. Find the area of a spherical polygon whose angles are 70, 105, 145, 125, 150. 4. Find the area of a spherical triangle whose angles are 85, 120, and 95 on a sphere whose radius is 6 inches. POLYEDRAL ANGLES 327 5. The area of a spherical triangle is 100 sq. in. and its angles are 100, 64, 200. What is the radius of the sphere on which the triangle lies ? (Board.) 6. Prove that the area of a spherical triangle is proportional to its spherical excess. Three complete great circles drawn on a sphere whose radius is 10 in. divide the surface of the sphere into eight spherical triangles; the angles of one of these triangles are 40, 80, and 120. Find the area in square inches of each of the eight triangles. (Harvard.) 7. On a sphere of radius 2 ft. the area of a certain triangle is 2 square yards. What is the perimeter of the polar triangle ? (Harvard.) Summary 355. The chapter has taught the meaning of the following terms : polyedral angle, triedral angle polar spherical triangles spherical polygon spherical excess spherical triangle congruent and symmetrical similar polyedrons figures spherical angle lune right, birectangular, and tri- spherical degree rectangular spherical triangles 356. The following theorems have been studied: 1. The sum of two face angles of a triedral angle is greater than the third. 2. The sum of two sides of a spherical triangle is greater than the third side. 3. The shortest line that can be drawn between two given points on the surface of a sphere is the minor arc of the great circle which passes through the two points. 328 THIRD-YEAR MATHEMATICS 4. The sum of the face angles of a convex polyedral angle is less than four right angles. 5. The sum of the sides of a convex spherical polygon is less thqn 360. 6. There cannot be more than five kinds of regular polyedrons. 7. Two tetraedrons having a triedral angle of one equal to a triedral angle of the other are to each other as the products of the edges including the equal triedral angles. 8. Two similar polyedrons are to each other as the cubes of two corresponding edges. 9. A spherical angle is measured by the arc of a great circle having the vertex as a pole^and included between the sides produced if necessary. 10. A spherical angle is equal to the diedral angle formed by the planes of the sides. 11. If a spherical triangle is the polar triangle of another, then the second is the polar triangle of the first. 12. In two polar spherical triangles each angle of the one is the supplement of that side of the other of which it is the pole. 13. The sum of the angles of a spherical triangle is less than six and greater than two right angles. 14. If two triedral angles have the corresponding face angles equal, the corresponding diedral angles are equal. 15. (1) If the face angles of one triedral angle are equal respectively to the face angles of another; POLYEDRAL ANGLES 329 (2) If two face angles and the included diedral angle are equal respectively to the corresponding parts of the other; (3) If two diedral angles and the included face angle are equal respectively to the corresponding parts of the other; (4) If the diedral angles of one are equal respectively to the corresponding parts of the other; The triedral angles are congruent if the parts are arranged in the same order, and symmetrical if they are arranged in the reverse order. v 16. Two spherical triangles are congruent, or symmetri- cal, if they have the following corresponding parts equal: (1) Three sides; (2) three angles; (3) two sides and the included angle; (4) two angles and the included side. 17. If two spherical triangles on the same or equal spheres are mutually equiangular, they are mutually equilateral, and conversely. 18. The base angles of an isosceles spherical triangle are equal, and conversely. 19. If two angles of a spherical triangle are unequal, the sides opposite are unequal, and conversely. 20. The diameters of a sphere drawn through the vertices of a spherical triangle meet the surface of the sphere in points which are the vertices of a triangle symmetrical to the given triangle. 21. Two symmetrical spherical triangles are equal. 357. The following constructions were taught: 1. To construct a sphere passing through four given points not all in the same plane. 2. To inscribe a sphere in a given tetraedron. 3. To determine the diameter of a given material sphere. 330 THIRD-YEAR MATHEMATICS 358. The following formulas have been proved: 1. The area of a lune, L = 2(A) spherical degrees, where A is the number of degrees in the angle of the lune. 2. The area of a spherical triangle, T = (A+B+C -180) spherical degrees, where A, B, and C are the number of degrees in the angles of the triangle. This formula may be stated, T=E, the spherical excess. 3. The area of a spherical polygon, P = (s (n 2) 180) spherical degrees, where s is the number of degrees in the sum of the angles of the polygon. This formula may be stated, P=E, the spherical excess. CHAPTER XV SUMMARY OF THE ASSUMPTIONS AND THEOREMS OF GEOMETRY GIVEN IN THE COURSES OF THE FIRST AND SECOND YEARS 359. For the convenience of the student a complete list of the assumptions and theorems studied in the first two courses is given below. References in the foregoing chapters are made to this list to save the student the time of looking them up in other textbooks. The numbers in the parentheses ( ) refer to the sections in First-Year Mathematics, those in brackets [ ] to the sections in Second- Year Mathematics in which these statements were given for the first time. Preliminary Assumptions 360. Through two points one and only one straight line can be drawn. (20) 361. A straight line two of whose points lie in a plane lies entirely in the plane. (204) 362. The shortest distance between two points is the straight line-segment joining the points. (21) 363. Two straight lines intersect in one and only one point. (25) 364. A line-segment, or an angle, is equal to the sum of all its parts. (33) %h. [482] 496. The area of a triangle is equal to one-half the product of the base and altitude, A = \b-h. [465] 497. The area of a triangle is equal to one-half the j product of two sides by the sine of the included angle, A = \ab sin C. [466] 498. The area of a triangle is equal to one-half the perimeter times the radius of the inscribed circle, A = lp'r=sr. [467] 499. The area of a triangle is equal to the product of the three sides divided by four times the radius of the circumscribed circle, A = <. [468] 500. The area of a triangle is equal to A = Vs(s-a)(s-b){s-c). [469] 501. The area of an equilateral triangle is one-fourth the square of a side times the square root of 3, 4 = |V3. [471] 502. The area of a trapezoid is equal to one-half the product of the altitude by the sum of the bases, T = \h(bi+b$. [483] 346 THIRD-YEAR MATHEMATICS 503. The area of a regular inscribed polygon is equal to the product of one-half the perimeter and the apothem. [484] 504. The area of a regular circumscribed polygon is equal to the product of one-half the perimeter and the radius. [485] *-/y fc R'73fi 505. The area of a circle is one-half the product of the length of the circle and the radius, i.e., A = \cr. [489] 506. The area of a circle is given by the formula A = -nr\ [489] 507. The area of a sector is given by the formula A = \a'r. [490] 508. The area of a segment of a circle is given by the formulas H-'-U-t or A = l<*'r-\r 2 sin x. [491] Proportionality of Areas 509. In a proportion the product of the means is equal to the product of the extremes. (259) 510. The areas of two rectangles are in the same ratio as the products of their dimensions. (260) 511. Two rectangles having equal bases are in the same ratio as the altitudes. (261) 512. Two rectangles having equal altitudes are in the same ratio as the bases. (262) ASSUMPTIONS AND THEOREMS OF GEOMETRY 347 513. The areas of parallelograms are in the same ratio as the products of the bases and altitudes. (263) i 514. The areas of triangles are in the same ratio as the products of the bases and altitudes. (264) 515. The areas of parallelograms having equal bases are in the same ratio as the altitudes. (265) 516. The areas of triangles having equal bases are in the same ratio as the altitudes. (266) 517. The areas of two similar triangles are to each other as the squares of any two corresponding sides. [497] 518. The areas' of two similar polygons are to each other as the squares of two corresponding sides. [498] Lines and Planes in Space 519. The following conditions determine the position of a plane in space : 1. A straight line and a point not in that line. 2. Three points not in the same straight line. 3. Two intersecting straight lines. 4. Two parallel straight lines. [139] 520. If two planes intersect, the intersection is a straight line. [143] 521. Two planes perpendicular to the same line are parallel. [178] 522. If two parallel planes are cut by a third plane, the intersections are parallel. [179] 523. Parallel line-segments intercepted by parallel planes are equal. [180] 348 THIRD-YEAR MATHEMATICS 524. If three or more parallel planes are cut by two transversals, the corresponding segments of the trans- versals are in proportion. [181] 525. The projection upon a plane, of a straight line not perpendicular to the plane, is a straight line. [355] 526. The projection upon a plane, of a straight line perpendicular to the plane, is a point. [356] 527. The acute angle formed by a given line and its projection upon a plane is smaller than the angle which it makes with any other line in the plane passing through the point of intersection of the given line and the plane. [357] 528. The perpendicular is the shortest distance from a point to a plane. [342] 529. Oblique lines drawn from a point to a plane, meeting the plane at points equidistant from the foot of the perpendicular, are equal. [344] 530. Oblique lines drawn from a point to a plane, meeting the plane at points unequally distant from the foot of the perpendicular, are unequal, the more remote being the greater. [345] 531. Equal oblique lines drawn from a point to a plane meet the plane at points equidistant from the foot of the perpendicular. [346] 532. Of two unequal oblique lines drawn from a point to a plane the greater meets the plane at the greater distance from the foot of the perpendicular. [347] 533. If a line is perpendicular to each of two inter- secting lines, it is perpendicular to the plane determined by these lines. [364] ASSUMPTIONS AND THEOREMS OF GEOMETRY 349 534. All the perpendiculars to a given line at a given point lie in a plane perpendicular to the given line at the point. [366] 535. Only one plane can be constructed perpendicular to a given line at a given point. [367] 536. Only one plane can be constructed perpendicular to a given line from a point outside of the line. [368] 537. Only one line can be constructed perpendicular to a given plane at a given point. [370] 538. From a point outside of a given plane only one line can be constructed perpendicular to the plane. [372] 539. Lines perpendicular to a plane are parallel. [373] 540. If one of two parallel lines is perpendicular to a plane, the other is perpendicular to the same plane. [374] 541. Two lines parallel to the same line are parallel to each other. [375] 542. If two lines are parallel, a plane containing one of them and not the other is parallel to the other. [376] 543. If one of two parallel planes is perpendicular to a line, the other is also. [377] 544. If two intersecting lines are parallel to a given plane, their plane is parallel to the given plane. [378] 545. If two angles not in the same plane have their sides parallel and running in the same direction, the angles are equal and their planes are parallel. [379] 546. All plane angles of a diedral angle are equal. [380] 547. If two diedral angles are equal, their plane angles are equal. [381] 350 THIRD-YEAR MATHEMATICS 548. Two diedral angles are equal if the plane angles are equal. [381] 549. If a line is perpendicular to a plane, every plane through this line is perpendicular to the plane. [382] 550. If two planes are perpendicular to each other, a line drawn in one of them perpendicular to the intersection is perpendicular to the other. [383] 551. If two planes are perpendicular to each other, a line perpendicular to one of them at a point of the inter- section must lie in the other. [383] 552. If from a point in one of two perpendicular planes a line is drawn perpendicular to the other, it must lie in the first plane. [383] 553. If a plane is perpendicular to each of two planes, it is perpendicular to their intersection. [384] 554. Through a line not perpendicular to a given plane one plane and only one may be passed perpendicular to the given plane. [385] 555. The section of a sphere made by a plane is a circle. [389] 556. The axis of a circle passes through the center. [390] 557. The diameter of a sphere passing through the center of a circle is perpendicular to the plane of the circle. [390] 558. All great circles of a sphere are equal. [390] 559. Every great circle bisects the surface of the sphere. [390] 560. Through two points on the surface of a sphere, not the endpoints of a diameter, only one great circle can be drawn. [390] ASSUMPTIONS AND THEOREMS OF GEOMETRY 351 561. All points on a circle of a sphere are equidistant from its poles. [392] 562. The polar distance of a great circle is a quadrant. [395] 563. If a point on the surface of a sphere is at the distance of a quadrant from each of two given points on the surface, it is a pole of the great circle passing through the given points. [396] 564. The intersection of the surfaces of two spheres is a circle whose plane is perpendicular to the line of centers of the spheres, and whose center is in that line. [397] 565. A plane tangent to a sphere is perpendicular to the radius at the point of contact. [399] 566. A plane perpendicular to a radius of a sphere at the outer extremity is tangent to the sphere. [400] Constructions 567. Through a given point in a given line pass a plane perpendicular to the given line. [365] 568. From a given point outside of a given line con- struct a plane perpendicular to the given line. [368] 569. At a given point in a given plane construct a per- pendicular to the plane. [369] 570. From a point outside of a plane construct a line perpendicular to the plane. [371] 571. To pass a plane perpendicular to a given plane, that shall contain a line not perpendicular to the given plane. [385] LOGARITHMIC AND TRIGONOMETRIC TABLES AND MATHEMATICAL FORMULAS # CONTENTS TABLE PAGE I. Common Logakithms of Numbers ... 1 II. Common Logarithms of the Trigonometric Functions 21 III. Values of the Natural Trigonometric Functions 73 IV. Tables of Powers and Roots .... 99 V. Formulas 108 VI. Equivalents and Logarithms of Important Constants ' . Ill VII. Reductions 115 TABLE I Common Logarithms of Numbers from 1 to 10000 to Five Decimal Places I] 1000 Common Logarithms of Numbers 1500 3 N 1 2 3 4 5 6 7 8 9 .PP 100 00 000 043 087 18 *30 ; 17-& 217 260 303 346 389 101 432 475* 604 647 689 732 775 817 102 860 903 945 988 *030 *072 *115 *157 *199 *242 44 43 4*2 103 01 284 326 368 4m 828 *453 494 536 578 620 662 104" 703 745 787 870 912 953 995 *036 *078 1 a 3 4.4 4.3 4X 8.8 8.6 8.4 13.2 12.9 12.6 105 02 119 160 202 243 284 325 366 407 449 490 10Q 531 572 612 653 694 735 776 816 857 898 4 17.6 17.2 16.8 107 938 979 *019 *060 *100 *141 *181 *222 *262 *302 5 22.0 21.5 21.0 108 03 342 383 423 463 503 543 583 623 663 703 6 26.4 25.8 25.2 109 743 782 822 862 902 941 981 *021 *060 *100 7 8 30.8 30.1 29.4 35.2 34,4 33.6 39.6 3S;7 3%.*8 110 04 139 179 218 258 297 336 376 415 454 493 111 532 571 610 650 689 727 766 805 844 883 112 922 961 999 *038 *077 *115 *154 *192 *231 *269 41 40 39 113 05 308 346 385i 423 461 500 538 576 614 652 1 4 1 4 3 9 114 690 729 767 805 843 881 918 956 994 *032 2 8.2 8.0 7.8 115 06 070 108 145 183 221 258 296 333 371 408 3 4 12.3 12.0 11.7 16.4 16.0 15.6 116 446 483 521 558 595 633 670 707 744 781 5 20.5 20.0 19.5 117 819 856 893 930 967 *004 *041 *078 *115 *151 6 24.6 24.0 23.4 118 07 188 225 262 298 335 372 408 445 482 518 7 28 7 28 27 3 119 555 591 628 664 700 737 773 809 846 882 8 32.8 32.0 31.2 120 918 954 990 *027 *063 *099 *135 *171 *207 *243 Q 36.9 36.0 35.1 121 OS 279 314 350 386 422 458 m 529 565 600 38 37 36 122 636 ^672 707 743 778 814 884 920 955 123 991 *026 *061 *096 *132 *167 *202 *237 *272 *307 1 6. 8 3.7 3.6 124 09 342 377 412 447 482 517 552 587 621 656 2 3 7.6 7.4 7.2 11 4 11 1 10 8 125 691 726 760 795 830 864 899 934 968 *003 4 15.2 14.8 14.4 126 10 037 072 106 140 175 209 243 278 312 346 5 19.0 18.5 18.0 127 380 415 449 483 517 551 585 619 958 653 687 6 22.8 22.2 21.6 128 721 755 789 823 857 890 924 992 *025 7 26.6 25.9 25.2 129 11 059 093 126 160 193 227 261 294 .327 361 8 9 30.4 29.6 28.8 34.2 33.3 32.4 130 394 428 461 494 528 561 594 628 661 694 131 727 760 793 826 860 893 926 959 992 *024 35 34 33 132 12 057 090 123 156 189 222 254 287 320 352 1 3.5 3.4 3.3 133 385 418 450 483 516 548 581 613 646 678 2 7.0 6.8 6.6 134 710 743 775 808 840 872 905 937 969 *001 3 4 10.5 10.2 9.9 14.0 13.6 13.2 135 13 033 066 098 130 162 194 226 258 290 609 322 5 17.5 17.0 16.5 136 354 386 418 450 481 513 545 577 640 6 21.0 20.4 19.8 137 672 1 704 735 767 799 830 862 893 925 956 7 24.5 23.8 23.1 138 988 *019 *051 *082 *114 *145 *176 *208 *239 *270 8 28.0 27.2 26.4 139 14 301 333 364 395 426 457 489 520 551 582 9 31.5 30.6 29.7 140 613 644 675 706 737 768 799 829 860 891 32 31 30 141 922 953 983 *014 *045 *076 *106 *137 *168 *198 1 3.2 3.1 3.0 6.4 6.2 6.0 9.6 9.3 9.0 12.8 12.4 12.0 142 15 229 259 290 320 351 381 412 442 473 503 2 3 4 143 534 564 594 625 655 685 715 746 776 806 144 836 866 897 927 957 987 *017 *047 *077 *107 145 16 137 167 197 227 256 286 316 346 376 406 5 6 7 8 9 16.0 15.5 15.0 19.2 18.6 18.0 22.4 21.7 21.0 25.6 24.8 24.0 28.8 27.9 27.0 146 435 465 495 524 554 584 613 643 673 702 147 732 761 791 820 850 879 909 938 967 997 148 17 026 056 085 114 143 173 202 231 260 289 49 319 348 377 406 435 464 493 522 551 580 150 609 638 667 696 725 754 782 811 840 869 N 1 2 3 4 5 6 7 8 9 PP 4 1500 Common Logarithms of Numbers 2000 [I N 1 2 3 4 5 6 7 8 9 PP 150 17 609 638 667 696 725 754 782 811 840 869 15V 898 926 955 984 *013 *041 *070*099 *127 *156 29 28 152 18 184 213 241 270 298 327 355 384 412 441 153 469 498 526 554 583 611 639 667 696 724 1 2.9 2.8 154 752 780 808 837 865 893 921 949 977 *005 2 3 5.8 5.6 8.7 8.4 155 19 033 061 089 117 145 173 201 229 n 257 285 4 11.6 11.2 156 312 340 368 396 424 451 479 507 535 562 5 14.5 14.0 157 590 618 645 673 700 728 756 783 811 838 6 17.4 16.8 158 866 893 921 948 976 *003 *030 *058 *085 *112 7 20.3 19.6 159 20 140 167 194 222 249 276 303 330 358 385 8 9 23.2 22.4 26.1 25.2 160 412 439 466 493 520 548 575 602 629 656 161 683 710 737 763 790 817 844 871 898 925 27 26 162 952 978 *005 *032 *059 *085 *112 *139 *165 *192 163 21 219 245 272 299 325 352 378 405 431 458 1 2.7 2.6 164 484 511 537 564 590 617 643 669 696 722 2 a 5.4 5.2 8.1 7.8 165 748 775 801 827 854 880 906 932 958 985 4 10.8 10.4 166 22 Oil 037 063 089 115 141 167 194 220 246 5 13.5 13.0 167 272 298 324 350 376 401 427 453 479 505 6 16.2 15.6 168 531 557 583 608 634 660 686 712 737 763 7 18.9 18.2 169 789 814 840 866 891 917 943 968 994 *019 8 9 21.6 20.8 24.3 23.4 170 23 045 070 096 121 147 172 198 223 249 274 171 300 325 350 376 401 426 452 477 502 528 25 172 553 578 603 629 654 679 704 729 754 779 173 805 830 855 880 905 930 955 980 *005 *030 1 2.5 174 24 055 080 105 130 155 180 204 229 254 279 2 3 4 5.0 7.5 10.0 175 304 329 353 378 403 428 452 477 502 527 176 551 576 601 625 650 ' 674 699 724 748 773 5 12.5 177 797 822 846 871 895 920 944 969 993 *018 6 15.0 178 25 042 066 091 115 139 164 188 212 237 261 7 17.5 179 285 310 334 358 382 406 431 455 479 503 8 9 20.0 22.5 ISO 527 551 575 600 624 648 672 696 720 744 181 768 792 816 840 864 888 912 935 959 983 24 23 182 26 007 031 055 079 102 126 150 174 198 221 183 245 269 293 316 340 364 387 411 435 458 1 2.4 2.3 184 482 505 529 553 576 600 623 647 670 694 2 3 4.8 4.6 7.2 6.9 185 717 741 764 788 811 834 858 881 905 928 4 9.6 9.2 185 951 975 998 *021 *045 *068 *091 *114 *138 *161 5 12.0 11.5 187 27 184 207 231 254 277 300 323 346 370 393 G 14.4 13.8 188 416 439 462 485 508 531 554 577 600 623 7 16.8 16.1 189 646 669 692 715 738 761 784 807 830 852 8 9 19.2 18.4 21.6 20.7 190 875 898 921 944 967 989 *012 *035 *058 *081 191 28 103 126 149 171 194 217 240 262 285 307 22 21 192 330 353 375 398 421 443 466 488 511 533 193 556 578 601 623 646 668! 691 713 735 758 1 2.2 2.1 194 780 803 825 847 870 892 914 937 959 981 2 a 4.4 4.2 6.6 6.3 195 29 003 026 048 070 092 115 137 159 181 203 4 8.8 8.4 196 226 248 270 292 314 336 358 380 403 425 6 11.0 10.5 197 447 469 491 513 535 557 579 601 623 645 6 13.2 12.6 198 667 688 710 732 754 776 798 820 842 863 7 15.4 14.7 199 885 907 929 951 973 994 *016 *038 *060 *081 8 9 17.6 <3.b 19.8 18.9 200 30 103 125 146 168 190 211 233 255 276 298 N 1 2 3 4 5 6 7 8 9 PP I] 2000 Common Logarithms of Numbers 2500 ; N 1 2 3 4 5 6 7 8 9 PP 200 30 103 125 146 168 190 211 233 255 276 298 201 320 341 363 384 406 428 449 471 492 514 22 21 202 535 557 578 600 621 643 664 685 707 728 203 750 771 792 814 835 856 878 899 920 942 1 2.2 2.1 204 963 984 *006 *027 *048 *069 *091 *112 *133 *154 2 3 4.4 4.2 6.6 6.3 205 31 175 197 218 239 260 281 302 323 345 366 4 8.8 8.4 206 387 408 429 450 471 492 513 534 555 576 5 11.0 10.5 207 597 618 639 660 681 702 723 744 765 785 6 13.2 12.6 208 806 827 848 869 890 911 931 952 973 994 7 15.4 14.7 209 32 015 035 056 077 098 118 139 160 181 201 8 9 17.6 16.8 19.8 18.9 210 222 243 263 284 305 325 346 366 387 408 211 428 449 469 490 510 531 552 572 593 613 20 212 634 654 675 695 715 736 756 777 797| 818 213 838 858 879 899 919 940 960 980 *001 *021 1 2.0 214 33 041 062 082 102 122 143 163 183 203 224 2 3 4- 4.0 6.0 8.0 215 244 264 284 304 325 345 365 385 405 425 216 445 465 486 506 526 546 566 586 606 626 5 10.0 217 646 666 686 706 726 746 766 786 806 826 6 12.0 218 846 866 885 905 925 945 965 985 *005 *025 7 14.0 219 34 044 064 084 104 124 143 163 183 203 223 8 9 16.0 18.0 220 242 262 282 301 321 341 361 380 400 420 221 439 459 479 498 518 537 557 577 596 616 19 222 635 655 674 694 713 733 753 772 792 811 223 830 850 869 889 908 928 947 967 986 *005 1 1.9 224 35 025 044 064 083 102 122 141 160 180 199 2 3 4 3.8 5.7 7.6 225 218 238 257 276 295 315 334 353 372 392 226 411 430 449 468 488 507 526 545 564 583 5 9.5 227 603 622 641 660 679 698 717 736 755 774 6 11.4 228 793 813 832 851 870 889 908 927 946 965 7 13.3 229 984 *003 *021 *040 *059 *078 *097 *116 *135 *154 8 9 15.2 17.1 230 36 173 192 211 229 248 267 286 305 324 342 231 361 380 399 418 436 455 474 493 511 530 18 232 549 568 586 605 624 642 661 680 698 717 233 736 754 773 791 810 829 847 866 884 903 1 1.8 234 922 940 959 977 996 *014 *033 *051 *070 *088 2 3 4 3.6 5.4 7.2 235 37 107 125 144 162 181 199 218 236 254 273 236 291 310 328 346 365 383 401 420 438 457 5 9.0 237 475 493 511 530 548 566 585 603 621 639 6 10.8 238 658 676 694 712 731 749 767 785 803 822 7 12.6 239 840 858 876 894 912 931 949 967 985 *003 8 9 14.4 16.2 240 38 021 039 057 075 093 112 130 148 166 184 241 202 220 238 256 274 292 31Q 328 346 364 17 242 382 399 417 435 453 471 489 507 525 543 243 56, 1 578 596 614 632 65O 668 686 703 721 1 1.7 244 739 757 775 792 810 828 846 863 881 899 2 3 4 3.4 5.1 6.8 245 917 934 952 970 987 *005 *023 *041 *058 *076 246 39 094 111 129 146 164 182 199 217 235 252 5 8.5 247 270 287 305 322 340 358 375 393 410 428 6 10.2 248 445 463 480 498 515 533 550 568 585 602 7 11.9 249 620 637 655 672 690 707 724 742 759 777 8 9 13.6 15.3 250 794 811 829 846 863 881 898 915 933 950 N 1 2 3 4 5 6 7 8 9 PP 6 2500 Common Logarithms of Numbers 3000 [I N 1 2 3 4 5 6 7 8 9 PP 250 39 794 811 829 846 863 881 898 915 933 950 251 967 985 *002 *019 *037 *054 *071 *088 *106 *123 18 252 40 140 157 175 192 209 226 243 261 278 295 253 312 329 346 364 381 398 415 432 449 466 1 1.8 254 483 500 518 535 552 569 586 603 620 637 2 3 4 3.6 5.4 7.2 255 654 671 688 705 722 739 756 773 790 807 256 824 841 858 875 892 909 926 943 960 976 5 9.0 257 993 *010 *027 *044 *061 *078 *095 *111 *128 *145 6 10.8 258 41 162 179 196 212 229 246 263 280 296 313 7 12.6 259 330 347 363 380 397 414 430 447 464 481 8 9 14.4 16.2 26 497 514 531 547 564 581 597 614 631 647 261 664 681 697 714 731 747 764 780 797 814 17 262 830 847 863 880 896 913 929 946 963 979 263 996 *012 *029 *045 *062 *078 *095 *111 *127 *144 1 1.7 264 42 160 177 193 210 226 243 259 275 292 308 2 3 4 3.4 5.1 6.8 265 325 341 357 374 390 406 423 439 455 472 266 488 504 521 537 553 570 586 602 619 635 5 8.5 267 651 667 684 700 716 732 749 765 781 797. 6 10.2 268 813 830 846 862 878 894 911 927 943 959 7 11.9 269 975 991 *008 *024 *040 *056 *072 *088 *104 *120 8 9 13.6 15.3 270 43 136 152 169 185 201 217 233 249 265 281 271 297 313 329 345 361 377 393 409 425 441 16 272 457 473 489 505 521 537 553 569 584 600 273 616 632 648 664 680 696 712 727 743 759 1 1.6 274 775 791 807 823 838 854 870 886 902 917 2 3 4 3.2 4.8 6.4 275 933 949 965 981 996 *012 *028 *044 *059 *075 276 44 091 107 122 138 154 170 185 201 217 232 5 8.0 277 248 264 279 295 311 326 342 358 373 389 6 9.6 278 404 420 436 451 467 483 498 514 529 545 7 11.2 279 560 576 592 607 623 638 654 669 685 700 8 9 12.8 14.4 2SO 716 731 747 762 778 793 809 824 840 855 281 871 886 902 917 932 948 963 979 994 *010 15 282 45 025 040 056 071 086 102 117 133 148 163 283 179 194 209 225 240 255 271 286 301 317 1 1.5 284 332 347 362 378 393 408 423 439 454 469 2 3 4 3.0 4.5 6.0 285 484 500 515 530 545 561 576 591 606 621 288 637 652 667 682 697 712 728 743 758 773 5 7.5 287 788 803 818 834 849 864 879 894 909 924 6 9.0 288 939 954 969 984 *000 *015 *030 *045 *060 *075 7 10.5 289 46 090 105 120 135 150 165 180 195 210 225 8 9 12.0 13.5 290 240 255 270 285 300 315 330 345 359 374 291 389 404 419 434 449 464 479 494 509 523 14 292 538 553 568 583 598 613 627 642 657 672 293 687 702 716 731 746 761 776 790 805 820 1 1.4 294 835 850 864 879 894 909 923 938 953 967 2 3 4 2.8 4.2 5.6 295 982 997 *012 *026 *041 *056 *070 *085 *100 *114 296 47 129 144 159 173 188 202 217 232 246 261 5 7.0 297 276 290 305 319 334 349 363 378 392 407 6 8.4 298 422 436 451 465 480 494 509 524 538 553 7 9.8 299 567 582 596 611 625 640 654 669 683 698 8 9 11.2 12.6 300 712 727 741 756 770 784 799 813 828 842 N 1 2 3 4 5 6 7 8 9 PP I] 3000 Common Logarithm s of Numbers 3500 7 N 1 2 3 4 5 6 7 8 9 PP 300 47 712 727 741 756 770 784 799 813 828 842 301 857 871 885 900 914 929 943 958 972 986 302 48 001 015 029 044 058 073 087 101 116 130 303 144 159 173 187 202 216 230 244 259 273 15 304 287 302 316 330 344 359 373 387 401 416 1 2 1.5 3.0 305 430 444 458 473 487 501 515 530 544 558 306 572 586 601 615 629 643 657 671 686 700 3 4.5 307 714 728 742 756 770 785 799 813 827 841 4 6.0 308 855 869 883 897 911 926 940 954 968 982 5 7.5 309 996 *010 *024 *038 *052 *066 *080 *094 *108 *122 6 7 8 9.0 10.5 12.0 310 49 136 150 164 178 192 206 220 234 248 262 311 276 290 304 318 332 346 360 374 388 402 9 13.5 312 415 429 443 457 471 485 499 513 527 541 313 554 568 582 596 610 624 638 651 665 679 314 693 707 721 734 748 762 776 790 803 817 315 831 845 859 872 886 900 914 927 941 955 14 316 969 982 996 *010 024 *037 *051 *065 *079 *092 1 1.4 317 50 106 120 133 147 161 174 188 202 215 229 2 2.8 318 243 256 270 284 297 311 325 338 352 365 3 4!2 319 379 393 406 420 433 447 461 474 488 501 4 5!6 320 515 529 542 556 569 583 596 610 623 637 5 6 7.0 8.4 321 651 664 678 691 705 718 732 745 759 772 7 9^8 322 786 799 813 826 840 853 866 880 893 907 8 11.2 323 920 934 947 961 974 987 *001 *014 *028 *041 9 12^6 324 51 055 068 081 095 108 121 135 148 162 175 325 188 202 215 228 242 255 268 282 295 308 326 322 335 348 362 375 388 402 415 428 441 327 455 468 481 495 508 521 534 548 561 574 13 328 587 601 614 627 640 654 667 680 693 706 329 720 733 746 759 772 786 799 812 825 838 2 1.3 2.6 330 851 865 878 891 904 917 930 943 957 970 3 3.9 331 983 996 *009 *022 *035 *048 *061 *075 *088 *101 4 5.2 332 52 114 127 140 153 166 179 192 205 218 231 5 6.5 333 244 257 270 284 297 310 323 336 349 362 6 7.8 334 375 388 401 414 427 440 453 466 479 492 7 8 9.1 10.4 335 504 517 530 543 556 569 582 595 608 621 9 11.7 336 634 647 660 673 686 699 711 724 737 750 337 763 776 789 802 815 827 840 853 866 879 338 892 905 917 930 943 956 969 982 994 *007 339 53 020 033 046 058 071 084 097 110 122 135 12 340 148 161 173 186 199 212 224 237 250 263 341 275 288 301 314 326 339 352 364 377 390 1 1.2 342 403 415 428 441 453 466 479 491 504 517 2 2.4 343 529 542 555 567 580 593 605 618 631 643 3 3.6 344 656 668 681 694 706 719 732 744 757 769 4 5 6 4.8 6.0 7.2 345 782 794 807 820 832 845 857 870 882 895 346 908 920 933 945 958 970 983 995 008 020 7 8.4 347 54 033 045 058 070 083 095 108 120 133 145 8 9.6 348 158 170 183 195 208 220 233 245 258 270 9 10.8 349 283 293 307 320 332 345 357 370 382 394 350 407 419 432 444 456 469 481 494 506 518 N 1 2 3 4 5 6 7 8 9 PP 8 3500 Common Logarithms of Numbers 4000 [I N 1 2 3 4 5 6 7 8 9 PP 350 54 407 419 432 444 456 469 481 494 506 518 351 531 543 555 568 580 593 605 617 630 642 352 654 667 679 691 704 716 728 741 753 765 353 777 790 802 814 827 839 851 864 876 888 13 354 900 913 925 937 949 962 974 986 998 *011 1 1.3 355, 356 1 55 023 035 047 060 072 084 096 108 121 133 2 2.6 145 157 169 182 194 206 218 230 242 255 3 3.9 357 267 279 291 303 315. 328 340 352 364 376 4 5.2 358 388 400 413 425 437 449 461 473 485 497 5 6.5 359 509 522 534 546 558 570 582 594 606 618 6 7 7.8 9 1 360 630 642 654 666 678 691 703 715 727 739 8 10.4 361 751 763 775 787 799 811 823 835 847 859 9 11.7 362 871 883 895 907 919 931 943 955 967 979 363 991 *003 *0l5 027 *038 *050 *062 *074 *086 *098 364 56 110 122 134 146 158 170 182 194 205 217 365 229 241 253 265 277 289 301 312 324 336 12 366 348 360 372 384 396 407 419 431 443 455 367 467 478 490 502 514 526 538 549 561 573 1 1.2 368 585 597 608 620 632 644 656 667 679 691 2 2.4 369 703 714 726 738 750 761 773 785 797 808 3 4 3.6 4.8 370 820 832 844 855 867 879 891 902 914 926 5 6.0 371 937 949 961 972 984 996 *008 *019 *031 *043 6 7.2 372 57 054 066 078 089 101 113 124 136 148 159 7 8.4 373 171 183 194 206 217 229 241 252 264 276 8 9.6 374 287 299 310 322 334 345 357 368 380 392 9 10.8 375 403 415 426 438 449 461 473 484 496 507 376 519 530 542 553 565 576 588 600 611 623 377 634 646 657 669 680 692 703 715 726 738 378 749 761 772 784 795 807 818 830 841 852 11 379 864 875 887 898 910 921 933 944 955 967 1 1.1 880 978 990 *001 *013 *024 *035 *047 *058 *070 *081 2 2.2 381 58 092 104 115 127 138 149 161 172 184 195 3 3.3 382 206 218 229 240 252 263 274 286 297 309 4 4.4 383 320 331 343 354 365 377 388 399 410 422 5 5.5 384 433 444 456 467 478 490 501 512 524 535 6 7 6.6 7.7 385 546 557 569 580 591 602 614 625 636 647 8 8.8 386 659 670 681 692 704 715 726 737 749 760 9 9.9 387 771 782 794 805 816 827 838 850 861 872 388 883 894 906 917 928 939 950 961 973 984 389 995 *006 *017 *028 *040 *051 *062 *073 084 095 390 59 106 118 129 140 151 162 173 184 195 207 10 391 218 229 240 251 262 273 284 295 306 318. 392 329 340 351 362 373 384 395 406 417 428 1 l.U 393 439 450 461 472 483 494 506 517 528 539 2 2.0 394 550 561 572 583 594 605 616 627 638 649 3 4 3.0 4.0 395 660 671 682 693 704 715 726 737 748 759 5 5.0 396 770 780 791 802 813 824 835 846 857 868 6 6.0 397 879 890 901 912 923 934 945 956 966 977 7 7.0 398 988 999 010 021 032 043 *054 *065 *076 *086 8 8.0 399 60 097 108 119 130 141 152 163 173 184 195 9 9.0 400 206 217 228 239 249 260 271 282 293 304 N 1 2 3 4 5 6 7 8 9 PP I] 4000 Common Logarithm s of Numbers 4500 9 N 1 2 3 4 5 6 7 8 9 PP 400 60 206 217 228 239 249 260 271 282 293 304 401 314 325 336 347 358 369 379 390 401 412 402 423 433 444 455 466 477 487 498 509 520 403 531 541 552 563 574 584 595 606 617 627 404 638 649 660 670 681 692 703 713 724 735 405 746 756 767 778 788 799 810 821 831 842 406 853 863 874 885 895 906 917 927 938 949 n 407 959 970 981 991 *002 *013 *023 *034 *045 *055 408 61 066 077 087 098 109 119 130 140 151 162 i 1.1 409 172 183 194 204 215 225 236 247 257 268 2 3 4 2.2 3.3 4.4 410 278 289 300 310 321 331 342 352 363 374 411 384 395 405 416 426 437 448 458 469 479 5 5.5 412 490 500 511 521 532 542 553 563 574 584 6 6.6 413 595 606 616 627 637 648 658 669 679 690 7 7.7 414 700 711 721 731 742 752 763 773 784 794 8 9 8.8 9.9 415 805 815 826 836 847 857 868 878 888 899 416 909 920 930 941 951 962 972 982 993 *003 417 62 014 024 034 045 055 066 076 086 097 107 418 118 128 138 149 159 170 180 190 201 211 419 221 232 242 252 263 273 284 294 304 315 420 325 335 346 356 366 377 387 397 408 418 421 428 439 449 459 469 480 490 500 511 521 10 422 531 542 552 562 572 583 593 603 613 624 423 634 644 655 665 675 685 696 706 716 726 1 1.0 424 737 747 757 767 778 788 798 808 818 829 2 3 4 2.0 3.0 4.0 425 839 849 859 870 880 890 900 910 921 931 426 941 951 961 972 982 992 *002 *012 *022 *033 5 5.0 427 63 043 053 063 073 083 094 104 114 124 134 6 6.0 428 144 155 165 175 185 195 205 215 225 236 7 7.0 429 246 256 266 276 286 296 306 317 327 337 8 9 8.0 9.0 430 347 357 367 377 387 397 407 417 428 438 431 448 458 468 478 488 498 508 518 528 538 432 548 558 568 579 589 599 609 619 629 639 433 649 659 669 679 689 699 709 719 729 739 434 749 759 769 779 789 799 809 819 829 839 435 849 859 869 879 889 899 909 919 929 939 436 949 959 969 979 988 998 *008 *018 *028 *038 9 437 64 048 058 068 078 088 098 108 118 128 137 438 147 157 167 177 187 197 207 217 227 237 1 0.9 439 246 256 266 276 286 296 306 316 326 335 2 3 4 1.8 2.7 3.6 440 345 355 365 375 385 395 404 414 424 434 441 444 454 464 473 483 493 503 513 523 532 5 4.5 442 542 552 562 572 582 591 601 611 621 631 6 5.4 443 640 650 660 670 680 689 699 709 719 729 7 6.3 444 738 748 758 768 777 787 797 807 816 826 8 9 7.2 8.1 445 836 846 856 865 875 885 895 904 914 924 446 933 943 953 963 972 982 992 002 *011 *021 447 65 031 040 050 060 070 079 089 099 108 118 448 128 137 147 157 167 176 186 196 205 215 449 225 234 244 254 263 273 283 292 302 312 450 321 331 341 350 360 369 379 389 398 408 N 1 2 3 4 5 6 7 8 9 PP 10 4500 Common Logarithm s of Numbers 5000 [I N 1 2 3 4 5 6 7 8 9 PP 450 65 321 331 341 350 360 369 379 389 398 408 451 418 427 437 447 456 466 475 485 495 504 452 514 523 533 543 552 562 571 581 591 600 453 610 619 629 639 648 658 667 677 686 696 454 706 715 725 734 744 753 763 772 782 792 455 801 811 820 830 839 849 858 868 877 887 456 896 906 916 925 935 944 954 963 973 982 10 457 992 *001 *011 *020 *030 *039 *049 *058 *068 *077 458 66 087 096 106 115 124 134 143 153 162 172 i 1.0 459 181 191 200 210 219 229 238 247 257 266 2 3 4 2.0 3.0 4.0 460 276 285 295 304 314 323 332 342 351 361 461 370 380 389 398 408 417 427 436 445 455 5 5.0 462 464 474 483 492 502 511 521 530 539 549 6 6.0 463 558 567 577 586 596 605 614 624 633 642 7 7.0 464 652 661 671 680 689 699 708 717 727 736 8 9 8.0 9.0 465 745 755 764 773 783 792 801 811 820 829 466 839 848 857 867 876 885 894 904 913 922 467 932 941 950 960 969 978 987 997 *006 *015 468 67 025 034 043 052 062 071 080 089 099 108 469 117 127 136 145 154 164 173 182 191 201 470 210 219 228 237 247 256 265 274 284 293 471 302 311 321 330 339 348 357 367 376 385 i 9 472 394 403 413 422 431 440 449 459 468 477 473 486 495 504 514 523 532 541 550 560 569 1 0.9 474 578 587 596 605 614 624 633 642 651 660 2 3 4 1.8 2.7 3.6 475 669 679 688 697 706 715 724 733 742 752 476 761 770 779 788 797 806 815 825 834 843 5 4.5 477 852 861 870 879 888 897 906 916 925 934 6 5.4 478 943 952 961 970 979 988 997 *006 *015 *024 7 6.3 479 68 034 043 052 061 070 079 088 097 106 115 8 9 7.2 8.1 ISO 124 133 142 151 160 169 178 187 196 205 481 215 224 233 242 251 260 269 278 287 296 482 305 314 323 332 341 350 359 368 377 386 483 395 404 413 422 431 440 449 458 467 476 484 485 494 502 511 520 529 538 547 556 565 485 574 583 592 601 610 619 628 637 646 655 486 664 673 681 690 699 708 717 726 735 744 8 487 753 762 771 780 789 797 806 815 824 833 488 842 851 860 869 878 886 895 904 913 922 1 0.8 489 931 940 949 958 966 975 984 993 *002 *01.1 2 3 4 1.6 2.4 3.2 490 69 020 028 037 046 055 064 073 082 090 099 491 108 117 126 135 144 152 161 170 179 188 5 4.0 492 197 205 214 223 232 241 249 258 267 276 6 4.8 493 285 294 302 311 320 329 338 346 355 364 7 5.6 494 373 381 390 399 408 417 425 434 443 452 8 9 6.4 7.2 495 461 469 478 487 496 504 513 522 531 539 496 548 557 566 574 583 592 601 609 618 627 497 636 644 653 662 671 679 688 697 705 714 498 723 732 740 749 758 767 775 784 793 801 499 810 819 827 836 845 854 862 871 880 888 500 897 906 914 923 932 940 949 958 966 975 N 1 2 3 4 5 6 7 8 9 PP I] 5000 Common Logarithms of Numbers 550C 1 11 N 1 2 3 4 5 6 7 8 9 PP 500 69 897 906| 914 923 932 940 949 958 966 975 501 984 992 *001 *010 *018 *027 *036 *044 *053 *062 502 70 070 0791 088 096 105 114 122 131 140 148 503 157 165 174 183 191 200 209 217 226 234 504 243 252 260 269 278 286 295 303 312 321 505 329 338 346 355 364 372 381 389 398 406 506 415 424 432 441 449 458 467 475 484 492 9 507 501 500 518 526 535 544 552 561 569 578 508 586 595 603 612 621 629 638 646 655 663 1 0.9 509 672 680 689 697 706 714 723 731 740 749 2 3 i 4 1.8 2.7 3.6 510 757 766 774 783 791 800 808 817 825 834 511 842 851 859 868 876 885 893 902 910 919 5 4.5 512 927 935 944 952 961 969 978 986 995 *003 6 5.4 513 71 012 020 029 037 046 054 063 071 079 088 7 6.3 514 096 105 113 122 130 139 147 155 164 172 8 9 7.2 8.1 515 181 189 198 206 214 223 231 24 l 248 257 516 265 273 282 290 299 307 315 324 332 341 517 349 357 366 374 383 391 399 408 416 425 518 433 441 450 458 466 475 483 492 500 508 519 517 525 533 542 550 559 567 575 584 592 520 600 609 617 625 634 642 550 659 667 675 521 684 692 700 709 717 725 734 742 750 759 8 522 767 775 784 792 800 809 817 825 834 842 523 850 858 867 875 883 892 900 908 917 925 1 0.8 524 933 941 950 958 966 975 983 991 999 *008 2 3 4 1.6 2.4 3.2 525 72 016 024 032 041 049 057 066 074 082 090 526 099 107 115 123 132 140 148 156 165 173 5 4.0 527 181 189 198 206 214 222 230 239 247 255 6 4.8 528 263 272 280 288 296 304 313 321 329 337 7 5.6 529 346 354 362 370 378 387 395 403 411 419 8 9 6.4 7.2 530 428 436 444 452 460 469 477 485 493 501 531 509 518 526 534 542 550 558 567 575 583 532 591 599 607 616 624 632 640 648 656 665 533 673 681 689 697 705 713 722 730 738 746 534 754 762 770 779 787 795 803 811 819 827 535 835 843 852 860 868 876 884 892 900 908 536 916 925 933 941 949 957 965 973 981 989 7 537 997 *006 *014 *022 *030 *038 *046 *054 *062 *070 538 73 078 086 094 102 111 119 127i 135 143 151 1 0.7 539 159 167 175 183 191 199 207 215 223 231 2 3 4 1.4 2.1 2.8 540 239 247 255 263 272 280 288 296 304 312 541 320 328 336 344 352 360 368 376 384 392 5 3.5 542 400 408 416 424 432 440 448 456 464 472 6 4.2 543 480 488 496 504 512 520 528 536 544 552 7 4.9 544 560 568 576 584 592 600 608 616 624 632 8 9 5.6 6.3 545 640 648 656 664 672 679 687 695 703 711 546 719 727 735 743 751 759 767 775 783 791 547 799 807 815 823 830 838 846 854 862 870 548 878 886 894 902 910 918 926 933 941 949 549 957 965 973 981 989 997 *005 *013 *020 *028 550 74 036 044 052 060 068 076 084 092 099 107 N 1 2 3 4 5 6 7 8 9 PP 12 5500 Common Logarithm s of Numbers 6000 [I N 1 2 3 4 5 6 7 8 9 PP 550 74 036 044 052 060 068 076 084 092 099 107 551 115 123 131 139 147 155 162 170 178 186 552 194 202 210 218 225 233 241 249 257 265 553 273 280 288 296 304 312 320 327 335 343 554 351 359 367 374 382 390 398 406 414 421 555 429 437 445 453 461 468 476 484 492 500 556 507 515 523 531 539 547 554 562 570 578 557 586 593 601 609 617 624 632 640 648 656 558 663 671 679 687 695 702 710 718 726 733 559 741 749 757 764 772 780 788 796 803 811 560 819 827 834 842 850 858 865 873 881 889 8 561 896 904 912 920 927 935 943 950 958 966 562 974 981 989 997 074 *005 *012 *020 *028 *035 *043 1 0.8 563 75 051 059 066 082 089 097 105 113 120 2 1.6 564 128 136 143 151 159 166 174 182 189 197 3 4 5 2.4 3.2 4.0 565 205 213 220 228 236 243 251 259 266 274 566 282 289 297 305 312 320 328 335 343 351 6 4.8 567 358 366 374 381 389 397 404 412 420 427 7 5.8 568 435 442 450 458 465 473 481 488 496 504 8 6.4 569 511 519 526 534 542 549 557 565 572 580 9 7.2 570 587 595 6Q3 610 618 626 633 641 648 656 571 664 671 679 686 694 702 709 717 724 732 572 740 747 755 762 770 778 785 793 800 808 573 815 823 831 838 846 853 .861 868 876 884 574 891 899 906 914 921 929 937 944 952 959 575 967 974 982 989 997 *005 *012 *020 *027 *035 576 76 042 050 057 065 072 080 087 095 103 110 577 118 125 133 140 148 155 163 170 1781 185 578 193 200 208 215 223 230 238 245 253 260 579 26j8 275 283 290 298 305 313 320 328 335 58 343 350 358 365 373 380 388 395 403 410 7 581 418 425 433 440 448 455 462 470 477 485 582 492 500 507 515 522 530 537 545 552 559 1 0.7 583 567 574 582 589 597 604 612 619 626 634 2 1.4 584 641 649 656 664 671 678 686 693 701 708 3 4 5 2.1 2.8 3.5 585 716 723 730 738 745 753 760 768 775 782 586 790 797 805 812 819 827 834 842 849 856 6 4.2 587 864 871 879 886 893 901 908 916 923 930 7 4.9 588 938 945 953 960 967 975 982 989 997 *004 8 5.6 589 7J_012 019 026 034 041 048 056 063 070 078 9 6.3 590 085 093 100 107 115 122 129 137 144 151 591 159 166 173 181 188 195 203 210 217 225 592 232 240 247 254 262 269 276 283 291 298 593 305 313 320 327 335 342 349 357 364 371 594 379 386 393 401 408 415 422 430 437 444 595 452 459 466 474 481 488 495 503 510 517 596 525 532 539 546 554 561 568 576 583 590 597 597 605 612 619 627 634 641 648 656 663 598 670 677 685 692 699 706 714 721 728 735 599 743 750 757 764 772 779 786 793 801 808 600 815 ^822 830 837 844 851 859 866 873 880 N 1 2 3 4 5 6 7 8 9 PP I] 6000 Common Logarithms of Numbers 6500 13 N 1 2 3 4 5 6 7 8 | 9 PP 600 77 815 822 830 837 844 851 859 866 873! 880 601 887 895 902 909 916 924 931 938 945 952 602 960 967 974 981 988 996 *003 *O10 *017 *025 603 78 032 039 046 053 061 068 075 082 089 097 604 104 111 118 125 132 140 147 154 161 168 605 176 183 190 197 204 211 219 226 233 240 606 247 254 262 269 276 283 290 297 305 312 8 607 319 326 333 340 347 355 362 369 376 383 608 390 398 405 412 419 426 433 440 447 455 1 0.8 609 462 469 476 483 490 497 504 512 519 526 2 3 4 1.6 2.4 3.2 610 533 540 547 554 561 569 576 583 590 597 611 604 611 618 625 633 640 647 654 661 668 5 4.0 612 675 682 689 696 704 711 718 725 732 739 6 4.8 613 746 753 760 767 774 781 789 796 803 810 7 5.6 614 817 824 831 838 845 852 859 866 873 880 8 9 6.4 7.2 615 888 895 902 909 916 923 930 937 944 951 616 958 965 972 979 986 993 *000 *007 *014 *021 617 79 029 036 043 050 057 064 071 078 O85 092 618 099 106 113 120 127 134 141 148 155 162 619 169 176 183 190 197 204 211 218 225 232 620 239 246 253 260 267 274 281 288 295 302 621 309 316 323 330 337 344 351 358 365 372 7 622 379 386 393 400 407 414 421 428 435 442 623 449 456 463 470 477 484 491 498 505! 511 1 0.7 624 518 525 532 539 546 553 560 567 574 581 2 3 4 1.4 2.1 2.8 625 588 595 602 609 616 623 630 637 644 650 626 657 664 671 678 685 692 699 706 713 720 5 3.5 627 727 734 741 748 754 761 768 775 782 789 6 4.2 628 796 803 810 817 824 831 837 844 851 858 7 4.9 629 865 872 879 886 893 900 906 913 920, 927 8 9 5.6 6.3 630 934 941 948 955 962 969 975 982 989! 996 631 80 003 010 017 024 030 037 044 051 058 065 632 072 079 085 092 099 106 113 120 127 134 633 140 147 154 161 168 175 182 188 195 202 634 209 216 223 229 236 243 250 257 264' 271 635 277 284 291 298 305 312 318 325 332 339 635 346 353 359 366 373 380 387 393 400 407 6 637 414 421 428 434 441 448 455 462 468 475 638 482 489 496 502 509 516 523 530 536 543 1 0.6 639 / 550 557 564 570 577 584 591 598 604, 611 2 3 4 1.2 1.8 2.4 640 618 625 632 638 645 652 659 665 672 ! 679 641 686 693 699 706 713 720 726 733 740 747 5 3.0 642 754 760 767 774 781 787 794 801 808 814 6 3.6 643 821 828 835 841 848 855 862 868 875 882 7 4.2 644 889 895 902 909 916 922 929 936 943 949 8 9 4.8 5.4 645 956 963 969 976 983 990 996 *003 *010*017 646 81 023 030 037 043 050 057 064 070 077 084 647 090 097 104 111 117 124 131 137 144 151 648 158 164 171 178 184 191 198 204 211 218 649 224 231 238 245 251 258 265 271 278 285 650 291 298 305 311 318 325 331 338 345 351 N 1 2 3 4 5 6 7 8 9 PP 14 6500 Common Logari thm s of Numbers 7000 [I N 1 2 3 4 5 6 7 8 9 PP 650 81 291 298 305 311 318 325 331 338 345 351 651 358 365 371 378 385 391 398 405 411 418 652 425 431 438 445 451 458 465 471 478 485 653 491 498 505 511 518 525 531 538 544 551 654 558 564 571 578 584 591 598 604 611 617 655 624 631 637 644 651 657 664 671 677 684 656 690 697 704 710 717 723 730 737 743 750 657 757 763 770 776 783 790 796 803 809 816 658 823 829 836 842 849 856 862 869 875 882 659 889 895 902 908 915 921 928 935 941 948 660 954 961 968 974 981 987 994 *000 *007 *014 7 661 82 020 027 033 040 046 053 060 066 073 079 662 086 092 099 105 112 119 125 132 138 145 1 0.7 663 151 158 164 171 178 184 191 197 204 210 2 1.4 664 217 223 230 236 243 249 256 263 269 276 3 2.1 665 282 289 295 302 308 315 321 328 334 341 4 5 2.8 3.5 666 347 354 360 367 373 380 387 393 400 406 6 4.2 667 413 419 426 432 439 445 452 458 465 471 7 4.9 668 478 484 491 497 504 510 517 523 530 536 8 5.6 669 543 549 556 562 569 575 582 588 595 601 9 6.3 67 607 614 620 627 633 640 646 653 659 666 671 672 679 685 692 698 705 711 718 724 730 672 737 743 750 756 763 769 776 782 789 795 673 802 808 814 821 827 834 840 847 853 860 674 866 872 879 885 892 898 905 911 918 824 675 930 937 943 950 956 963 969 975 982 988 676 995 *001 *008 *014 *020 *027 *033 *040 *046 *052 677 83 059 065 072 078 O85 091 097 104 110 117 678 123 129 136 142 149 155 161 168 174 181 679 187 193 200 206 213 219 225 232 238 245 680 251 257 264 270 276 283 289 296 302 308 681 315 321 327 334 340 347 353 359 366 372 6 682 378 385 391 398 404 410 417 423 429 436 683 442 448 455 461 467 474 480 487 493 499 1 0.6 684 506 512 518 525 531 537 544 550 556 563 2 3 4 1.2 1.8 2.4 685 569 575 582 588 594 601 607 613 620 626 686 632 639 645 651 658 664 670 677 683 689 5 3.0 687 696 702 708 715 721 727 734 740 746 753 6 3.6 688 759 765 771 778 784 790 797 803 809 816 7 4.2 689 822 828 835 841 847 853 860 866 872 879 8 9 4.8 5.4 690 885 891 897 904 910 916 923 929 935 942 691 948 954 960 967 973 979 985 992 998 *004 692 84 Oil 017 023 029 036 042 048 055 061 067 693 073 080 086 092 098 105 111 117 123 130 694 136 142 148 155 161 167 173 180 186 192 695 198 205 211 217 223 230 236 242 248 255 696 261 267 273 280 286 292 298 305 311 317 i 697 323 330 336 342 348 354 361 367 373 379 698 386 392 398 404 410 417 423 429 435 442 699 448 454 460 466 473 479 485 491 497 504 700 510 516 522 528 535 541 547 553 559 566 N 1 2 3 4 5 6 7 8 9 PP I] 7000 Common Logarithm s of Numbers 7500 15 N 1 2 3 4 5 6 7 8 9 PP 700 84 510 516 522 528 535 541 547 553 559 566 701 572 578 584 590 597 603 609 615 621 628 702 634 640 646 652 658 665 671 677 683 689 703 696 702 708 714 720 726 733 739 745 751 704 757 763 770 776 782 788 794 800 807 813 75 819 825 831 837 844 850 856 862 868 874 706 880 887 893 899 905 911 917 924 930 936 7 707 942 948 954 960 967 973 979 985 991 997 708 85 003 009 016 022 028 034 040 046 052 058 1 0.7 709 065 071 077 083 089 095 101 107 114 120 2 3 4 1.4 2.1 2.8 71 126 132 138 144 150 156 163 169 175 181 711 187 193 199 205 211 217 224 230 236 242 5 3.5 712 248 254 260 266 2?2 278 285 291 297 303 6 4.2 713 309 315 321 327 333 339 345 352 358 364 7 4.9 714 370 376 382 388 394 400 406 412 418 425 8 9 5.6 6.3 715 431 437 443 449 455 461 467 473 479 485 716 491 497 503 509 516 522 528 534 540 546 717 552 558 564 570 576 582 588 594 600 606 718 612 618 625 631 637 643 649 655 661 667 719 673 679 685 691 697 703 709 715 721 727 720 733 739 745 751 757 763 769 775 781 788 721 794 800 806 812 818 824 830 836 842 848 6 722 854 860 866 872 878 884 890 896 902 908 723 914 920 926 932 938 944 950 956 962 968 1 0.6 724 974 980 986 992 998 *004 *010 *016 *022 *028 2 3 4 1.2 1.8 2.4 725 86 034 040 046 052 058 064 070 076 082 088 726 094 100 106 112 118 124 130 136 141 147 5 3.0 727 153 159 165 171 177 183 189 195 201 207 6 3.6 728 213 219 225 231 237 243 249 255 261 267 7 4.2 729 273 279 285 291 297 303 308 314 320 326 8 9 4.8 5.4 730 332 338 344 350 356 362 368 374 380 386 731 392 398 404 410 415 421 427 433 439 445 732 451 457 463 469 475 481 487 493 499 504 733 510 516 522 528 534 540 546 552 558 564 734 570 576 581 587 593 599 605 611 617 623 735 629 635 641 646 652 658 664 670 676 682 736 688 694 700 705 711 717 723 729 735 741 5 737 747 753 759 764 770 776 782 788 794 800 738 806 812 817 823 829 835 841 847 853 859 1 0.5 739 864 870 876 882 888 894 900 906 911 917 2 3 4 1.0 1.5 2.0 740 923 929 935 941 947 953 958 964 970 976 741 982 988 994 999 *005 *011 *017 *023 *029 *035 5 2.5 742 87 040 046 052 058 064 070 075 081 087 093 6 3.0 743 099 105 111 116 122 128 134 140 146 151 7 3.5 744 157 163 169 175 181 186 192 198 204 210 8 9 4.0 4.5 745 216 221 227 233 239 245 251 256 262 268 746 274 280 286 291 297 303 309 315 320 326 747 332 338 344 349 355 361 367 373 379 384 748 390 396 402 408 413 419 425 431 437 442 749 448 454 460 466 471 477 483 489 495 500 750 506 512 518 523 529 535 541 547 552 558 N 1 2 3 4 5 6 7 8 9 PP 16 7500 Common Logarithms of Numbers 8000 [I N 1 2 3 4 5 6 7 8 9 PP 750 87 506 512 518 523 529 535 541 547 552 558 751 564 570 576 581 587 593 599 604 610 616 752 622 628 633 639 645 651 656 662 668 674 753 679 685 691 697 703 708 714 720 726 731 754 737 743 749 754 760 766 772 777 783 789 755 795 800 806 812 818 823 829 835 841 846 756 852 858 864 869 875 881 887 892 898 904 757 910 915 921 927 933 938 944 950 955 961 758 967 973 978 984 990 996 *001 *007 *013 *018 759 88 024 030 036 041 047 053 058 064 070 076 760 081 087 093 098 104 110 116 121 127 133 761 138 144 150 156 161 167 173 178 184 190 6 762 195 201 207 213 218 224 230 235 241 247 763 252 258 264 270 275 281 287 292 298 304 1 0.6 764 309 315 321 326 332 338 343 349 355 360 2 1.2 765 366 372 377 383 389 395 400 406 412 417 3 4 1.8 2.4 766 423 429 434 440 446 451 457 463 468 474 5 3.0 767 480 485 491 497 502 508 513 519 525 530 6 3.6 768 536 542 547 553 559 564 570 576 581 587 7 4.2 769 593 598 604 610 615 621 627 632 638 643 8 4.8 770 649 655 660 666 672 677 683 689 694 700 9 5.4 771 705 711 717 722 728 734 739 745 750 756 772 762 767 773 779 784 790 795 801 807 812 773 818 824 829 835 840 846 852 857 863 868 774 874 880 885 891 897 902 908 913 919 925 775 930 936 941 947 953 958 964 969 975 981 776 986 992 997 *003 *009 *014 *020 *025 *031 *037 777 89 042 048 053 059 064 070 076 081 087 092 778 098 104 109 115 120 126 131 137 143 148 779 154 159 165 170 176 182 187 193 198 204 780 209 215 221 226 232 237 243 248 254 260 781 265 271 276 282 287 293 298 304 310 315 5 782 321 326 332 337 343 348 354 360 365 371 783 376 382 387 393 398 404 409 415 421 426 1 0.5 784 432 437 443 448 454 459 465 470 476 481 2 3 4 1.0 1.5 2.0 785 487 492 498 504 509 515 520 526 531 537 786 542 548 553 559 564 570 575 581 586 592 5 2.5 787 597 603 609 614 620 625 631 636 642 647 6 3.0 788 653 658 664 669 675 680 686 691 697 702 7 3.5 789 708 713 719 724 730 735 741 746 752 757 8 9 4.0 4.5 790 763 768 774 779 785 790 796 801 807 812 791 818 823 829 834 840 845 851 856 862 867 792 873 878 883 889 894 900 905 911 916 922 793 927 933 938 944 949 955 960 966 971 977 794 982 988 993 998 *004 *009 *015 *020 *026 *031 795 90 037 042 048 053 059 064 069 075 080 086 796 091 097 102 108 113 119 124 129 135 140 797 146 151 157 162 168 173 179 184 189 195 798 200 206 211 217 222 227 233 238 244 249 799 255 260 266 271 276 282 287 293 298 304 800 309 314 320 325 331 336 342 347 352 358 N 1 2 3 4 5 6 7 8 9 PP I] 8000 Common Logarithms of Numbers I 3500 17 N 1 2 3 4 5 6 7 8 9 PP soo 90 309 314 320 325 331 336 342 347 352 358 801 363 369 374 380 385 390 396 401 407 412 802 417 423 428 ! 434 439 445 450 455 461 466 803 472 477 4821 488 493 499 504 509 515 520 804 526 531 536 542 547 553 558 563 569 574 805 580 585 590 596 601 607 612 617 623 628 806 634 639 644 650 655 660 666 671 677 682 807 687 693 698 703 709 714 720 725 730 736 808 741 747 752 757 763 768 773 779 784| 789 809 795 800 806 811 816 822 827 832 838j 843 810 849 854 859 865 870 875 881 886 891 897 811 902 907 913 918 924 929 934 940 945 950 6 812 956 961 966 972 977 982 988 993 998 *004 813 91 009 014 020 025! 030 036 041 046 052 057 1 0.6 814 062 068 073 078 084 089 094 100 105 110 2 3 4 1.2 1.8 2.4 815 116 121 126 132 137 142 148 153 158 164 816 169 . 174 228 180 185 190 196 201 206 212 217 5 3.0 817 222 233 238 243 249 254 259 265 270 6 3.6 818 275 281 286 291 297 302 307 312 318 323 7 4.2 819 328 334 339 344 350 355 360 365 371! 376 8 9 4.8 5.4 820 381 387 392 397 403 408 413 418 424 429 821 434 440 445 450 455 461 466 471 477 482 822 487 492 498 503 508 514 519 524 529 535 823 540 545 551 556 561 566 572 577 582! 587 824 593 598 603 609 614 619 624 630 635 640 825 645 651 656 661 666 672 677 682 687 693 826 698 703 709 714 719 724 730 735 740 745 827 751 756 761 766 772 777 782 787 793 798 828 803 808 814 819 824 829 834 840 845 850 829 855 861 866 871 876 882 887 892 897 903 830 908 913 918 924 929 934 939 944 950 955 831 960 965 971 976 981 986 991 997 *002*007 5 832 92 012 018 023 028 033 038 044 049 054 059 833 065 070 075 080 085 091 096 101 106 111 1 0.5 834 117 122 127 132 137 143 148 153 158! 163 2 3 4 1.0 1.5 2.0 835 169 174 179 184 189 195 200 205 210 215 836 221 226 231 236 241 247 252 257 262 267 5 2.5 837 273 278 283 288 293 298 304 309 314 319 6 3.0 838 324 330 335 340 345 350 355 361 366 371 7 3.5 839 376 381 387 392| 397 402 407 412 418 423 8 9 4.0 4.5 840 428 433 438 443 449 454 459 464 469 474 841 480 485 490 495 500 505 511 516 521! 526 842 531 536 542 547! 552 557 562 567 572 578 843 583 588 593 598, 603 609 614 619 624 629 844 634 639 645 650 655 660 665 670 675 681 845 686 691 696 701 706 711 716 722 727 732 846 737 742 747 752 758 763 768 773 778 783 847 788 793 799 804 809 814 819 824 829 834 848 840 845 850 855 860 865 870 875 8811 886 849 891 896 901 906 911 916 921 927 932 937 850 942 947 952 957 962 967 973 978 983 988 | N 1 2 3 4 5 6 7 8 | 9 | PP 18 8500 Common Logarithm s of Numbers 9000 [I N 1 2 3 4 5- ' 6 7 8 9 PP 850 92 942 947 952 957 962 967 973 978 983 988 851 993 998 *0Q3 *008 *013 *018 *024 *029 *034 *039 852 93 044 049 054 059 064 069 075 080 085 090 853 095 100 105 110 115 120 125 131 136 141 854 146 151 156 161 166 171 176 181 186 192 i 855 197 202 207 212 217 222 227 232 237 242 856 247 252 258 263 268 273 278 283 288 293 6 857 298 303 308 313 318 323 328 334 339 344 858 349 354 359 364 369 374 379 384 389 394 1 0.6 859 399 404 409 414 420 425 430 435 440 445 2 3 4 1.2 1.8 2.4 860 450 455 460 465 470 475 480 485 490 495 861 500 505 510 515 520 526 531 536 541 546 5 3.0 862 551 556 561 566 571 576 581 586 591 596 6 3.6 863 601 606 611 616 621 626 631 636 641 646 7 4.2 864 651 656 661 666 671 676 682 687 692 697 8 9 4.8 865 702 707 712 717 722 727 732 737 742 747 5.4 866 752 757 762 767 772 777 782 787 792 797 867 802 807 812 817 822 827 832 837 842 847 868 852 857 862 867 872 877 882 887 892 897 869 902 907 912 917 922 927 932 937 942 947 870 952 957 962 967 972 977 982 987 992 997 871 94 002 007 012 017 022 027 032 037 042 047 5 872 052 057 062 067 072 077 082 086 091 096 873 101 106 111 116 121 126 131 136 141 146 1 0.5 874 151 156 161 166 171 176 181 186 191 196 2 3 4 1.0 1.5 2.0 875 201 206 211 216 221 226 231 236 240 245 876 250 255 260 265 270 275 280 285 290 295 5 2.5 877 300 305 310 315 320 325 330 335 340 345 6 3.0 878 349 354 359 364 369 374 379 384 389 394 7 3.5 879 399 404 409 414 419 424 429 433 438 443 8 9 4.0 4.5 880 448 453 458 463 468 473 478 483 488 493 881 498 503 507 512 517 522 527 532 537 542 882 547 552 557 562 567 571 576 581 586 591 883 596 601 606 611 616 621 626 630 635 640 884 645 650 655 660 665 670 675 680 685 689 885 694 699 704 709 714 719 724 729 734 738 886 743 748 753 758 763 768 773 778 783 787 4 887 792 797 802 807 812 817 822 827 832 836 888 841 846 851 856 861 866 871 876 880 885 1 0.4 889 890 895 900 9051 910 915 919 924 929 934 2 3 4 0.8 1.2 1.6 890 939 944 949 954 959 963 968 973 978 983 891 988 993 998 *002 *007 *012 *017 *022 *027 *032 5 2.0 892 95 036 041 046 051 056 061 066 071 075 080 6 2.4 893 085 090 095 100 105 109 114 119 124 129 7 2.8 894 134 139 143 148 153 158 163 168 173 177 8 9 3.2 3.6 895 182 187 192 197 202 207 211 216 221 226 896 231 236 240 245 250 255 260 265 270 274 897 279 284 289 294 299 303 308 313 318 323 898 328 332 337 342 347 352 357 361 366 371 899 376 381 386 390 395 400 405 410 415 419 900 424 429 434 439 444 448 453 458 463 468 N 1 2 3 4 5 6 7 8 9 PP I] 9000 Common Logarithms of Numbers 9500 19 N 1 2 3 4 5 6 7 8 9 PP 900 95 424 429 434 439 487 444 448 453 458 ' 463 468 901 472 477 482 492 497 501 506 511 516 902 521 525 530 535 540 545 550 554 559 564 903 569 574 578 583 588 593 598 602 607 612 904 617 622 626 631 636 641 646 650 655 660 905 665 670 674 679 684 689 694 698 703 708 906 713 718 722 727 732 737 742 746 751 756 907 761 766 770 775 780 785 789 794 799 804 908 809 813 818 823 828 832 837 842 847 852 909 856 861 866 871 875 880 885 890 895 899 91 904 909 914 918 923 928 933 938 942 947 911 952 957 961 966 971 976 980 985 990 995 5 912_ 999 *004 *009 *014) oef *019 *023 *028 *033> 080 *038 *042 913 96 047 Wo8 052 057 066 071 118 076 085 090 1 0.5 914 099 104 019" 114 123 128 133 137 2 3 4 1.0 1.5 2.0 , 915 142 147 152 156 161 166 171 175 180 185 916 190 194 199 204 209 213 218 223 227 232 5 2.5 917 237| 242 246 251 256 261 265 270 275 280 6 3.0 918 284! 289 294 298 303 308 313 317 322 327 7 3.5 919 332, 336 341 346 350 355 360 365 369 374 8 9 4.0 4.5 929 379 384 388 393 398 402 407 412 417 421 921 426[ 431 435 440 445 450 454 459 464 468 922 473 478 483 487 492 497 501 506 511 515 923 520 525 530 534 539 544 548 553 558 562 924 567 572 577 581 586 591 595 600 605 609 925 614 619 624 628 633 638 642 647 652 656 926 661 666 670 675 680 685 689 694 699 703 927 708 713 717 722 727 731 736 741 745 750 928 755 759 764 769 774 778 783 788 792 797 929 802 806 811 816 820 825 830 834 839 844 930 848 853 858 862 867 872 876 881 886 890 931 895 900| 904 909 914 918 923 928 932 937 4 932 942 946 951 956 960 965 970 974 979 984 933 988 993 039 997 *002 *007 *011 *016 *021 *025 *030 1 0.4 934 97 035 044 049 053 058 063 067 072 077 2 3 4 0.8 1.2 1.6 935 081 086 090 095 100 104 109 114 118 123 936 128 132 137 142 146 151 155 160 165 169 5 2.0 937 174, 179 183 188 192 197 202 206 211 216 6 2.4 938 220 225 230 234 239 .243 248 253 257 262 7 2.8 939 267 271 276 280 285 290 294 299 304 308 8 9 3.2 3.6 940 313 317 322 327 331 336 340 345 350 354 941 359 364 368 373 377 382 387 391 396; 400 942 405 410 414 419 424 428 433 437 442 447 943 451 456 460 465 470 474 479 1 483 488 493 944 497 502 506 511 516 520 525 529 534, 539 945 543 548 552 557 562 566 571 575 5801 585 946 589 594 598 603 607 612 617 621 626! 630 947 635 640 644 649 653 658 663 i 667 672 676 948 681 685 690 1 695 699 704 708 713 717 722 949 727 731 736 740 745 749 754 759 763 768 959 772 777 782 786 791 795 800 804 80S 813 N 1 2 3 4 5 6 7 8 | 9 PP 20 9500 Common Logarithms of Numbers 10000 [I N 1 2 3 4 5 6 7 8 9 PP 95 97 772 777 782 786 832 791 795 800 804 809 813 951 818 823 827 836 841 845 850 855 859 952 864 868 873 877 882 886 891 896 900 905 953 909 914 918 923 928 932 937 941 946 950 954 955 959 964 968 973 978 982 987 991 996 955 98 000 005 009 014 019 023 028 032 037 041 956 046 050 055 059 064 068 073 078 082 087 957 091 096 100 105 109 114 118 123 127 132 958 137 141 146 150 155 159 164 168 173 177 959 182 186 191 195 200 204 209 214 218 223 960 227 232 236 241 245 250 254 259 263 268 961 HI - 277 281 286 290 295 299 304 308 313 5 962 322 327 331 336 340 345 349 354 358 963 363 367 372 376 381 385 390 394 399 403 1 0.5 964 408 412 417 421 426 430 435 439 444 448 2 3 4 1.0 1.5 2.0 965 453 457 462 466 471 475 480 484 489 493 966 498 502 507 511 516 520 525 529 534 538 5 2.5 967 543 547 552 556 561 565 570 574 579 583 6 3.0 968 588 592 597 601 605 610 614 619 623 628 7 3.5 969 632 637 641 646 650 655 659 664 668 673 8 9 4.0 4.5 97 677 682 686 691 695 700 704 709 713 717 971 722 726 731 735 740 744 749 753 758 762 972 767 771 776 780 784 789 793 798 802 807 973 811 816 820 825 829 834 838 843 847 851 974 856 860 865 869 874 878 883 887 892 896 975 900 905 909 914 918 923 927 932 936 941 976 945 949 954 958 963 967 972 976 981 985 977 989 994 998 *003 *007 *012 *016 *021 *025 *029 978 99 034 038 043 047 052 056 061 065 069 074 979 078 083 087 092 096 100 105 109 114 118 980 123 127 131 136 140 145 149 154 158 162 981 167 171 176 180 185 189 193 198 202 207 4 982 211 216 220 224 229 233 238 242 247 251 983 255 260 264 269 273 277 282 286 291 .295 1 0.4 984 300 304 308 313 317 322 326 330 335 339 2 3 4 0.8 1.2 1.6 985 344 348 352 357 361 366 370 374 379 383 986 388 392 396 401 405 410 414 419 423 427 5 2.0 987 432 436 441 445 449 454 458 463 467 471 6 2.4 988 476 480 484 489 493 498 502 506 511 515 7 2.8 989 520 524 528 533 537 542 546 550 555 559 8 9 3.2 3.6 990 564 568 572 577 581 585 590 594 599 603 991 607 612 616 621 625 629 634 638 642 647 992 651 656 660 664 669 673 677 682 686 691 993 695 699 704 708 712 717 721 726 730 734 994 739 743 747 752 756 760 765 769 7741 778 995 782 787 791 795 800 804 808 813 817 822 996 826 830 835 839 843 848 852 856 861 865 997 870 874 878 883 887 891 896 900 904 909 998 913 917 922 926 930 935 939 944 948 952 999 957 961 965 970 974 978 983 987 991 996 lOOO 00 000 004 009 013 017 022 026 030 035 039 N 1 2 3 4 5 6 7 8 9 PP TABLE II Common Logarithms of the Trigonometric Functions For each second from 0' to 3' and from 89 57 ' to 90 For every ten seconds from 3' to 2 and from 88 to 89 57' For each minute from 2 to 88 to Five Decimal Places II] 0'- -Logarithms of Trigonometric Functions -3' 23 L Sin and L Tan L Sin and L Tan // 0' V 2' // n 0' V 2' // o 1 2 3 4 4.68 557 4.98 660 5.16 270 5.28 763 6.46 373 6.47 090 6.47 797 6.48 492 6.49 175 6.76 476 6.76 836 6.77 193 6.77 548 6.77 900 60 59 58 57 56 80 31 32 33 34 6.16 270 6.17 694 6.19 072 6.20 409 6.21 705 6.63 982 6.64 462 6.64 936 6.65 406 6.65 870 6.86 167 6.86 455 6.86 742 6.87 027 6.87 310 30 29 28 27 26 5 6 7 8 9 5.38 454 5.46 373 5.53 067 5.58 866 5.63 982 6.49 849 6.50 512 6.51 165 6.51 808 6.52 442 6.78 248 6.78 595 6.78 938 6.79 278 6.79 616 55 54 53 52 51 35 36 37 38 39 6.22 964 6.24 188 6.25 378 6.26 536 6.27 664 6.66 330 6.66 785 6.67 235 6.67 680 6.68 121 6.87 591 6.87 870 6.88 147 6.88 423 6.88 697 25 24 23 22 21 io 11 12 13 14 5.68 557 5.72 697 5.76 476 5.79 952 5.83 170 6.53 067 6.53 683 6.54 291 6.54 890 6.55 481 6.79 952 6.80 285 6.80 615 6.80 943 6.81 268 50 49 48 47 46 40 41 42 43 44 6.28 763 6.29 836 6.30 882 6.31 904 6.32 903 6.68 557 6.68 990 6.69 418 6.69 841 6.70 261 6.88 969 6.89 240 6.89 509 6.89 776 6.90 042 20 19 18 17 16 15 16 17 18 19 5.86 167 5.88 969 5.91 602 5.94 085 5.96 433 6.56 064 6.56 639 6.57 207 6.57 767 6.58 320 6.81 591 6.81 911 6.82 230 6.82 545 6.82 859 45 44 43 42 41 45 46 47 48 49 6.33 879 6.34 833 6.35 767 6.36 682 6.37 577 6.70 676 6.71 088 6.71 496 6.71 900 6.72 300 6.90 306 6.90 568 6.90 829 6.91 088 6.91 346 15 14 13 12 11 20 21 22 23 24 5.98 660 6.00 779 6.02 800 6.04 730 6.06 579 6.58 866 6.59 406 6.59 939 6.60 465 6.60 985 6.83 170 6.83 479 6.83 786 6.84 091 6.84 394 40 39 38 37 36 50 51 52 53 54 6.38 454 6.39 315 6.40 158 6.40 985 6.41 797 6.72 697 6.73 090 6.73 479 6.73 865 6.74 248 6.91 602 6.91 857 6.92 110 6.92 362 6.92 612 IO 9 8 7 6 25 26 27 28 29 6.08 351 6.10 055 6.11 694 6.13 273 6.14 797 6.61 499 6.62 007 6.62 509 6.63 006 6.63 496 6.84 694 6.84 993 6.85 289 6.85 584 6.85 876 35 34 33 32 31 55 56 57 58 59 6.42 594 6.43 376 6.44 145 6.44 900 6.45 643 6.74 627 6.75 003 6.75 376 6.75 746 6.76 112 6.92 861 6.93 109 6.93 355 6.93 599 6.93 843 5 4 3 2 1 30 6.16 270 6.63 982 6.86 167 30 OO 6.46 373 6.76 476 6.94 085 O rr 59' 58' 57' // // 59' 58' 57' // L Cos and L Cot L Cos and L Cot 89 57 , Logarithms of Trigonometric Functions 90 c 10 should be written after every logarithm taken from this page. 24 3' Logarithms of Trigonometric Functions- -20' [II / // LSin LCos LTan // / 10 7.46 373 0.00 0007.46 373 50 10 7.47 090 0.00 0007.47 091 50 20 7.47 797 0.00 0007.47 797 40 30 7.48 491 0.00 0007.48 492 30 Logarithms of functions of angle sless 40 7.49 175 0.00 0007.49 176 20 than 3' or greater than 8957' are found 50 7.49 8490. 10 on the preceding page. 11 7.50 5120.00 49 10 7.51 1650.00 0007.51 165 50 20 7.51 808 0.00 0007.51 809 40 30 7.52 442 0.00 0007.52 443 30 40 7.53 067 0.00 0007.53 067 20 50 7.53 683 0.00 000 7.53 683 10 12 7.54 291 0.00 0007.54 291 48 10 7.54 8900.00 0007.54 890 50 20 30 7.55 481 7.56 064 0.00 0007.55 481 40 0.00 000 7.56 064 30 / // LSin LCos LTan // / 40 7.56 639 0.00 000 7.56 639 20 50 13 7.57 206 7.57 767 0.00 000 0.00 000 7.57 207 7.57 767 10 47 3 6.94 085 0.00 000 6.94 085 57 10 6.96 433 0.00 000 6.96 433 50 10 7.58 3200.00 000 7.58 320 50 20 6.98 660 0.00 000 6.98 661 40 20 7.58 866 0.00 000 7.58 867 40 30 7.00 779 0.00 000 7.00 779 30 30 7.59 4060.00 000 7.59 406 30 40 7.02 800 0.00 000 7.02 800 20 40 7.59 939 0.00 000 7.59 939 20 50 7.04 730 0.00 000 7.04 730 10 50 7.60 465 0.00 000 7.60 466 10 4 7.06 579 0.00 000 7.06 579 56 14 7.60 985 0.00 000 7.60 986 46 10 7.08 3510.00 000 7.08 352 50 10 7.61 499 0.00 000)7.61 500 50 20 7.10 055l0.00 000 7.10 055 40 20 7.62 007 0.00 000 7.62 008 40 30 7.11 694 0.00 000 7.11 694 30 30 7.62 509 0.00 0007.62 510 30 40 7.13 273 0.00 000 7.13 273 20 40 7.63 006 0.00 000 7.63 006 20 50 7.14 797 0.00 000 7.14 797 10 50 7.63 496 0.00 0007.63 497 10 5 7.16 27o'o.OO 000 7.16 270 55 15 7.62 982 0.00 0007.63 982 45 10 7.17 694 0.00 000 7.17 694 50 10 7.64 461 0.00 0007.64 462 50 20 7.19 0720.00 000 7.19 073 40 20 7.64 936 0.00 0007.64 937 40 30 7.20 409j 0.00 000 7.20 409 30 30 7.65 406|0.00 0007.65 406 30 40 7.21 705 0.00 000 7.21 705 20 40 |7.65 870 0.00 00017.65 871 20 50 7.22 964 ( 0.00 000 7.22 964 10 50 7.66 330 0.00 0007.66 330 10 6 7.24 188 0.00 000 7.24 188 54 16 7.66 784 0.00 0007.66 785 44 10 7.25 378 0.00 000 7.25 378 50 10 7.67 235 0.00 000 7.67 235 50 20 7.26 5360.00 000 7.26 536 40 20 7.67 680 0.00 0007.67 680 40 30 7.27 664 0.00 000 7.28 763 0.00 000 7.27 664 30 30 7.68 121 0.00 0007.68 121 30 40 7.28 764 20 40 7.68 657 9.99 999 7.68 558 20 50 7.29 836 0.00 000,7.29 836 10 50 7.68 989 9.99 999 7.68 990 10 7 7.30 882 0.00 0007.30 882 53 17 7.69 417 9.99 999 7.69 418 43 10 7.31 904 0.00 000 7.31 904 50 10 7.69 841|9.99 999 7.69 842 50 20 7.32 90310.00 000 7.32 903 50 20 7.70 2619.99 999 7.70 261 40 30 7.33 879|0.00 000 7.33 879 30 30 7.70 6769.99 9997.70 677 30 40 7.34 833 t.35 767 0.00 000 7.34 833 20 40 7.71 0889.99 999 7.71 088 20 50 0.00 000 7.35 767 10 50 7.71 496,9.99 999 7.71 496 10 8 7.36 682 0.00 000 7.36 682 52 18 7.71 900i9.99 9997.71 900 42 10 7.37 577 0.00 000 7.37 577 50 10 7.72 3009.99 999 7.72 301! 50 20 7.38 454 0.00 000 7.38 455 40 20 7.72 697 9.99 9997.72 697 40 7 . 73 090 9 . 99 999 7 . 73 090 30 30 7.39 3140.00 000 7.39 315 30 30 40 7.40 158 0.00 000 7.40 158 20 40 7.73 4799.99 9997.73 480 20 50 7.40 985 0.00 000 7.40 985 10 50 7.73 8659.99 999 7.73 866 10 9 7.41 797 0.00 000 7.41 797 51 19 7.74 248 9.99 999 7.74 248 41 10 7.42 594 0.00 000 7.42 594 50 10 7.74 627 9.99 999 7.74 628 50 20 7.43 3760.00 0007.43 376 40 20 7.75 003 9.99 999 7.75 004 40 30 7.44 145 0.00 000 7.44 145 30 30 7.75 376 9.99 999 7.75 377 30 40 7.44 9000.00 000 7.44 900 20 40 7.75 745 9.99 999 7.75 746 20 50 7.45 643 0.00 000 7.45 643 10 50 7.76 112 9.99 999 7.76 113 10 10 7.46 373 0.00 000 7.46 373 50 20 7.76 475 9.99 999 7.76 476 40 / // LCos LSin LCot // / / // LCos LSin LCot // / 8940' Logarithms of Trigonometric Functions 8957' 10 should be written after every logarithm taken from this page, except those Wat are 0.00000 II] 20'- -Logarithms of Trigonometric Functions 40' 25 1 If LSin LCos LTan // / / // LSin L Cos LTan // / 20 7.76 475 9.99 999 7.76 476 40 30 7.94 084 9.99 998 7.94 086 30 10 7.76 836 9.99 99917.76 837 50 10 7.94 3259.99 998 7.94 326 50 20 7.77 193 9.99 999 7.77 194| 40 20 7.94 5649.99 99817.94 566 40 30 7.77 548 9.99 999 7.77 549| 30 30 7.94 8029.99 99817.94 804 30 40 7.77 899 9.99 999 7.77 900 20 40 7.95 039 9.99 998 7.95 040 20 50 7.78 248j9.99 999 7.78 249 10 50 7.95 274 9.99 998 7.95 276 10 21 7.78 594 9.99 999 7.78 595 39 31 7.95 508 9.99 998 7.95 510 29 10 7.78 938 9.99 999 7.78 938 50 10 7.95 741 9.99 998 7.95 743 50 20 7.79 278 9.99 999 7.79 279 40 20 7.95 973 9.99 998:7.95 974| 40 30 7.79 616 9.99 99917.79 617 30 30 7.96 203 9.99 9987.96 205 30 40 7 . 79 952 9 . 99 999 7.79 952 20 40 7.96 432 9.99 998 7.96 434 20 50 7.80 284:9.99 999 7.80 285 10 50 7.96 660:9.99 998 7.96 662 10 22 7.80 615'9.99 999 7.80 615 38 32 7.96 887 9.99 998 7.96 889 28 10 7.80 942 9.99 999 7.80 943 50 10 7.97 113 9.99 998 7.97 114 50 20 7.81 268 9.99 999 7.81 269 40 20 7.97 337 9.99 998 7.97 339! 40 30 7.81 5919.99 999 7.81 591 30 30 7.97 560!9.99 998 7.97 562i 30 40 7.81 911 9.99 999 7.81 912 20 40 7.97 782 9.99 998 7.97 784 20 50 7.82 229,9.99 999 7.82 230 10 50 7.98 003 9.99 998 7 . 98 005 10 23 7.82 545 9.99 999 7.82 546 37 33 7.98 223 9.99 998 7.98 225 27 10 7.82 859 9.99 999 7.82 860 50 10 7.98 4429.99 998 7.98 444 50 20 7.83 170 9.99 999 7.83 171 40 20 7.98 660 9.99 998 7.98 662 40 30 7.83 479 9.99 999 7.83 480 30 30 7.98 876 9.99 998j7.98 878 30 40 7.83 786i9.99 999 7.83 787 20 40 7.99 092 9.99 998 7.99 094 20 50 7.84 091 9.99 999 7.84 092 10 50 7.99 306 9.99 998 7.99 308 10 24 7.84 393 9.99 999 7.84 394 36 34 7.99 520 9.99 998 7.99 522 26 10 7.84 694 9.99 999 7.84 695 50 10 7.99 732 9.99 998 7.99 734 50 20 7.84 992 9.99 999 7 . 84 994 40 20 7.99 943 9.99 998 7.99 946 40 30 7.85 289i9.99 999 7 . 85 290! 30 30 8.00 154 9.99 998 8.00 156 30 40 7.85 583 9.99 999 7.85 584 20 40 8.00 363 9.99 998 8.00 365 20 50 7.85 876 9.99 999 7.85 877 10 50 8.00 571 9.99 998 8.00 574 10 25 7.86 166 9.99 999 7.86 167 35 35 8.00 779 9.99 998 8.00 781 25 10 7.86 455 9.99 999 7.86 456 50 10 8.00 985 9.99 998 8.00 987 50 20 7.86 741 9.99 999 7.86 743 40 20 8.01 1909.99 9988.01 193 40 30 7.87 026 9.99 999 7.87 027 30 30 8.01 395 9.99 998 8.01 397 30 40 7.87 309 9.99 999 7.87 310 20 40 8.01 598 9.99 998 8.01 600 20 50 7.87 590 9.99 999 7.87 591 10 50 8.01 801 9.99 998j8.01 803 10 26 7.87 870 9.99 999 7.87 871 34 36 8.02 OO2I9.99 998 8.02 004 24 10 7.88 147 9.99 999 7.88 148 50 10 8.02 203 9.99 998 8.02 205 50 20 7 . 88 423 9 . 99 999 7 . 88 424 40 20|8.02 402 9.99 998 8.02 405 40 30 7.88 6979.99 999 7.88 698 30 .30|8.02 601|9.99 9988.02 604 30 40 7.88 969 9.99 999 7.88 970 20 40 8.02 799 9.99 998 8.02 801 20 50 7.89 240 9.99 999 7.89 241 10 50 8.02 996 9.99 998 8.02 998 10 27 7.89 509 9.99 999 7.89 510 33 37 8.03 192 9.99 997 8.03 194 23 10 7.89 776 9.99 99917.89 777 50 10 8.03 387 9.99 997 8.03 390 50 20 7.90 041 9.99 999 7.90 043 40 20 8.03 581 9.99 997 8.03 584 40 30 7.90 305 9.99 999 7.90 307 30 30 8.03 775 9.99 997|8.03 777 30 40 7.90 568 9.99 999 7.90 569 20 40 8.03 967 9.99 99718.03 970 20 50 7.90 829 9.99 999 7.90 830 10 50 8.04 159 9.99 997 8.04 162 10 28 7.91 088 9.99 999 7.91 089 32 38 8.04 350 9.99 997 8.04 353 22 10 7.91 346 9.99 999 7.91 347 50 10 8.04 540 9.99 997 8.04 543 50 20 7.91 602 9.99 999 7.91 603 40 20 8.04 729 9.99 997 8.04 732 40 30 7.91 857 9.99 999 7.91 858 30 30 8.04 918 9.99 997 8.04 921 30 40 7.92 110 9.99 998 7.92 111 20 40 8.05 105 9.99 997 8.05 108 20 50 7.92 362 9.99 998 7.92 363 10 50 8.05 292 9.99 997 8.05 295 10 29 7.92 612 9.99 998 7.92 613 31 39 8.05 478 9.99 997 8.05 481 21 10 7.92 861 9.99 998 7.92 862 50 10 8.05 663 9.99 997 8.05 666 50 20 7.93 108 9.99 998 7.93 110 40 20 8.05 848 9.99 997 8.05 851 40 30 7.93 354 9.99 998 7.93 356 30 30 8.06 031 9.99 997 8.06 034 30 40 7.93 599 9.99 998 7.93 601 20 40 8.06 214 9.99 9978.06 217 20 50 7.93 842 9.99 998 7.93 844 10 50 8.06 396 9.99 997 8.06 399 10 30 7.94 084 9.99 998 7.94 086 30 40 8.06 578 9.99 997 8.06 581 20 / // LCos LSin LCot // / / // LCos LSin LCot // / 8920' Logarithms of Trigonometric Functions 8940' - 10 should be written after every logarithm taken from this page. 26 40'- -Logarithms of Trigonometric Functions 1 [II /. // LSin LCos LTan // / / // LSin 1 L Cos I L Tan // / 40 8. OS 57? 9.99 997 8.06 581 20 50 8.16 268 9.99 995 8.16 273 10 10 8.06 758 9.99 997 8.06 761 50 10 8.16 413 9.99 995 8.16 417 50 20 8.06 9389.99 997 8.06 941 40 20 8.16 557 9.99 995 8.16 561 40 30 8.07 117,9.99 997 8.07 120 30 30 8.16 700 9.99 995 8.16 705 30 40 8.07 29 9.99 997 8.07 299 20 40 8.16 84319.99 995 8.16 848 20 50 8.07 472 9.99 997 8.07 476 10 50 8.16 986 9.99 995 8.16 991 10 41 8.07 65C 9.99 997 8.07 653 19 51 8.17 128 9.99 995 8.17 133 9 10 8.07 826 9.99 997(8.07 829 50 10 8.17 270 9.99 995(8.17 275 50 20 8.08 002 9.99 997 8.08 005 40 20 8.17 4119.99 9958.17 416 40 30 8.08 176 9.99 997 8.08 180 30 30 8.17 552 9.99 995 8.17 557 30 40 8.08 35019.99 997 8.08 354 20 40 8.17 692 9.99 995 8.17 697 20 50 8.08 524 9.99 997 8.08 527 10 50 8.17 832 9.99 995 8.17 837 10 42 8.08 696 9.99 997 8.08 700 18 52 8.17 971 9.99 995 8.17 976 8 10 8.08 868 9.99 997 8.08 872 50 10 8.18 110 9.99 995 8.18 115 50 20 8.09 040 9.99 997 8.09 043 40 20 8.18 249 9.99 995 8.18 254 40 30 8.09 210 9.99 997 8.09 214 30 30 8.18 387 9.99 995 8.18 392 30 40 8.09 380 9.99 997 8.09 384 20 40 8.18 524 9.99 995 8.18 530 20 50 8.09 550|9.99 997 8.09 553j 10 50 8.18 662 9.99 995 8.18 667 10 43 8.09 718 9.99 99718.09 722 17 53 8.18 798 9.99 995 8.18 804 7 10 8.09 886 9.99 9978.09 89/) 50 10 8.18 935 9.99 995 8.18 940 50 20 8.10 054 9.99 997 8.10 057 40 20 8.19 071 9.99 995J8.19 076 40 30 8.10 220 9.99 997 8.10 224 30 30 8.19 206 9.99 995 8.19 212 30 40 8.10 386 9.99 997 8.10 390 20 40 8.19 341 9.99 995 8.19 347 20 50 8.10 552 9.99 996 8.10 555 10 50 8.19 476 9.99 995 8.19 481 10 44 8.10 717 9.99 996 8.10 720 16 54 8.19 610 9.99 995 8.19 616 6 10 8.10 881;9.99 996 8.10 884 50 10 8.19 744 9.99 995 8.19 749 50 20 8.11 044 9.99 996 8.11 048 40 20 8.19 877 9.99 995|8.19 883 40 30 8.11 207!9.99 9968.11 211 30 30 8.20 010 9.99 995 8.20 016 30 40 8.11 370 9.99 996 8.11 373 20 40 8.20 143 9.99 995 8.20 149 20 50 8.11 531 9.99 996 8.11 535 10 50 8.20 275 9.99 994 8.20 281 10 45 8.11 693 9.99 996 8.11 696 15 55 8.20 407 9.99 994 8.20 413 5 10 8.11 853 9.99 996 8.11 857 50 10 8.20 538 9.99 994 8.20 544 50 20 8.12 0139.99 996:8.12 017 40 20 8.20 669 9.99 994 8.20 675 40 30 8.12 1729.99 996 8.12 176 30 30 8.20 800 9.99 994 8.20 806 30 40 8.12 331 9.99 996 8.12 335 20 40 8.20 930 9.99 994 8.20 936 20 50 8.12 489 9.99 996 8.12 493 10 50 8.21 060 9.99 994 8.21 066 10 46 8.12 647 9.99 996 8.12 651 14 56 8.21 189 9.99 994 8.21 195 4 10 8.12 804 9.99 996 8.12 808 50 10 8.21 319 9.99 994 8.21 324 50 20 8.12 961 9.99 996 8.12 965 40 20 8.21 447 9.99 994 8.21 453 40 30 8.13 117 9.99 996 8.13 121 30 30 8.21 576 9.99 994 8.21 581 30 40 8.13 272 9.99 996 8.13 276 20 40 8.21 703 9.99 994 8.21 709 20 50 8.13 427 9.99 996 8.13 431 10 50 8.21 831 9.99 994 8.21 837 10 47 8.13 581 9.99 996 8.13 585 13 57 8.21 958 9.99 994 8.21 964 3 10 8.13 735 9.99 996 8.13 739 50 10 8.22 085 9.99 994 8.22 091 50 20 8.13 888 9.99 996 8.13 892 40 20 8.22 211 9.99 994 8.22 217 40 30 8.14 041 9.99 996 8.14 045 30 30 8.22 337 9.99 994 8.22 343 30 40 8.14 193 9.99 996 8.14 197 20 40 8.22 463 9.99 994 8.22 469 20 50 8.14 344 9.99 996 8.14 348 10 50 8.22 588 9.99 994 8.22 595 10 48 8.14 495 9.99 996 8.14 500 12 58 8.22 713 9.99 994 8.22 720 2 10 8.14 646 9.99 99* i 8.14 650 50 10 8.22 838 9.99 994 8.22 844 50 20 8.14 796 9.99 996 8.14 800 40 20 8.22 962 9.99 994 8.22 96S 40 30 8.14 945 9.99 996,8.14 950 30 30 8.23 086 9.99 994 8.23 092 30 40 8.15 094 9.99 996 8.15 099 20 40 8.23 210 9.99 994 8.23 216 20 50 8.15 243 9.99 996 8.15 247 10 50 8.23 333 9.99 994 8.23 339 10 49 8.15 391 9.99 996 8.15 395 11 59 8.23 456 9.99 994 8.23 462 1 10 8.15 538 9.99 996 8.15 543 50 10 S.23 578 9.99 994 8.23 585 50 20 8.15 685 9.99 996 8.15 690 40 20 S.23 700 9.99 9<>4 8.23 707 40 30 8.15 832 9.99 996 8.15 836 30 30 18.23 822 9.99 993 8.23 829 30 40 8.15 978 9.99 995 8.15 982 20 40 S.23 944 9.99 993 8.23 950 20 50 8.16 123 9.99 995 8.16 128 10 50 S.24 065 9.99 993 8.24 071 10 60 o 8.16 268 9.99 995 8.16 273 10 60 S.24 186 9.99 993 8.24 192 / // LCos LSin L Cot " ' / // LCos LSin LCot u f 89 Logarithms of Trigon ometric Functions 8920' -10sh ould be w ritten aft( jr ever j r logaril ,hm taken from this 5 page. II] 1 Logarithms of Trigonometric Functions -120' 27 / // LSin LCos LTan // / / // LSin LCos LTan // / 8.24 186 9.99 993 8.24 192 60 10 8.30 879 9.99 991 8.30 888 50 10 8.24 306 9.99 993 8.24 313 50 10 8.30 983 9.99 991 8.30 992( 50 20 8.24 426 9.99 993 8.24 433 40 20 8.31 0869.99 991:8.31 095j 40 30 8.24 546 9.99 993 8.24 553 30 30 8.31 188 9.99 9918.31 198 30 40 8.24 665 9.99 9938.24 672 20 40 8.31 2919.99 9918.31 300 20 50 8.24 785 9.99 993 8.24 791 10 50 8.31 393 9.99 991J8.31 403 10 1 8.24 903 9.99 993 8.24 910 59 11 8.31 4959.99 99118.31 505 49 10 8.25 022 9.99 993 8.25 029 50 10 8.31 597 9.99 991 8.31 606 50 20 8.25 1409.99 993 8.25 147 40 . 20 8.31 699 9.99 9918.31 708 40 30 8.25 258 9.99 993 8.25 265 30 30 8.31 800 9.99 9918.31 809 30 40 8.25 375 9.99 993J8.25 382 20 40 8.31 9019.99 9918.31 911 20 50 8.25 493.9.99 993 8.25 500 10 50 8.32 0029.99 9918.32 012 10 2 8.25 609 9.99 9938.25 616 58 12 8.32 103 9.99 9908.32 112 48 10 8.25 7269.99 993 8.25 733 50 10 8.32 203 9.99 9908.32 213 50 20 8.25 84219.99 993 8.25 849 40 20 8.32 303 9.99 990 8.32 313 40 30 8.25 9589.99 9938.25 965 30 30 8.32 403 9.99 990 8.32 413 30 40 8.26 074:9.99 993j8.26 081 20 40 8.32 503 9.99 990 8.32 513 20 50 8.26 189 8.99 993.8.26 196 10 50,8.32 602 9.99 990 8.32 612 10 1 3 8.26 304 9.99 993 8.26 312 57 13 8.32 702 9.99 990 8.32 711 47 10 8.26 419 9.99 993 8.26 426 50 10 8.32 8019.99 990 8.32 811 50 20 8.26 5339.99 993 8.26 541 40 20(8.32 899 9.99 990 8.32 909 40 30 8.26 648 9.99 993 8.26 655 30 30 8.32 998 9.99 990 8.33 008: 30 40 8.26 7619.99 993 8.26 769 20 40 8.33 096 9.99 9908.33 106 20 50 8.26 875 9.99 993 8.26 882 10 50 8.33 195 9.99 990 8.33 205 10 4 8.26 988 9.99 992 8.26 996 56 14 8.33 292 9.99 990'8.33 302 46 10 8.27 101 9.99 992 8.27 109 50 10 8.33 390(9.99 990 8.33 400| 50 20 8.27 214 9.99 99218.27 221 40 20 8.33 488 9.99 9908.33 498 40 30 8.27 326 9.99 992 8.27 334 30 30 8.33 5859.99 990 8.33 595 30 40 8.27 438,9.99 992 8.27 446! 20 40 8.33 682 9.99 9908.33 692 20 50 8.27 550 9.99 992 8.27 558 10 50 8.33 779 9.99 990,8.33 789 10 5 8.27 661 9.99 992 8.27 669 55 15 8.33 87519.99 990 8.33 886 45 10 8.27 773 9.99 992(8.27 780 50 10 18.33 972(9.99 990 8.33 982 50 20 8.27 883(9.99 992|8.27 891 ! 40 20 8.34 068 9.99 990 8.34 078 40 30 8.27 994 9.99 992(8.28 002| 30 30 18.34 164(9.99 990(8.34 174 30 40 8.28 104 9.99 992 8.28 112 20 40 |8.34 260(9.99 989(8.34 270 20 50 8.28 215 9.99 992 8.28 223 10 50 8.34 355 9.99 989i8.34 366 10 6 8.28 324 9.99 992 8.28 332| 54 16 8.34 450 9.99 989'8.34 461 44 10 8.28 434 9.99 992 8.28 4421 50 10 8.34 546 9.99 989 8.34 556 50 20 8.28 5439.99 992 8.28 551140 20 8.34 640 9.99 989 8.34 651 40 30 8.28 65219.99 9928.28 660 30 30 8.34 735 9.99 989 8.34 746 30 40 8.28 761 9. 99 992 8.28 769 20 40 8.34 830 9.99 989 8.34 840 20 50 8.28 869i9.99 992 8.28 877 10 50 8.34 924 9.99 989j8.34 935 10 7 8.28 977 9.99 992 8.28 986 53 17 8.35 018 9.99 989 8.35 029 43 10 8.29 085 9.99 992(8.29 094 50 10 8.35 112J9.99 9898.35 123 50 20 8.29 193 9.99 992(8.29 201 40 20 8.35 206 9.99 989 8.35 217 40 30 8.29 300 9.99 992 8.29 309 30 30 8.35 299 9.99 989 8.35 310 30 40 8.29 407 9.99 992 8.29 416 20 40 8.35 392 9.99 989 8.35 403 20 50 8.29 514 9.99 992 8.29 523 10 50 8.35 485 9.99 989 8.35 497 10 8 8.29 621 9.99 992 8.29 629 52 18 8.35 578 9.99 989 8.35 590 42 10 8.29 727 9.99 991 8.29 736 50 10 8.35 671 9.99 989 8.35 682 50 20 8.29 833 9.99 9918.29 842i 40 20 8.35 764 9.99 989 8.35 775 40 30 18.29 939 9.99 9918.29 947 30 30 8.35 85619.99 989 8.35 867 30 40i8.30 044 9.99 9918.30 053 20 40 8.35 948 9.99 989 8.35 959 20 50 8.30 150|9.99 991j8.30 158 10 50 8.36 040 9v99 989 8.36 051 10 9 8.30 255 9.99 991 8.30 263 51 19 8.36 131 9.99 989 8.36 143 41 10 8.30 359 9.99 9918.30 368 50 10 8.36 223 9.99 988 8.36 235 50 20 8.30 464 9.99 9918.30 473 40 20 8.36 314 9.99 988(8.36 326 40 30 8.30 568 9.99 9918.30 577 30 30 8.36 405 9.99 98818.36 417 30 40 8.30 6729.99 991 8.30 681 20 40 8.36 496 9.99 988 8.36 508 20 50 8.30 776 9.99 991 j 8. 30 785 10 50 8.36 587 9.99 988 8.36 599 10 10 8.30 879 9.99 991 8.30 888 50 20 8.36 678 9.99 988 8.36 689 40 / // LCos LSin LCot // / / // LCos LSin LCot // / 8840' Logarithms of Trigonometric Functions 8860 / 10 should be written after every logarithm taken from this page. 28 120'- -Logarithms of Trigonometric Functions 140' [II t // LSin LCos LTan // f / // LSin LCos LTan // / 20 8.36 678 9.99 988 8.36 689 40 30 8.41 792 9.99 985 8.41 807 30 10 18.36 7689.99 9888.36 780 50 10 8.41 87219.99 985 8.41 887 50 20 8.30 S58 9.99 988 8.36 870 40 20 8.41 952!9.99 985 8.41 967 40 30 8 . 36 948 9 . 99 988 8 . 36 960 30 30 8.42 032 9.99 985 8.42 048 30 40 8.37 038 9.99 988 8.37 050 20 40 8.42 112 9.99 985 8.42 127 20 50 8.37 128,9.99 988,8.37 140 10 50 8.42 192 9.99 985 8.42 207 10 21 8.37 217 9.99 98818.37 229 39 31 8.42 272 9.99 985 8.42 287 29 10 8.37 306 9.99 988 8.37 318 50 10 8.42 351 9.99 985 8.42 366 50 20 8.37 395 9.99 988 8.37 408 40 20 8.42 430 9.99 985 8.42 446 40 30 18 . 37 484 9 . 99 988 8 . 37 497 30 30 8.42 51019.99 985 8.42 525 30 40 8.37 573 9.99 988 8.37 585 20 40 8.42 589 ! 9.99 985 8.42 606 20 50 8.37 6629.99 988j8.37 674 10 50 8.42 667 9.99 985 8.42 683 10 22 0i8.37 75o'9.99 988'8.37 762 38 32 8.42 746 9.99 984 8.42 762 28 10 8.37 838 9.99 988 8.37 850 50 10 8.42 825 9.99 984,8.42 840 50 20 8.37 926 9.99 988 8.37 938 40 20 8.42 903 9.99 984 8.42 919 40 30 8.38 014 9.99 987 8.38 026 30 30 18.42 982 9.99 984 8.42 997 30 40 8.38 1019.99 987 8.38 114 20 40 8.43 060 9.99 984 8.43 075 20 50 8.38 189,9.99 987 8.38 202 10 50 8.43 138 9.99 984 8.43 154 10 23 8.38 276^9.99 987 8.38 289 37 33 8.43 216 9.99 984 8.43 232 27 10 8.38 363 9.99 987 8.38 376 50 10 8.43 293 9.99 984 8.43 309 50 20 8.38 450 9.99 987 8.38 463 40 20 8.43 371 9.99 984 8.43 387 40 30 18.38 537 9.99 987S8.38 550: 30 30 8.43 448 9.99 98418.43 464 30 40 i8.38 624 9.99 987 8.38 636 20 40 8.43 526 9.99 984 8.43 542 20 50i8.38 710|9.99 987 8.38 723 10 50 8.43 603 9.99 984 8.43 619 10 24 '8.38 796 9.99 987 8.38 809 36 34 8.43 680 9.99 984 8.43 696 26 10 8.38 882 9.99 987 8.38 895 50 10 8.43 757 9.99 984 8.43 773 50 20 8.38 968 9.99 987 8.38 981; 40 20 8.43 834 9.99 98418.43 850 40 30 8 . 39 054 9 . 99 987 8.39 067| 30 30 8.43 910 9.99 984 8.43 927 30 40 |8.39 139 9.99 987 8.39 153 20 40 8.43 987 9.99 984 .44 003 20 50 8.39 225 9.99 987 8.39 238 10 50 8.44 063 9.99 983 8.44 080 10 25 '8.39 310 9.99 987 8.39 323 35 35 8.44 139 9.99 983 8.44 156 25 10 8.39 395J9.99 987 8.39 408 50 10 18.44 216 9.99 983 8.44 232 50 20 8 . 39 480 9 . 99 987 8 . 39 493i 40 20 8.44 292 9.99 983 8.44 308 40 30 8.39 565:9.99 987 8.39 587 30 30 8.44 367 9.99 983 8.44 384 30 40 8.39 649 9.99 987 8.39 663i 20 40 |8.44 443 9.99 983 8:44 460 20 50 8.39 734 9.99 986 8.39 747 10 50 8.44 519 9.99 983 8.44 536 10 26 8.39 818 9.99 986 8.39 832 34 36 18 . 44 594 9.99 983 8.44 611 24 10 8.39 902 9.99 986 8.39 916 50 10 8 . 44 669 9.99 983 8.44 686 50 20 8.39 986 9.99 9868. 40 000 40 20 8 . 44 745 9 . 99 983 8.44 762 40 30 8 . 40 070 9 . 99 986 8 . 40 083' 30 30 8 . 44 820 9.99 983 8.44 837 30 40 8.40 153 9.99 986 8.40 167 20 40 8.44 895 9.99 983 8.44 912 20 50 8.40 237 9.99 986 8.40 251 10 50 8.44 969 9.99 983 8.44 987 10 27 8.40 320 9.99 986 8.40 334 33 37 8.45 044 9.99 983 8.45 061 23 10 8.40 403 9.99 986 8.40 417 50 10 8.45 119 9.99 983 8.45 136 50 20 8.40 486 9.99 98618.40 500 40 20 8.45 193 9.99 983 8.45 210 40 30 8.40 569 9.99 986:8.40 583 30 30 8.45 26719.99 983 8.45 285 30 40 8.40 651 9.99 986 8.40 665 20 40 8.45 3419.99 982 8.45 359 20 50 8.40 734 9.99 986 8.40 748 10 50 8.45 415 9.99 982 8.45 433 10 28 8.40 816 9.99 986 8.40 830 32 38 8.45 489 9.99 982 8.45 507 22 10 8.40 898 9.99 986 8.40 913 50 10 8.45 563 9.99 982 8.45 581 50 20 8.40 980-9.99 986 8 . 40 995 40 20 8.45 637 9.99 982|8.45 655 40 30 8.41 062 9.99 986 8.41 077! 30 30 8.45 7109.99 9828.45 728 30 40 8.41 144 9.99 986 8.41 158 20 40 8.45 784 9.99 982 8.45 802 20 50 8.41 225 9.99 986 8.41 240, 10 50 8.45 857,9.99 982 8.45 875 10 29 8.41 307 9.99 985 8.41 321 31 39 8.45 930 9.99 982 8.45 948 21 10 8.41 388 9.99 985|8.41 403 50 10 8.46 003 9.99 982 8.46 021 50 20 8.41 469 9.99 9858.41 484 40 20 8 . 46 076 9.99 982 8 . 46 094 40 30 8.41 550 9.99 985 8.41 565 30 30 8.46 149 9.99 982 8.46 167 30 40 8.41 6319.99 985 8.41 646 20 40 8.46 222 9.99 982 8.46 240 20 50 8.41 711j9.99 985 8.41 726 10 50 8.46 294 9.99 982|8.46 312 1 10 30 8.41 792 9.99 985 8.41 807 30 40 8.46 366 9.99 982 8.40 385 20 / // l LCos LSin LCot // / I n LCos LSin LCot // / 8820' Logarithms of Trigonometric Functions 8840' 10 should be written after every logarithm taken from this page. II] 140'- -Logarithms of Trigonometric Functions 2 29 / // LSin LCos L Tan ! " ' ' " j L Sin LCos LTan // / 40 8.46 366 9.99 982 8.46 385| 20 50 8.50 504 9.99 978 8.50 527 10 10 8.46 439 9.99 982 8.40 457 50 10 8.50 570 9.99 978 8.50 593 50 20 8.46 51119.99 982J8.46 529 40 20 8.50 636 9.99 978 8.50 658 40 30 8.46 5839.99 98118.46 602 30 30 8.50 701 9.99 978 8.50 724 30 40 8.46 655|9.99 981 8.46 674 20 40 8.50 767 9 . 99 977 8 . 50 789 20 50 8.46 72719.99 981 8.46 745 10 50,8.50 8329.99 977 8.50 855 10 41 8.46 79919.99 98118.46 817 19 51 18.50 8979.99 977 8.50 920 9 10 8.46 870 9.99 98l|8.46 889 50 10|8.50 963 9.99 977 8.50 985 50 20 8.46 9429.99 981J8.46 960 40 20 8.51 028 9.99 977 8.51 050 40 30 8.47^013 9.99 9818.47 032 30 30J8.51 092 9.99 977 8.51 015 30 40 8.47 084 9.99 981 8.47 103 20 40 8.51 157 9.99 977 8.51 180 20 50 8.47 155,9.99 981 8.47 174 10 50 8.51 222i9.99 977,8.51 245 10 42 8.47 226:9.99 981 8.47 245 18 52 8.51 287'9.99 9778.51 310 8 10 8.47 29719.99 981 8.47 316 50 10 8.51 3519.99 977 8.51 374 50 20 8.47 368,9.99 981 8.47 387 40 20 8.51 416 9.99 977 8.51 439j 40 30 8.47 439 9.99 981J8.47 458 30 30 8.51 480 9.99 977 8.51 503 30 40 8.47 509 9.99 981i8.47 528 20 40 8.51 544 9.99 977 8.51 568' 20 50 8.47 580 9.99 981 8.47 599 10 50 8.51 609,9.99 977.8.51 632 10 43 8.47 650 9.99 981 8.47 669 17 53 8.51 673'9.99 977 8.51 696 7 10 8.47 720 9.99 980 8.47 740 50 10 8.51 737|9.99 976 8.51 760 50 20 8.47 790 9.99 980 8.47 810 40 20 8.51 8019.99 976 8.51 824 40 30 8.47 860 9.99 980 8.47 880| 30 30 8.51 864 9.99 976 8.51 888 30 40 8.47 930 9.99 980 8.47 950 20 40 8.51 928 9.99 976 8.51 952 20 50 8.48 000 9.99 980 8.48 020 10 50 8.51 992 9.99 976 8.52 015 10 44 8.48 096 9.99 980 8.48 090 16 54 8.52 055 9.99 976 8.52 079 6 10 8.48 139J9.99 980 8.48 159 50 10 8.52 119 9.99 976 8.52 143 50 20 8.48 20819.99 980 8.48 228 40 20 8.52 1829.99 9768.52 206 40 30 8.48 278 9.99 980 8.48 298 30 30 8.52 245 9.99 976 8.52 269 30 40 8.48 347 9.99 980 8.48 367 20 40 8.52 3089.99 976 8.52 332 20 50 8.48 416 9.99 980 8.48 436 10 50 8.52 371 9.99 976 8.52 396 10 45 8. 48 485 9.99 980 8.48 505 15 55 8.52 434 9.99 976 8.52 459 5 10 8.48 554 9.99 980 8.48 574 50 10 8.52 49719.99 976 8.52 522 50 20 8.48 622 9.99 980 8.48 643 40 20 8.52 560|9.99 976 8.52 584 40 30 8.48 691 9.99 980 8.48 711 30 30 8.52 623 9.99 975 8.52 647 30 40 8 . 48 760 9.99 979 8.48 780! 20 40 8 . 52 685 9 . 99 975 8.52 710 20 50 8.48 828 9.99 979 8.48 849 10 50 8.52 748 9.99 975 8.52 772 10 46 8.48 896 9.99 979 8.48 917 14 56 0'8.52 810 9.99 975 8.52 835 4 10 8.48 965 9.99 979 8.48 985 50 10 8.52 872 9.99 975 8.52 897 50 20 8.49 033 9.99 979 8.49 053 40 20 8.52 935 9.99 975 8.52 960 40 30 8.49 101 9.99 979 8.49 121 30 30 8.52 997 9.99 975 8.53 022 30 40 8.49 169 9.99 979 8.49 189 20 40 8.53 059 9.99 975 8.53 084 20 50 8.49 236 9.99 979 8.49 257 10 50 8.53 121 9.99 975 8.53 146 10 47 8.49 304 9.99 979 8.49 325 13 57 8.53 183 9.99 975 8.53 208 3 10 8.49 372 9.99 979 8.49 393 50 10 8.53 245 9.99 975 8.53 270 50 20 8.49 439 9.99 979 8.49 460 40 20 8.53 306 9.99 975 8.53 332 40 30 8.49 506 9.99 979 8.49 528 30 30 8.53 368 9.99 975 8.53 393 30 40 8.49 574 9.99 979 8.49 595 20 40 8.53 429 9.99 975 8.53 455 20 50 8.49 641 9.99 979 8.49 662 10 50 8.53 491 9.99 974 8.53 516 10 48 8.49 708 9.99 979 8.49 729 12 58 8.53 552 9.99 974 8.53 578 2 10 8.49 775 9.99 979 8.49 796 50 10 8.53 614 9.99 974 8.53 639 50 20 8.49 842 9.99 978 8.49 863 40 20 8.53 675 9.99 974 8.53 700 40 30 8.49 908 9.99 978 8.49 930 30 30 8.53 736 9.99 974 8.53 762 30 40 8.49 975 9.99 978 8.49 997 20 40 18.53 79719.99 974 8.53 823 20 50 8.50 042 9.99 978 8.50 063 10 50 8.53 858 9.99 974 8.53 884 10 49 8.50 108 9.99 978 8.50 130 11 59 8.53 919'9.99 974 8.53 945 1 10 8.50 174 9.99 978 8.50 196 50 10 8.53 979 9.99 974|8.54 005 50 20 8.50 241 9.99 978 8.50 263 40 20 8.54 04019.99 9748.54 066 40 30 8.50 307 9.99 97818.50 329 30 30 8.54 1019.99 974 8.54 127 30 40 8.50 373 9.99 978 8.50 395 20 40 8.54 16119.99 974 8.54 187 20 50 8.50 439 9.99 978 8.50 461 10 50 8.54 2229.99 974 8.54 248 10 J 50 8.50 504 9.99 978 8.50 527 10 60 8.54 282 9.99 974 8.54 308 LI LCos LSin L Cot " i n L Cos | L Sin LCot // / 88 Logarithms of Trigonometric Functions 8820 / 10 should be written after every logarithm taken from this page. 30 2 Logarithms of Trigonometric Functions LSin LTan cd L Cot L Cos PP o 1 2 3 4 5 6 7 8 9 lO 11 12 13 14 15 16 17 18 19 2 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 8.54 282 8.54 642 8.54 999 8.55 354 8.55 705 8.56 054 8.56 400 8.56 743 8.57 084 8.57 421 8.57 757 8.58 089 8.58 419 8.58 747 8.59 072 8.59 395 8.59 715 8.60 033 8.60 349 8.60 662 8.60 973 8.61 282 8.61 589 8.61 894 8.62 196 8.62 497 8.62 795 8.63 091 8.63 385 8.63 678 8.63 968 8.64 256 8.64 543 8.64 827 8.65 110 8.65 391 8.65 670 8.65 947 8.66 223 8.66 497 8.66 769 8.67 039 8.67 308 8.67 575 8.67 841 8.68 104 8.68 367 8.68 627 8.68 886 8.69 144 8.69 400 8.69 654 8.69 907 8.70 159 8.70 409 8.70 658 8.70 905 8.71 151 8.71 395 8.71 638 8.71 880 360 357 355 351 349 346 343 341 337 336 332 330 328 325 323 320 318 316 313 311 309 307 305 302 301 298 296 294 293 290 288 287 284 283 281 279 277 276 274 272 270 269 267 266 263 263 260 259 258 256 254 253 252 250 249 247 246 244 243 242 8.54 308 8.54 669 8.55 027 8.55 382 8.55 734 8.56 083 8.56 429 8.56 773 8.57 114 8.57 452 8.57 788 8.58 121 8.58 451 8.58 779 8.59 105 8.59 428 8.59 749 8.60 068 8.60 384 8.60 698 8.61 009 8.61 319 8.61 626 8.61 931 8.62 234 8.62 535 8.62 834 8.63 131 8.63 426 8.63 718 8.64 009 8.64 298 8.64 585 8.64 870 8.65 154 8.65 435 8.65 715 8.65 993 8.66 269 8.66 543 8.66 816 8.67 087 8.67 356 8.67 624 8.67 890 8.68 154 8.68 417 8.68 678 8.68 938 8.69 196 8.69 453 8.69 708 8.69 962 8.70 214 8.70 465 8.70 714 8.70 962 8.71 208 8.71 453 8.71 697 8.71 940 361 358 355 352 349 346 344 341 338 336 333 330 328 326 323 321 319 316 314 311 310 307 305 303 301 299 297 295 2*92 291 289 287 285 284 281 280 278 276 274 273 271 269 268 264 263 261 260 258 257 255 254 252 251 249 248 246 245 244 243 1.45 692 1.45 331 1.44 973 1.44 618 1.44 266 1.43 917 1.43 571 1.43 227 1.42 886 1.42 548 1.42 212 1.41 879 1.41 549 1.41 221 1.40 895 1.40 572 1.40 251 1.39 932 1.39 616 1.39 302 1.38 991 1.38 681 1.38 374 1.38 069 1.37 766 1.37 465 1.37 166 1.36 869 1.36 574 1.36 282 1.35 991 1.35 702 1.35 415 1.35 130 1.34 846 1.34 565 1.34 285 1.34 007 1.33 731 1.33 457 1.33 184 1.32 913 1.32 644 1.32 376 1.32 110 1.31 846 1.31 583 1.31 322 1.31 062 1.30 804 1.30 547 1.30 292 1.30 038 1.29 786 1.29 535 1.29 286 1.29 038 1.28 792 1.28 547 1.28 303 1.28 060 9.99 974 9.99 973 9.99 973 9.99 972 9.99 972 9.99 971 9.99 971 9.99 970 9.99 970 9.99 969 9.99 969 9.99 968 9.99 968 9.99 967 9.99 967 9.99 967 9.99 966 9.99 966 9.99 965 9.99 964 9.99 964 9.99 963 9.99 963 9.99 962 9.99 962 9.99 961 9.99 961 9.99 960 9.99 960 9.99 959 9.99 959 9.99 958 9.99 958 9.99 957 9.99 956 9.99 956 9.99 955 9.99 955 9.99 954 9.99 954 9.99 953 9.99 952 9.99 952 9.99 951 9.99 951 9.99 950 9.99 949 9.99 949 9.99 948 9.99 948 9.99 947 9.99 946 9.99 946 9.99 945 9.99 944 9.99 944 9.99 943 9.99 942 9.99 942 9.99 941 9.99 940 <;c> 89 58 57 66 55 54 53 52 51 BO 19 48 47 46 15 44 43 42 41 to 39 38 37 36 34 33 32 31 SO 29 28 27 26 25 24 23 22 21 00 19 18 17 16 15 14 13 12 11 10 9 360 350 340 II KJ 1H6 170 201 1 36 35 :>. 72 70 8 108 105 4 144 140 5 180 175 a 216 210 7 252 245 S 288 280 9 324 315 330 33 272 306 320 310 4 132 128 R 165 160 fl 198 192 7 231 224 R 264 256 9 297 288 31 62 93 124 155 186 217 248 279 300 290 28S 28.5 57.0 85.5 114.0 142.5 171.0 199.5 228.0 256.5 280 275 270 1 30 29 2 60 5H 8 90 87 4 120 J 150 / 116 5 145 180 174 7 210 203 S 240 232 9 270 261 28.0 56.0 84.0 112.0 140.0 168.0 196.0 224.0 252.0 27.5 55.0 82.5 110.0 137.5 165.0 192.5 220.0 247.5 26.5 53.0 79.5 106.0 132.5 159.0 185.5 212.0 238.5 26.0 52.0 78.0 104.0 130.0 156.0 182.0 208.0 234.0 25.0 50.0 75.0 100.0 125.0 150.0 175.0 200.0 225.0 24.5 49.0 73.5 198.0 122.5 147.0 171.5 196.0 220.5 27.0 54.0 81.0 108.0 135.0 162.0 189.0 216.0 243.0 265 260 255 25.5 51.0 76.5 102.0 127.5 153.0 178.5 204.0 229.5 250 245 240 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 L Cos LCot cd LTan LSin PP 87 Logarithms of Trigonometric Functions -10 should be written after every logarithm taken mns_ In the remaining part of table II frnin t.h first-. fl*vnniT n.nH fnnrt.h nnli 3 Logarithms of Trigonometric Functions 31 LSin d L Tan c d L Cot LCos PP 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 a4 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 8.71 880 8.72 120 8.72 359 8.72 597 8.72 834 8.73 069 8.73 303 8.73 535 8.73 767 8.73 997 8.74 226 8.74 454 8.74 680 8.74 906 8.75 130 8.75 353 8.75 575 8.75 795 8.76 015 8.76 234 8.76 451 8.76 667 8.76 883 8.77 097 8.77 310 8.77 522 8.77 733 8.77 943 8.78 152 8.78 360 8.78 568 8.78 774 8.78 979 8.79 183 8.79 386 8.79 588 8.79 789 8.79 990 8.80 189 8.80 388 8.80 585 8.80 782 8.80 978 8.81 173 8.81 367 8.81 560 8.81 752 8.81 944 8.82 134 8.82 324 8.82 513 8.82 701 8.82 888 8.83 075 8.83 261 8.83 446 8.83 630 8.83 813 8.83 996 8.84 177 8.84 358 240 239 238 237 235 234 232 232 230 229 228 226 226 224 223 222 220 220 219 217 216 216 214 213 212 211 210 209 208 208 206 205 204 203 202 201 201 199 199 197 197 196 195 194 193 192 192 190 190 189 188 187 187 186 185 184 183 183 181 181 8.71 940 8.72 181 8.72 420 8.72 659 8.72 896 8.73 132 8.73 366 8.73 600 8.73 832 8.74 063 8.74 292 8.74 521 8.74 748 8.74 974 8.75 199 8.75 423 8,75 645 8.75 867 8.76 087 8.76 306 8.76 525 8.76 742 8.76 958 8.77 173 8.77 387 8.77 600 8.77 811 8.78 022 8.78 232 8.78 441 8.78 649 8.78 855 8.79 061 8.79 266 8.79 470 8.79 673 8.79 875 8.80 076 8.80 277 8.80 476 8.80 674 8.80 872 8.81 068 8.81 264 8.81 459 8.81 653 8.81 846 8.82 038 8.82 230 8.82 420 8.82 610 8.82 799 8.82 987 8.83 175 8.83 361 8.83 547 8.83 732 8.83 916 8.84 100 8.84 282 8.84 464 241 239 239 237 236 234 234 232 231 229 229 227 226 225 224 ,222 222 220 219 219 217 216 215 214 213 211 211 210 209 208 206 206 205 204 203 202 201 201 199 198 198 196 196 195 194 193 192 192 190 190 189 188 188 186 186 185 184 184 182 182 1.28 060 1.27 819 1.27 580 1.27 341 1.27 104 1.26 868 1.26 634 1.26 400 1.26 168 1.25 937 1.25 708 1.25 479 1.25 252 1.25 026 1.24 801 1.24 577 1.24 355 1.24 133 1.23 913 1.23 694 1.23 475 1.23 258 1.23 042 1.22 827 1.22 613 1.22 400 1.22 189 1.21 978 1.21 768 1.21 559 1.21 351 1.21 145 1.20 939 1.20 734 1.20 530 1.20 327 1.20 125 1.19 924 1.19 723 1.19 524 1.19 326 1.19 128 1.18 932 1.18 736 1.18 541 18 347 18 154 17 962 17 770 17 580 1.17 390 1.17 201 1.17 013 1.16 825 1.16 639 1.16 453 1.16 268 1.16 084 1.15 900 1.15 718 1.15 536 9.99 940 9.99 940 9.99 939 9.99 938 9.99 938 9.99 937 9.99 936 9.99 936 9.99 935 9.99 934 9.99 934 9.99 933 9.99 932 9.99 932 9.99 931 9.99 930 9.99 929 9.99 929 9.99 928 9.99 927 9.99 926 9.99 926 9.99 925 9.99 924 9.99 923 9.99 923 9.99 922 9.99 921 9.99 920 9.99 920 9.99 919 9.99 918 9.99 917 9.99 917 9.99 916 9.99 915 9.99 914 9.99 913 9.99 913 9.99 912 9.99 911 9.99 910 9.99 909 9.99 909 9.99 908 9.99 907 9.99 906 9.99 905 9.99 904 9.99 904 9.99 903 9.99 902 9.99 901 9.99 900 9.99 899 9.99 898 9.99 898 9.99 897 9.99 896 9.99 895 9.99 894 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 20 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 241 239 237 236 234 1 24.1 239 23.7 23.6 23.4 :' 48.2 47.8 47.4 47.2 46.8 3 72.3 71.7 71.1 70.8 70.2 4 96.4 95.6 94.8 94.4 93.6 .> 120.5 119.5 118.5 118.0 117.0 144.6 143.4 142.2 141.6 140.4 7 168.7 167.3 165.9 165.2 163.8 H 192.8 191.2 189.6 188.8 187.2 9 216.9 215.1 213.3 212.4 210.6 232 231 229 227 226 1 23.2 23.1 22.9 22.7 22.6 '..' 46.4 46.2 45.8 45.4 45.2 H 69.6 69.3 68.7 68.1 67.8 4 92.8 92.4 91.6 90.8 90.4 .-> 116.0 115.5 114.5 113.5 113.0 fi 139.2 138.6 137.4 136.2 135.6 7 162.4 161.7 160.3 158.9 158.2 8 185.6 184.8 183.2 181.6 180.8 9 208.8 207.9 206.1 204.3 203.4 224 222 220 219 217 22.4 44.8 67.2 89.6 112.0 134.4 156.8 179.2 201.6 22.2 44.4 2-2.0 44.0 66.6 66.0 88.8 88.0 111.0 110.0 133.2 132.0 155.4 154.0 177.6 176.0 199.8 198.0 21.9 21.7 43.8 43.4 65.7 65.1 87.6 86.8 109.5 108.5 131.4 130.2 153.3 151.9 175.2 173.6 197.1 195.3 216 214 213 211 209 21.6 21.4 21.3 21.1 20.9 43.2 42.8 42.6 ^2.2 41.8 64.8 64.2 63.9 63.3 62.7 86.4 85.6 85.2 84.4 83.6 108.0 107.0 106.5 105.5 104.5 129.6 128.4 127.8 126.6 125.4 151.2 149.8 149.1 147.7 146.3 172.8 171.2 170.4 168.8 167.2 194.4 192.6 191.7 189.9 188.1 208 206 203 201 199 20.8 20.6 20.3 20.1 19.9 41.6 41.2 40.6 40.2 39.8 62.4 61.8 60.9 60.3 59.7 83.2 82.4 81.2 80.4 79.6 104.0 103.0 101.5 100.5 99.5 124.8 1216 121.8 120.6 119.4 145.6 144.2 142.1 140.7 139.3 166.4 164.8 162.4 160.8 159.2 187.2 185.4 182.7 180.9 179.1 198 196 194 192 190 19.8 19.6 19.4 19.2 19.0 39.6 39.2 38.8 38.4 38.0 59.4 58.8 58.2 57.6 57.0 79.2 78.4 77.6 76.8 76.0 99.0 98.0 97.0 96.0 95.0 118.8 117.6 116.4 115.2 114.0 138.6 137.2 135.8 134.4 133.0 158.4 156.8 155.2 153.6 152.0 178.2 176.4 174.6 172.8 171.0 188 186 184 182 181 18.8 18.6 18.4 37.6 37.2 36.8 56.4 55.8 55.2 75.2 74.4 73.6 94.0 93.0 92.0 112.8 111.6 110.4 131.6 130.2 128.8 18.2 18.1 36.4 36.2 54.6 54.3 72.8 72.4 91.0 90.5 109.2 108.6 127.4 126.7 150.4 148.8 147.2 145.6 144.8 169.2 167.4 165.6 163.8 162.9 LCos d LCot cd LTan LSin PP 86 Logarithms of Trigonometric Functions 32 4 Logarithms of Trigonometric Functions III t LSin d LTan cd LCot L Cos PP 8.84 358 181 179 179 178 8.84 464 182 180 180 179 1.15 536 9.99 894 oo 1 8.84 539 8.84 646 1.15 354 9.99 893 59 182 181 180 179 178 2 8.84 718 8.84 826 1.15 174 9.99 892 58 1 18.2 18.1 18.0 17.9 17 8 3 8.84 897 8.85 006 1.14 994 9.99 891 57 > 36.4 36.2 36.0 35.8 36.6 4 8.85 075 8.85 185 1.14 815 9.99 891 56 3 54.6 54.3 54.0 53.7 53.4 177 178 4 72.8 72.4 72.0 71.6 71.2 5 6 8.85 252 8.85 429 177 176 8.85 363 8.85 540 -477 177 1.14 637 1.14 460 9.99 890 9.99 889 55 54 8 7 91.0 90.5 90.0 89.5 89.0 109.2 108.6 108.0 107.4 106.8 127.4 126.7 126.0 125.3 124.6 7 8.85 605 175 175 8.85 717 176 176 1.14 283 9.99 888 53 8 145.6 144.8 144.0 143.2 142.4 8 8.85 780 8.85 893 1.14 107 9.99 887 52 9 163.8 162.9 162.0 161.1 160.2 9 8.85 955 8.86 069 1.13 931 9.99 886 51 io 8.86 128 173 173 173 171 8.86 243 174 174 174 172 172 1.13 757 9.99 885 50 177 176 175 174 173 11 8.86 301 8.86 417 1.13 583 9.99 884 49 1 2 3 17.7 17.6 17.5 17.4 17.3 35.4 85.2 35.0 34.8 34.6 53.1 52 8 52 5 52 2 51 9 12 8.86 474 8.86 591 1.13 409 9.99 883 48 13 8.86 645 171 8.86 763 1.13 237 9.99 882 47 4 70.8 70.4 70.0 69.6 69! 2 14 8.86 816 8.86 935 1.13 065 9.99 881 46 "> 88,5 88.0 87.5 87.0 86.5 171 171 6 106.2 105.6 105.0 104.4 103.8 15 16 8.86 987 8.87 156 169 169 8.87 106 8.87 277 171 170 1.12 894 1.12 723 9.99 880 9.99 879 45 44 7 K 9 123.9 123.2 122.5 121.8 121.1 141.6 140.8 140.0 139.2 138.4 159.3 158.4 157.5 156.6 155.7 17 8.87 325 169 167 8.87 447 169 169 1.12 553 9.99 879 43 18 8.87 494 8.87 616 1.12 384 9.99 878 42 19 8.87 661 8.87 785 1.12 215 9.99 877 41 172 171 170 169 168 168 168 1 17.2 17.1 17.0 16.9 16.8 20 8.87 829 166 166 165 8.87 953 167 167 166 1.12 047 9.99 876 40 2 34.4 34.2 34.0 33.8 33.6 21 8.87 995 8.88 120 1.11 880 9.99 875 39 3 51.6 51.3 51.0 50.7 50.4 22 8.88 161 8.88 287 1.11 713 9.99 874 38 4 5 68.8 68.4 68.0 67.6 67.2 86 85 5 85 84 5 84 23 8.88 326 164 8.88 453 165 1.11 547 9.99 873 37 6 103^ 10216 102i0 10L4 10018 24 8.88 490 8.88 618 1.11 382 9.99 872 36 7 120.4 119.7 119.0 118.3 117.6 164 165 8 137.6 136.8 136.0 135.2 134.4 25 8.88 654 163 163 162 8.88 783 165 163 163 163 1.11 217 9.99 871 35 9 154.8 153.9 153.0 152.1 151.2 26 8.88 817 8.88 948 1.11 052 9.99 870 34 27 8.88 980 8.89 111 1.10 889 9.99 869 33 167 166 165 164 163 28 29 8.89 142 8.89 304 162 8.89 274 8.89 437 1.10 726 1.10 563 9.99 868 9.99 867 32 31 1 2 16.7 16.6 16.5 16.4 16.3 33.4 33.2 33.0 32.8 32.6 160 161 3 50.1 49.8 49.5 49.2 48.9 30 8.89 464 161 159 159 159 8.89 598 162 160 160 160 159 158 158 157 157 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055 9.18 560 0.81 440 9.99 495 17 5 42.0 41.5 44 9.18 137 9.18 644 0.81 356 9.99 494 16 6 50.4 49.8 83 84 7 58.8 58.1 45 9.18 220 82 81 82 82 9.18 728 84 84 83 84 0.81 272 9.99 492 15 8 67.2 66.4 46 9.18 302 9.18 812 0.81 188 9.99 490 14 9 75.6 74.7 47 9.18 383 9.18 896 0.81 104 9.99 488 13 48 9.18 465 9.18 979 0.81 021 9.99 486 12 49 9.18 547 9.19 063 0.80 937 9.99 484 11 81 83 82 81 80 50 9.18 628 9.19 146 83 83 83 83 0.80 854 9.99 482 IO 51 9.18 709 81 9.19 229 0.80 771 9.99 480 9 1 8.2 8.1 8.0 52 9.18 790 81 9.19 312 0.80 688 9.99 478 8 2 16.4 16.2 16.0 53 9.18 871 81 9.19 395 0.80 605 9.99 476 7 3 24.6 24.3 24.0 54 9.18 952 81 9.19 478 0.80 522 9.99 474 6 4 32.8 32.4 32.0 81 83 5 11.0 40.5 40.0 55 9.19 033 9.19 561 82 82 0.80 439 9.99 472 5 6 19.2 48.6 48.0 66 9.19 113 80 9.19 643 0.80 357 9.99 470 4 7 37.4 56.7 56.0 57 9.19 193 80 9.19 725 0.80 275 9.99 468 3 8 35.6 64.8 64.0 58 9.19 273 80 9.19 807 82 82 0.80 193 9.99 466 2 9 73.8 72.9 72.0 59 9.19 353 80 9.19 889 0.80 111 9.99 464 1 80 82 60 9.19 433 9.19 971 0.80 029 9.99 462 O LCos d LCot cd LTan L Sin / PP 81 Logarithms of Trigonometric Functions 9 Logarithms of Trigonometric Functions 37 1 LSin d LTan c d LCot LCos PP o 9.19 433 80 79 80 79 9.19 971 82 81 82 81 0.80 029 9.99 462 60 1 9.19 513 9.20 053 0.79 947 9.99 460 59 2 9.19 592 9.20 134 0.79 866 9.99 458 58 3 9.19 672 9.20 216 0.79 784 9.99 456 57 4 9.19 751 9.20 297 0.79 703 9.99 454 56 79 81 - 5 9.19 830 79 79 79 78 9.20 378 81 81 81 80 0.79 622 9.99 452 55 82 81 80 6 9.19 909 9.20 459 0.79 541 9.99 450 54 7 9.19 988 9.20 540 0.79 460 9.99 448 53 1 8.2 8.1 8.0 8 9.20 067 9.20 621 0.79 379 9.99 446 52 o 16.4 16.2 16.0 9 9.20 145 9.20 701 0.79 299 9.99 444 51 3 24.6 24.3 24.0 78 81 4 32.8 32.4 32.0 io 9.20 223 79 78 78 77 9.20 782 80 80 80 80 0.79 218 9.99 442 50 5 41.0 40.5 40.0 11 9.20 302 9.20 862 0.79 138 9.99 440 49 6 49.2 48.6 48.0 12 9.20 380 9.20 942 0.79 058 9.99 438 48 7 57.4 56.7 56.0 13 9.20 458 9.21 022 0.78 978 9.99 436 47 8 65.6 64.8 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639 9.99 400 30 31 9.21 836 9.22 438 0.77 562 9.99 398 29 32 9.21 912 9.22 516 0.77 484 9.99 396 28 33 9.21 987 9.22 593 0.77 407 9.99 394 27 34 9.22 062 9.22 670 0.77 330 9.99 392 26 76 75 74 75 V 77 35 9.22 137 74 75 75 74 9.22 747 77 77 76 77 0.77 253 9.99 390 25 1 7.6 7.5 7.4 36 9.22 211 9.22 824 0.77 176 9.99 388 24 2 15.2 15.0 14.8 37 9.22 286 9.22 901 0.77 099 9.99 385 23 3 22.8 22.5 22.2 38 9.22 361 9.22 977 0.77 023 9.99 383 22 4 30.4 30.0 29.6 39 9.22 435 9.23 054 0.76 946 9.99 381 21 5 38.0 37.5 37.0 74 76 6 45.6 45.0 44.4 40 9.22 509 74 74 74 74 9.23 130 76 77 76 76 0.76 870 9.99 379 20 7 53.2 52.5 51.8 41 9.22 583 9.23 206 0.76 794 9.99 377 19 8 60.8 60.0 59.2 42 9.22 657 9.23 283 0.76 717 9.99 375 18 9 68.4 67.5 66.6 43 9.22 731 9.23 359 0.76 641 9.99 372 17 44 9.22 805 9.23 435 0.76 565 9.99 370 16 73 75 45 9.22 878 74 73 73 73 9.23 510 76 75 76 75 0.76 490 9.99 368 15 46 9.22 952 9.23 586 0.76 414 9.99 366 14 47 9.23 025 9.23 661 0.76 339 9.99 364 13 48 9.23 098 9.23 737 0.76 263 9.99 362 12 73 72 71 49 9.23 171 9.23 812 0.76 188 9.99 359 11 73 75 1 7.3 7.2 7.1 50 9.23 244 73 73 72 73 9.23 887 75 75 75 74 0.76 113 9.99 357 IO 2 14.6 14.4 14.2 51 9.23 317 9.23 962 0.76 038 9.99 355 9 3 21.9 21.6 21.3 52 9.23 390 9.24 037 0.75 963 9.99 353 8 4 29.2 28.8 28.4 53 9.23 462 9.24 112 0.75 888 9.99 351 7 5 36.5 36.0 35.5 54 9.23 535 9.24 186 0.75 814 9.99 348 6 G 43.8 43.2 42.6 72 75 7 51.1 50.4 49.7 55 9.23 607 72 73 71 72 9.24 261 74 75 74 74 0.75 739 9.99 346 5 8 58.4 57.6 56.8 56 9.23 679 9.24 335 0.75 665 9.99 344 4 9 65.7 64.8 63.9 57 9.23 752 9.24 410 0.75 590 9.99 342 3 58 9.23 823 9.24 484 0.75 516 9.99 340 2 59 9.23 895 9.24 558 0.75 442 9.99 337 1 72 74 60 9.23 967 9.24 632 0.75 368 9.99 335 O LCos d LCot cd LTan LSin / PP 80 Logarithms of Trigonometric Functions 38 10 Logarithms of Trigonometric Functions [II / LSin d LTan cd LCot LCos d PP o 9.23 967 72 71 71 72 9.24 632 74 73 74 73 0.75 368 9.99 335 2 2 3 2 60 1 9.24 039 9.24 706 0.75 294 9.99 333 59 2 9.24 110 9.24 779 0.75 221 9.99 331 58 3 9.24 181 9.24 853 0.75 147 9.99 328 57 4 9.24 253 9.24 926 0.75 074 9.99 326 56 71 74 2 5 9.24 324 71 71 70 71 9.25 000 73 73 73 73 0.75 000 9.99 324 2 3 2 o 55 74 73 72 6 9.24 395 9.25 073 0.74 927 9.99 322 54 7 9.24 466 9.25 146 0.74 854 9.99 319 53 1 7.4 7.3 7.2 8 9.24 536 9.25 219 0.74 781 9.99 317 52 2 14.8 14.6 14.4 9 9.24 607 9.25 292 0.74 708 9.99 315 51 3 22.2 21.9 21.6 70 73 2 4 29.6 29.2 28.8 io 9.24 677 71 70 70 70 9.25 365 72 73 72 73 0.74 635 9.99 313 3 2 2 2 50 5 37.0 36.5 36.0 11 9.24 748 9.25 437 0.74 563 9.99 310 49 6 44.4 43.8 43.2 12 9.24 818 9.25 51Q 0.74 490 9.99 308 48 7 51.8 51.1 50.4 13 9.24 888 9.25 582 0.74 418 9.99 306 47 8 59.2 58.4 57.6 14 9.24 958 9.25 655 0.74 345 9.99 304 46 9 66.6 65.7 64.8 70 72 3 15 9.25 028 70 70 69 70 9.25 727 72 72 72 72 0.74 273 9.99 301 2 2 3 2 45 16 9.25 098 9.25 799 0.74 201 9.99 299 44 17 9.25 168 9.25 871 0.74 129 9.99 297 43 18 9.25 237 9.25 943 0.74 057 9.99 294 42 19 9.25 307 9.26 015 0.73 985 9.99 292 41 71 70 69 69 71 2 SO 9.25 376 69 69 69 69 9.26 086 72 71 72 71 0.73 914 9.99 290 2 3 2 2 40 1 7.1 7.0 6.9 21 9.25 445 9.26 158 0.73 842 9.99 288 39 2 14.2 14.0 13.8 22 9.25 514 9.26 229 0.73 771 9.99 285 38 3 21.3 21.0 20.7 23 9.25 583 9.26 301 0.73 699 9.99 283 37 4 28.4 28.0 27.6 24 9.25 652 9.26 372 0.73 628 9.99 281 36 5 35.5 35.0 34.5 69 71 3 6 42.6 42.0 41.4 25 9.25 721 69 68 69 68 9.26 443 71 71 70 71 0.73 557 9.99 278 2 2 3 2 35 7 49.7 49.0 48.3 26 9.25 790 9.26 514 0.73 486 9.99 276 34 8 56.8 56.0 55.2 27 9.25 858 9.26 585 0.73 415 9.99 274 33 9 63.9 63.0 62.1 28 9.25 927 9.26 655 0.73 345 9.99 271 32 29 9.25 995 9.26 726 0.73 274 9.99 269 31 68 71 2 30 9.26 063 68 68 68 68 9.26 797 70 70 71 70 0.73 203 9.99 267 3 2 2 3 30 31 9.26 131 9.26 867 0.73 133 9.99 264 29 32 9.26 199 9.26 937 0.73 063 9.99 262 28 33 9.26 267 9.27 008 0.72 992 9.99 260 27 68 67 66 34 9.26 335 9.27 078 0.72 922 9.99 257 26 68 70 2 1 6.8 6.7 6.6 35 9.26 403 67 68 67 67 9.27 148 70 70 69 70 0.72 852 9.99 255 3 2 2 3 25 2 13.6 13.4 13.2 36 9.26 470 9.27 218 0.72 782 9.99 252 24 3 20.4 20.1 19.8 37 9.26 538 9.27 288 0.72 712 9.99 250 23 4 27.2 26.8 26.4 38 9.26 605 9.27 357 0.72 643 9.99 248 22 5 34.0 33.5 33.0 39 9.26 672 9.27 427 0.72 573 9.99 245 21 6 40.8 40.2 39.6 67 69 2 7 47.6 46.9 46.2 40 9.26 739 67 67 67 67 9.27 496 70 69 69 69 0.72 504 9.99 243 2 3 2 3 20 8 54.4 53.6 52.8 11 9.26 806 9.27 566 0.72 434 9.99 241 19 9 61.2 60.3 59.4 9.26 873 9.27 635 0.72 365 9.99 238 18 43 9.26 940 9.27 704 0.72 296 9.99 236 17 44 9.27 007 9.27 773 0.72 227 9.99 233 16 66 69 2 45 9.27 073 67 66 67 66 9.27 842 69 69 69 68 0.72 158 9.99 231 2 3 2 3 15 46 9.27 140 9.27 911 0.72 089 9.99 229 14 47 9.27 206 9.27 980 0.72 020 9.99 226 13 65 3 48 9.27 273 9.28 049 0.71 951 9.99 224 12 49 9.27 339 9.28 117 0.71 883 9.99 221 11 1 6.5 0.3 66 69 2 2 13.0 0.6 50 9.27 405 9.28 186 68 69 68 68 0.71 814 9.99 219 2 3 2 3 IO 3 19.5 0.9 51 9.27 471 66 9.28 254 0.71 746 9.99 217 9 4 26.0 1.2 52 9.27 537 66 9.28 323 0.71 677 9.99 214 8 5 32.5 1.5 53 9.27 602 65 9.28 391 0.71 609 9.99 212 7 6 39.0 1.8 54 9.27 668 66 9.28 459 0.71 541 9.99 209 6 7 45.5 2.1 66 68 2 8 52.0 2.4 55 9.27 734 65 9.28 527 68 67 68 68 0.71 473 9.99 207 3 2 2 3 5 9 58.5 2.7 56 9.27 799 9.28 595 0.71 405 9.99 204 4 57 9.27 864 65 66 65 0.28 662 0.71 338 9.99 202 3 58 9.27 930 9.28 730 0.71 270 9.99 200 2 59 9.27 995 9.28 798 0.71 202 9.99 197 1 65 67 2 60 9.28 060 9.28 865 0.71 135 9.99 195 O LCos d LCot cd LTan LSin d / PP 79 Logarithms of Trigonometric Functions II] 11 Logarithms of Trigonometric Functions 39 / LSin d LTan cd LCot LCos d PP o 9.28 060 65 65 64 65 9.28 865 68 67 67 67 0.71 135 9.99 195 3 2 3 2 60 1 9.28 125 9.28 933 0.71 067 9.99 192 59 2 9.28 190 9.29 000 0.71 000 9.99 190 58 3 9.28 254 9.29 067 0.70 933 9.99 187 57 4 9.28 319 9.29 134 0.70 866 9.99 185 56 65 67 3 5 9.28 384 64 64 65 64 9.29 201 67 67 67 66 0.70 799 9.99 182 2 3 2 3 55 68 67 66 6 9.28 448 9.29 268 0.70 732 9.99 180 54 7 9.28 512 9.29 335 0.70 665 9.99 177 53 1 6.8 6.7 6.6 8 9.28 577 9.29 402 0.70 598 9.99 175 52 2 13.6 13.4 13.2 9 9.28 641 9.29 468 0.70 532 9.99 172 51 3 20.4 20.1 19.8 64 67 2 4 27.2 26.8 26.4 io 9.28 705 64 64 63 64 9.29 535 66 67 66 66 0.70 465 9.99 170 3 2 3 2 50 5 34.0 33.5 33.0 11 9.28 769 9.29 601 0.70 399 9.99 167 49 6 40.8 40.2 39.2 12 9.28 833 9.29 668 0.70 332 9.99 165 48 7 47.6 46.9 46.2 13 9.28 896 9.29 734 0.70 266 9.99 162 47 8 54.4 53.6 52.8 14 9.28 960 9.29 800 0.70 200 9.99 160 46 9 61.2 60.3 59.4 64 66 3 15 9.29 024 63 63 64 63 9.29 866 66 66 66 66 0.70 134 9.99 157 2 3 . 2 3 45 16 9.29 087 9.29 932 0.70 068 9.99 155 44 17 9.29 150 9.29 998 0.70 002 9.99 152 43 18 9.29 214 9.30 064 0.69 936 9.99 150- 42 - 19 9.29 277 9.30 130 0.69 870 9.99 147 41 63 65 2 65 64 63 20 9.29 340 63 63 63 62 9.30 195 66 0.69 805 9.99 145 3 2 3 2 40 21 9.29 403 9.30 261 65 65 66 0.69 739 9.99 142 39 1 6.5 6.4 6.3 22 9.29 466 9.30 326 0.69 674 9.99 140 38 2 13.0 12.8 12.6 23 9.29 529 9.30 391 0.69 609 9.99 137 37 3 19.5 19.2 18.9 24 9.29 591 9.30 457 0.69 543 9.99 135 36 4 26.0 25.6 25.2 63 65 3 5 32.5 32.0 31.5 25 9.29 654 62 63 62 62 9.30 522 65 65 65 65 0.69 478 9.99 132 2 3 3 2 35 6 39.0 38.4 37.8 26 9.29 716 9.30 587 0.69 413 9.99 130 34 7 45.5 44.8 44.1 27 9.29 779 9.30 652 0.69 348 9.99 127 33 8 52.0 51.2 50.4 28 9.29 841 9.30 717 0.69 283 9.99 124 32 9 58.5 57.6 56.7 29 9.29 903 9.30 782 0.69 218 9.99 122 31 63 64 3 30 9.29 966 62 62 61 62 9.30 846 65 64 65 64 0.69 154 9.99 119 2 3 2 3 30 31 9.30 028 9.30 911 0.69 089 9.99 117 29 32 9.30 090 9.30 975 0.69 025 9.99 114 28 33 9.30 151 9.31 040 0.68 960 9.99 112 27 34 9.30 213 9.31 104 0.68 896 9.99 109 26 62 61 60 62 64 3 35 9.30 275 61 62 61 62 9.31 168 65 64 64 64 0.68 832 9.99 106 2 3 2 3 25 1 6.2 6.1 6.0 36 9.30 336 9.31 233 0.68 767 9.99 104 24 2 12.4 12.2 12.0 37 9.30 398 9.31 297 0.68 703 9.99 101 23 3 18.6 18.3 18.0 38 9.30 459 9.31 361 0.68 639 9.99 099 22 4 24.8 24.4 24.0 39 9.30 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0.67 564 9.99 054 3 3 2 3 5 8 47.2 2.4 56 9.31 549 9.32 498 0.67 502 9.99 051 4 9 53.1 2.7 57 9.31 609 9.32 561 0.67 439 9.99 048 3 58 9.31 669 9.32 623 0.67 377 9.99 046 2 59 9.31 728 9.32 685 0.67 315 9.99 043 1 60 62 3 60 9.31 788 9.32 747 0.67 253 9.99 040 o LCos d LCot cd LTan LSin d r PP 78 Logarithms of Trigonometric Functions 40 12 Logarithms of Trigonometric Functions [H LSin d L Tan c d LCot LCos PP o 1 2 3 4 5 6 7 8 9 lO 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 50 9.31 788 9.31 847 9.31 907 9.31 966 9.32 025 9.32 084 9.32 143 9.32 202 9.32 261 9.32 319 9.32 378 9.32 437 9.32 495 9.32 553 9.32 612 9.32 670 9.32 728 9.32 786 9.32 844 9.32 902 9.32 960 9.33 018 9.33 075 9.33 133 9.33 190 9.33 248 9.33 305 9.33 362 9.33 420 9.33 477 9.33 534 9.33 591 9.33 647 9.33 704 9.33 761 9.33 818 9.33 874 9.33 931 9.33 987 9.34 043 9.34 100 9.34 156 9.34 212 9.34 268 9.34 324 9.34 380 9.34 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4 6 9.41 300 9.42 805 0.57 195 9.98 494 O LCos d LCot cd LTan L Sin d / PP 75 Logarithms of Trigonometric Functions II] 15- Logarithms of Trigonometric Functions 43 1 LSin d LTan cd LCot LCos d PP o 9.41 300 47 47 47 47 9.42 805 5L 50 51 50 0.57 195 9.98 494 3 3 4 3 60 9.41 347 9.42 856 0.57 144 9.98 491 59 2 9.41 394 9.42 906 0.57 094 9.98 488 58 3 9.41 441 9.42 957 0.57 043 9.98 484 57 4 9.41 488 9.43 007 0.56 993 9.98 481 56 47 50 4 5 9.41 535 47 46 47 47 9.43 057 51 50 50 50 0.56 943 9.98 477 3 3 4 3 55 51 50 49 6 9.41 582 9.43 108 0.56 892 9.98 474 54 7 9.41 628 9.43 158 0.56 842 9.98 471 53 1 5.1 5.0 4.9 8 9.41 675 9.43 208 0.56 792 9.98 467 52 2 10.2 10.0 9.8 9 9.41 722 9.43 258 0.56 742 9.98 464 51 3 15.3 15.0 14.7 46 50 4 4 20.4 20.0 19.6 io 9.41 768 47 46 47 46 9.43 308 50 50 50 50 0.56 692 9.98 460 3 4 3 3 50 5 25.5 25.0 24.5 11 9.41 815 9.43 358 0.56 642 9.98 457 49 6 30.6 30.0 29.4 12 9.41 861 9.43 408 0.56 592 9.98 453 48 7 35.7 35.0 34.3 13 9.41 908- 9.43 458 0.56 542 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608 9.98 155 4 4 3 4 25 1 4.2 4.1 36 9.45 589 9.47 438 0.52 562 9.98 151 24 2 8.4 8.2 37 9.45 632 9.47 484 0.52 516 9.98 147 23 3 12.6 12.3 38 9.45 674 9.47 530 0.52 470 9.98 144 22 4 16.8 16.4 39 9.45 716 9.47 576 0.52 424 9.98 140 21 5 21.0 20.5 42 46 4 6 25.2 24.6 40 9.45 758 43 42 42 42 9.47 622 46 46 46 46 0.52 378 9.98 136 4 3 4 4 20 7 29.4 28.7 41 9.45 801 9.47 668 0.52 332 9.98 132 19 8 33.6 32.8 42 9.45 843 9.47 714 0.52 286 9.98 129 18 9 37.8 36.9 43 9.45 885 9.47 760 0.52 240 9.98 125 17 44 9.45 927 9.47 806 0.52 194 9.98 121 16 42 46 4 45 9.45 969 42 42 42 41 9.47 852 45 46 46 46 0.52 148 9.98 117 4 3 4 4 15 46 9.46 Oil 9.47 897 0.52 103 9.98 113 14 47 9.46 053 9.47 943 0.52 057 9.98 110 13 48 9.46 095 9.47 989 0.52 Oil 9.98 106 12 4 3 49 9.46 136 9.48 035 0.51 965 9.98 102 11 42 45 4 1 0.4 0.3 50 9.46 178 42 42 41 42 9.48 080 46 45 46 45 0.51 920 9.98 098 4 4 3 4 IO 2 0.8 0.6 51 9.46 220 9.48 126 0.51 874 9.98 094 9 3 1.2 0.9 52 9.46 262 9.48 171 0.51 829 9.98 090 8 4 1.6 1.2 53 9.46 303 9.48 217 0.51 783 9.98 087 7 5 2.0 1.5 54 9.46 345 9.48 262 0.51 738 9.98 083 6 6 2.4 1.8 41 45 4 7 2.8 2.1 55 9.46 386 42 41 42 41 9.48 307 46 45 45 46 0.51 693 9.98 079 4 4 4 4 5 8 3.2 2.4 56 9.46 428 9.48 353 0.51 647 9.98 075 4 9 3.6 2.7 57 9.46 469 9.48 398 0.51 602 9.98 071 3 58 9.46 511 9.48 443 0.51 557 9.98 067 2 59 9.46 552 9.48 489 0.51 511 9.98 063 1 42 45 3 60 9.46 594 9.48 534 0.51 466 9.98 060 O L Cos d LCot cd LTan LSin d / PP 73 Logarithms of Trigonometric Functions II] 17 Logarithms of Trigonometric Functions 45 / L Sin d LTan cd LCot LCos d PP o 9.46 594 41 41 41 41 9.48 534 45 45 45 45 * 0.51 466 9.98 060 4 4 4 4 60 1 9.46 635 9.48 579 0.51 421 9.98 056 59 2 9.46 676 9.48 624 0.51 376 9.98 052 58 3 9.46 717 9.48 669 0.51 331 9.98 048 57 4 9.46 758 9.48 714 0.51 286 9.98 044 56 42 45 4 5 9.46 800 41 41 41 41 9.48 759 45 45 45 45 0.51 241 9.98 040 4 4 3 4 55 6 9.46 841 9.48 804 0.51 196 9.98 036 54 7 9.46 882 9.48 849 0.51 151 9.98 032 53 8 9.46 923 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40 40 40 9.49 652 44 44 44 44 0.50 348 9.97 962 4 4 4 4 35 26 9.47 654 9.49 696 0.50 304 9.97 958 34 42 41 40 27 9.47 694 9.49 740 0.50 260 9.97 954 33 28 9.47 734 9.49 784 0.50 216 9.97 950 32 1 4.2 4.1 4.0 29 9.47 774 9.49 828 0.50 172 9.97 946 31 2 8.4 8.2 8.0 40 44 4 3 12.6 12.3 12.0 30 9.47 814 40 40 40 40 9.49 872 44 44 44 44 0.50 128 9.97 942 4 4 4 4 30 4 16.8 16.4 16.0 31 9.47 854 9.49 916 0.50 084 9.97 938 29 5 21.0 20.5 20.0 32 9.47 894 9.49 960 0.50 040 9.97 934 28 6 25.2 24.6 24.0 33 9.47 934 9.50 004 0.49 996 9.97 930 27 7 29.4 28.7 28.0 34 9.47 974 9.50 048 0.49 952 9.97 926 26 S 33.6 32.8 32.0 40 44 4 9 37.8 36.9 36.0 35 9.48 014 40 40 39 40 9.50 092 44 44 43 44 0.49 908 9.97 922 4 4 4 4 25 36 9.48 054 9.50 136 0.49 864 9.97 918 24 37 9.48 094 9.50 180 0.49 820 9.97 914 23 38 9.48 133 9.50 223 0.49 777 9.97 910 22 39 9.48 173 9.50 267 0.49 733 9.97 906 21 40 44 4 40 9.48 213 39 40 40 39 9.50 311 44 43 44 43 0.49 689 9.97 902 4 4 4 4 20 41 9.48 252 9.50 355 0.49 645 9.97 898 19 42 9.48 292 9.50 398 0.49 602 9.97 894 18 43 9.48 332 9.50 442 0.49 558 9.97 890 17 44 9.48 371 9.50 485 0.49 515 9.97 886 16 39 5 4 3 40 44 4 45 9.48 411 39 40 39 39 9.50 529 43 44 43 44 0.49 471 9.97 882 4 4 4 4 15 1 3.9 0.5 0.4 0.3 46 9.48 450 9.50 572 0.49 428 9.97 878 14 2 7.8 1.0 0.8 0.6 47 9.48 490 9.50 616 9.50 659 0.49 384 9.97 874 13 3 11.7 1.5 1.2 0.9 48 9.48 529 0.49 341 9.97 870 12 4 15.6 2.0 1.6 1.2 49 9.48 568 9.50 703 0.49 297 9.97 866 11 5 L9.5 2.5 2.0 1.5 39 43 5 6 23.4 3.0 2.4 1.8 50 9.48 607 40 39 39 39 9.50 746 43 44 43 43 0.49 254 9.97 861 4 4 4 4 IO 7 27.3 3.5 2.8 2.1 51 9.48 647 9.50 789 0.49 211 9.97 857 9 8 U.2 4.0 3.2 2.4 52 9.48 686 9.50 833 0.49 167 9.97 853 8 9 J5.14.5 3.6 2.7 53 9.48 725 9.50 876 0.49 124 9.97 849 7 54 9.48 764 9.50 919 0.49 081 9.97 845 6 39 43 4 55 9.48 803 39 39 39 39 9.50 962 43 43 44 43 0.49 038 9.97 841 4 4 4 4 5 56 9.48 842 9.51 005 0.48 995 9.97 837 4 57 9.48 881 9.51 048 0.48 952 9.97 833 3 58 9.48 920 9.51 092 0.48 908 9.97 829 2 59 9.48 959 9.51 135 0.48 865 9.97 825 1 39 43 4 60 9.48 998 9.51 178 0.48 822 9.97 821 O LCos d LCot cd LTan LSin d t PP 72 Logarithms of Trigonometric Functions 46 18 Logarithms of Trigonometric Functions [II r LSin d LTan cd LCot LCos d PP O 9.48 998 39 39 39 38 9.51 178 43 43 42 43 0.48 822 9.97 821 4 5 4 4 60 1 9.49 037 9.51 221 0.48 779 9.97 817 59 2 9.49 076 9.51 264 0.48 736 9.97 812 58 3 9.49 115 9.51 306 0.48 694 9.97 808 57 4 9.49 153 9.51 349 0.48 651 9.97 804 56 39 43 4 5 9.49 192 39 38 39 39 9.51 392 43 43 42 43 0.48 608 9.97 800 4 4 4 4 55 6 9.49 231 9.51 435 0.48 565 9.97 796 54 7 9.49 269 9.51 478 0.48 522 9.97 792 53 8 9.49 308 9.51 520 0.48 480 9.97 788 52 9 9.49 347 9.51 563 0.48 437 9.97 784 51 43 42 41 38 43 5 io 9.49 385 39 38 38 39 9.51 606 42 43 43 42 0.48 394 9.97 779 4 4 4 4 50 1 4.3 4.2 4.1 11 9.49 424 9.51 648 0.48 352 9.97 775 49 2 8.6 8.4 8.2 12 9.49 462 9.51 691 0.48 309 9.97 771 48 3 12.9 12.6 12.3 13 9.49 500 9.51 734 0.48 266 9.97 767 47 4 17.2 16.8 16.4 14 9.49 539 9.51 776 0.48 224 9.97 763 46 5 21.5 21.0 20.5 38 43 4 8 25.8 25.2 24.6 15 9.49 577 38 39 38 38 9.51 819 42 42 43 42 0.48 181 9.97 759 5 4 4 4 45 7 30.1 29.4 28.7 16 9.49 615 9.51 861 0.48 139 9.97 754 44 S 34.4 33.6 32.8 17 9.49 654 9.51 903 0.48 097 9.97 750 43 9 38.7 37.8 36.9 18 9.49 692 9.51 946 0.48 054 9.97 746 42 19 9.49 730 9.51 988 0.48 012 9.97 742 41 38 43 4 20 9.49 768 38 38 38 38 9.52 031 42 42 42 43 0.47 969 9.97 738 4 5 4 4 40 21 9.49 806 9.52 073 0.47 927 9.97 734 39 22 9.49 844 9.52 115 0.47 885 9.97 729 38 23 9.49 882 9.52 157 0.47 843 9.97 725 37 24 9.49 920 9.52 200 0.47 800 9.97 721 36 38 42 4 25 9.49 958 38 38 38 38 9.52 242 42 42 42 42 0.47 758 9.97 717 4 5 4 4 35 26 9.49 996 9.52 284 0.47 716 9.97 713 34 39 38 37 27 9.50 034 9.52 326 0.47 674 9.97 708 33 28 9.50 072 9.52 368 0.47 632 9.97 704 32 1 3.9 3.8 3.7 29 9.50 110 9.52 410 0.47 590 9.97 700 31 2 7.8 7.6 7.4 38 42 4 3 11.7 11.4 11.1 30 9.50 148 37 38 38 37 9.52 452 42 42 42 42 0.47 548 9.97 696 5 4 4 4 30 4 15.6 15.2 14.8 31 9.50 185 9.52 494 0.47 506 9.97 691 29 5 19.5 19.0 18.5 32 9.50 223 9.52 536 0.47 464 9.97 687 28 (i 23.4 22.8 22.2 33 9.50 261 9.52 578 0.47 422 9.97 683 27 7 27.3 26.6 25.9 34 9.50 298 9.52 620 0.47 380 9.97 679 26 8 31.2 30.4 29.6 38 41 5 9 35.1 34.2 33.3 35 9.50 336 38 37 38 37 9.52 661 42 42 42 42 0.47 339 9.97 674 4 4 4 5 25 36 9.50 374 9.52 703 0.47 297 9.97 670 24 37 9.50 411 9.52 745 0.47 255 9.97 666 23 38 9.50 449 9.52 787 0.47 213 9.97 662 22 39 9.50 486 9.52 829 0.47 171 9.97 657 21 37 41 4 IO 9.50 523 38 37 37 38 9.52 870 42 41 42 42 0.47 130 9.97 653 4 4 5 4 20 41 9.50 561 9.52 912 0.47 088 9.97 649 19 42 9.50 598 9.52 953 0.47 047 9.97 645 18 43 9.50 635 9.52 995 0.47 005 9.97 640 17 44 9.50 673 9 . 53 037 0.46 963 9.97 636 16 36 5 4 37 41 4 45 9.50 710 37 37 37 37 9.53 078 42 41 41 42 0.46 922 9.97 632 4 5 4 4 15 1 3.6 0.5 0.4 46 9.50 747 9.53 120 0.46 880 9.97 628 14 2 7.2 1.0 0.8 47 9.50 784 9.53 161 0.46 839 9.97 623 13 3 10.8 1.5 1.2 48 9.50 821 9.53 202 0.46 798 9.97 619 12 4 14.4 2.0 1.6 49 9.50 858 9.53 244 0.46 756 9.97 615 11 5 18.0 2.5 2.0 38 41 5 6 21.6 3.0 2.4 50 9.50 896 37 37 37 36 9.53 285 42 41 41 41 0.46 715 9.97 610 4 4 5 4 IO 7 25.2 3.5 2.8 51 9.50 933 9.53 327 0.46 673 9.97 606 9 8 28.8 4.0 3.2 52 9.50 970 9.53 368 0.46 632 9.97 602 8 9 32.4 4.5 3.6 53 9.51 007 9.53 409 0.46 691 9.97 597 7 54 9.51 043 9.53 450 0.46 550 9.97 593 6 37 42 4 55 9.51 080 37 37 37 36 9.53 492 41 41 41 41 0.46 508 9.97 589 _ 5 56 9.51 117 9.53 533 0.46 467 9.97 584 o 4 4 5 4 57 9.51 154 9.63 574 0.46 426 9.97 580 3 58 9.51 191 9.63 616 0.46 385 9.97 576 2 59 9.51 227 9.53 656 0.46 344 9.97 571 1 37 41 4 eo 9.51 264 9.53 697 0.46 303 9.97 567 O LCos d LCot cd LTan LSin d / PP 71 Logarithms of Trigonometric Functions II] 19 Logarithms of Trigonometric Functions 47 / LSin d LTan cd LCot LCos d PP o 9.51 264 37 37 36 37 9.53 697 41 41 41 41 0.46 303 9.97 567 4 5 4 4 60 1 9.51 301 9.53 738 0.46 262 9.97 563 59 2 9.51 338 9.53 779 0.46 221 9.97 558 58 3 9.51 374 9.53 820 0.46 180 9.97 554 57 4 9.51 411 9.53 861 0.46 139 9.97 550 56 36 41 5 5 9.51 447 37 36 37 36 9.53 902 41 41x 41 40 0.46 098 9.97 545 4 5 4 4 55 6 9.51 484 9.53 943 0.46 057 9.97 541 54 7 9.51 520 9.53 984 0.46 016 9.97 536 53 8 9.51 557 9.54 025 0.45 975 9.97 532 52 9 9.51 593 9.54 065 0.45 935 9.97 528 51 41 40 39 36 41 5 io 9.51 629 37 36 36 36 9.54 106 41 40 41 41 0.45 894 9.97 523 4 4 5 4 50 1 4.1 4.0 3.9 11 9.51 666 9.54 147 0.45 853 9.97 519 49 2 8.2 8.0 7.8 12 9.51 702 9.54 187 0.45 813 9.97 515 48 3 12.3 12.0 11.7 13 9.51 738 9.54 228 0.45 772 9.97 510 47 4 16.4 16.0 15.6 14 9.51 774 9.54 269 0.45 731 9.97 506 46 5 20.5 20.0 19.5 37 40 5 6 24.6 24.0 23.4 15 9.51 811 36 36 36 36 9.54 309 41 40 41 40 0.45 691 9.97 501 4 5 4 4 45 7 28.7 28.0 27.3 16 9.51 847 9.54 350 0.45 650 9.97 497 44 s 32.8 32.0 31.2 17 9.51 883 9.54 390 0.45 610 9.97 492 43 9 36.9 36.0 35.1 18 9.51 919 9.54 431 0.45 569 9.97 488 42 19 9.51 955 9.54 471 0.45 529 9.97 484 41 36 41 5 20 9.51 991 36 36 36 36 9.54 512 40 41 40 40 0.45 488 9.97 479 4 5 4 5 40 21 9.52 027 9.54 552 0.45 448 9.97 475 39 22 9.52 063 9.54 593 0.45 407 9.97 470 38 23 9.52 099 9.54 633 0.45 367 9.97 466 37 24 9.52 135 9.54 673 0.45 327 9.97 461 36 36 41 4 25 9.52 171 36 35 36 36 9.54 714 40 40 41 40 0.45 286 9.97 457 4 5 4 35 26 9.52 207 9.54 754 0.45 246 9.97 453 34 37 36 35 27 9.52 242 9.54 794 0.45 206 9.97 448 33 28 9.52 278 9.54 835 0.45 165 9.97 444 32 1 3.7 3.6 3.5 29 9.52 314 9.54 875 0.45 125 9.97 439 5 31 2 7.4 7.2 7.0 36 40 4 3 11.1 10.8 10.5 30 9.52 350 35 36 35 36 9.54 915 40 40 40 40 0.45 085 9.97 435 5 4 5 4 30 4 14.8 14.4 14.0 31 9.52 385 9.54 955 0.45 045 9.97 430 29 5 18.5 18.0 17.5 32 9.52 421 9.54 995 0.45 005 9.97 426 28 6 22.2 21.6 21.0 33 9.52 456 9.55 035 0.44 965 9.97 421 27 7 25.9 25.2 24.5 34 9.52 492 9.55 075 0.44 92g 9.97 417 26 S 29.6 28.8 28.0 35 40 5 9 33.3 32.4 31.5 35 9.52 527 36 35 36 35 9.55 115 40 40 40 40 0.44 885 9.97 412 4 5 4 5 25 36 9.52 563 9.55 155 0.44 845 9.97 408 24 37 9.52 598 9.55 195 0.44 8O5 9.97 403 23 38 9.52 634 9.55 235 0.44 765 9.97 399 22 39 9.52 669 9.55 275 0.44 72g 9.97 394 21 36 40 4 40 9.52 705 35 35 36 35 9.55 315 40 40 39 40 0.44 685 9.97 390 5 4 5 4 20 41 9.52 740 9.55 355 0.44 645 9.97 385 19 42 9.52 775 9.55 395 0.44 605 9.97 381 18 43 9.52 811 9.55 434 0.44 566 9.97 376 17 44 9.52 846 9.55 474 0.44 526 9.97 372 16 34 5 4 35 40 5 45 9.52 881 35 35 35 35 9.55 514 40 39 40 40 0.44 486 9.97 367 4 5 5 4 15 1 3.4 0.5 0.4 46 9.52 916 9.55 554 0.44 446 9.97 363 14 2 6.8 1.0 0.8 47 9.52 951 9.55 593 0.44 407 9.97 358 13 3 10.2 1.5 1.2 48 9.52 986 9.55 633 0.44 367 9.97 353 12 4 13.6 2.0 1.6 49 9.53 021 9.55 673 0.44 327 9.97 349 11 5 17.0 2.5 2.0 35 39 5 6 20.4 3.0 2.4 50 9.53 056 36 34 35 35 9.55 712 40 39 40 39 0.44 288 9.97 344 4 5 4 5 IO 7 23.8 3.5 2.8 51 9.53 092 9.55 752 0.44 248 9.97 340 9 8 27.2 4.0 3.2 52 9.53 126 9.55 791 0.44 209 9.97 335 8 9 30.6 4.5 3.6 53 9.53 161 9.55 831 0.44 169 9.97 331 7 54 9.53 196 9.55 870 0.44 130 9.97 326 6 35 40 4 55 9.53 231 35 35 35 34 9.55 910 39 40 39 39 0.44 090 9.97 322 5 5 4 5 5 56 9.53 266 9.55 949 0.44 051 9.97 317 4 57 9.53 301 9.55 989 0.44 Oil 9.97 312 3 58 9.53 336 9.56 028 0.43 972 9.97 308 2 59 9.53 370 9.56 067 0.43 933 9.97 303 1 35 40 4 60 9.53 405 9.56 107 0.43 893 9.97 299 O LCos d LCot cd LTan LSin d / PP 70 -Logarithms of Trigonometric Functions 48 20 Logarithms of Trigonometric Functions III / LSin LTan cd LCot LCos d PP o 9.53 405 35 35 34 35 9.56 107 39 39 39 40 0.43 893 9.97 299 5 5 4 5 GO 1 9.53 440 9.56 146 0.43 854 9.97 294 59 2 9.53 475 9.5.6 185 0.43 815 9.97 289 58 3 9.53 509 9.56 224 0.43 776 9.97 285 57 4 9.53 544 9.56 264 0.43 736 9.97 280 56 34 39 4 5 9.53 578 35 34 35 34 9.56 303 39 39 39 39 0.43 697 9.97 276 5 5 4 5 55 6 9.53 613 9.56 342 0.43 658 9.97 271 54 7 9.53 647 9.56 381 0.43 619 9.97 266 53 8 9.53 682 9.56 420 0.43 580 9.97 262 52 9 9.53 716 9.56 459 0.43 541 9.97 257 51 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LTan cd LCot LCos d PP o 9.04 184 20 20 20 20 9.68 818 32 ;*2 32 32 0.31 182 9.95 300 6 7 60 1 9.64 210 9.68 850 0.31 150 9.95 360 59 2 9.04 236 9.68 882 0.31 118 9.95 354 58 3 9.04 202 9.08 914 0.31 086 9.95 348 57 4 9.04 288 9.08 940 0.31 054 9.95 341 56 25 32 5 9.04 313 20 20 20 20 9.08 978 32 32 32 32 0.31 022 9.95 335 7 55 6 9.04 339 9.09 010 0.30 990 9.95 329 54 7 9.04 305 9.09 042 0.30 958 9.95 323 53 8 9.04 391 9.09 074 0.30 920 9.95 317 52 9 9.04 417 9.09 100 0.30 894 9.95 310 51 32 31 25 32 io 9.04 442 20 20 25 20 9.09 138 32 32 32 32 0.30 802 9.95 304 7 50 1 3.2 3.1 11 9.04 408 9.09 170 0.30 830 9.95 298 49 2 6.4 6.2 12 9.04 494 9.09 202 0.30 798 9.95 292 48 3 9.6 9.3 13 9.04 519 9.09 234 0.30 700 9.95 280 47 4 12.8 12.4 14 9.04 545 9.09 200 0.30 734 9.95 279 46 5 16.0 15.5 20 32 a 19.2 18.6 15 9.04 571 25 20 25 20 9.09 298 31 32 32 32 0.30 702 9.95 273 7 45 7 22.4 21.7 16 9.04 590 9.09 329 0.30 071 9.95 207 44 8 25.6 24.8 17 9.04 022 9.09 301 0.30 039 9.95 201 43 9 28.8 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25 26 25 25 9.09 932 31 32 31 32 0.30 008 9.95 148 i 7 6 6 7 25 36 9.05 104 9.09 903 0.30 037 9.95 141 24 37 38 9.05 130 9.05 155 9.09 995 9.70 020 0.30 005 0.29 974 9.95 135 9.95 129 23 22 39 9.05 180 9.70 058 0.29 942 9.95 122 21 25 31 6 40 9.05 205 25 25 26 25 9.70 089 32 31 32 31 0.29 911 9.95 110 6 7 6 7 20 41 9.05 230 9.70 121 0.29 879 9.95 110 19 42 9.05 255 9.70 152 0.29 848 9.95 103 18 43 9.05 281 9.70 184 0.29 816 9.95 097 17 44 9.05 300 9.70 215 0.29 785 9.95 090 16 7 6 25 32 6 45 9.05 331 25 25 25 25 9.70 247 31 31 32 31 0.29 753 9.95 084 6 7 6 6 15 1 0.7 0.6 46 9.05 350 9.70 278 0.29 722 9.95 078 14 2 1.4 1.2 47 9.05 381 9.70 309 0.29 091 9.95 071 13 3 2.1 1.8 48 9.05 400 9.70 341 0.29 059 9.95 085 12 4 2.8 2.4 49 9.05 431 9.70 372 0.29 028 9.95 0ij9 11 5 3.5 3.0 25 32 7 6 4.2 3.6 50 9.05 450 25 25 25 25 9.70 404 31 31 32 31 0.29 590 9.95 052 6 7 6 IO 7 4.9 4.2 51 9.05 481 9.70 435 0.29 505 9.95 040 9 8 5.6 4.8 52 9.05 500 9.70 400 0.29 534 9.95 039 8 9 6.3 5.4 53 9.05 531 9.70 498 0.29 502 9.95 033 7 54 9.05 550 9.70 529 0.29 471 9.95 027 6 24 31 7 55 9.05 580 25 25 25 25 9.70 500 32 31 31 31 0.29 440 9.95 020 7 6 6 5 56 9.05 005 9.70 592 0.29 408 9.95 014 4 57 9.05 630 9.70 023 0.29 377 9.95 007 3 58 9.05 055 9.70 054 0.29 340 9.95 001 2 59 9.05 080 9.70 085 0.29 315 9.94 995 1 25 32 7 60 9.05 705 9.70 717 0.29 283 9.94 988 O LCos d LCot cd LTan LSin d / PP 63 Logarithms of Trigonometric Functions II] 27 Logarithms of Trigonometric Functions 55 t L Sin d LTan cd LCot LCos d PP o 9.65 705 24 25 25 25 9.70 717 31 31 31 31 0.29 283 9.94 988 6 7 6 7 60 1 9.65 729 9.70 748 0.29 252 9.94 982 59 2 9.65 754 9.70 779 0.29 221 9.94 975 58 3 9.65 779 9.70 810 0.29 190 9.94 969 57 4 9.65 804 9.70 841 0.29 159 9.94 962 56 24 32 6 5 9.65 828 25 25 24 25 9.70 873 31 31 31 31 0.29 127 9.94 956 7 6 7 6 55 6 9.65 853 9.70 904 0.29 096 9.94 949 54 7 9.65 878 9.70 935 0.29 065 9.94 943 53 8 9.65 902 9.70 966 0.29 034 9.94 936 52 9 9.65 927 9.70 997 0.29 003 9.94 930 51 32 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952 9.94 707 17 44 9.66 779 9.72 078 0.27 922 9.94 700 16 7 6 24 31 6 45 9.66 803 24 24 24 24 9.72 109 31 30 31 30 0.27 891 9.94 694 7 7 6 7 15 1 0.7 0.6 46 9.66 827 9.72 140 0.27 860 9.94 687 14 2 1.4 1.2 47 9.66 851 9.72 170 0.27 830 9.94 680 13 3 2.1 1.8 48 9 . 66 .875 9.72 201 0.27 799 9.94 674 12 4 2.8 2.4 49 9.66 B99 9.72 231 0.27 769 9.94 667 11 5 3.5 3.0 23 31 7 6 4.2 3.6 50 9.66 922 24 24 24 24 9.72 262 31 30 31 30 0.27 738 9.94 660 6 7 7 6 IO 7 4.9 4.2 51 9.66 946 9.72 293 0.27 707 9.94 654 9 8 5.6 4.8 52 9.66 970 9.72 323 0.27 677 9.94 647 8 9 6.3 5.4 53 9.66 994 9.72 354 0.27 646 9.94 640 .7 54 9.67 018 9.72 384 0.27 616 9.94 634 6 24 31 7 55 9.67 042 24 24 23 24 9.72 415 30 31 30 31 0.27 585 9.94 627 7 6 7 7 5 56 9.67 066 9.72 445 0.27 555 9.94 620 4 # 9.67 090 9.72 476 0.27 524 9.94 614 3 9.67 113 9.72 506 0.27 494 9.94 607 2 59 9.67 137 9.72 537 0.27 463 9.94 600 1 24 30 7 60 9.67 161 9.72 567 0.27 433 9.94 593 O LCos d LCot cd LTan LSin d / PP 62 Logarithms of Trigonometric Functions 56 28- Logarithms of Trigonometric Functions [II / LSin d LTan cd LCot LCos d PP o 9.67 161 24 23 24 24 9.72 567 31 30 31 30 0.27 433 9.94 593 6 7 7 6 60 1 9.67 185 9.72 598 0.27 402 9.94 587 59 2 9.67 208 9.72 628 0.27 372 9.94 580 58 3 9.67 232 9.72 659 0.27 341 9.94 573 57 4 9.67 256 9.72 689 0.27 311 9.94 567 56 24 31 7 5 9.67 280 23 24 23 24 9.72 720 30 30 31 30 0.27 280 9.94 560 7 7 6 7 55 6 9.67 303 9.72 750 0.27 250 9.94 553 54 7 9.67 327 9.72 780 0.27 220 9.94 546 53 8 9.67 350 9.72 811 0.27 189 9.94 540 52 9 9.67 374 9.72 841 0.27 159 9.94 533 51 31 30 29 24 31 7 io 9.67 398 23 24 23 24 9.72 872 30 30 31 30 0. 27 128 9.94 526 7 6 7 7 50 1 3.1 3.0 2.9 11 9.67 421 9.72 902 0.27 098 9.94 519 49 2 6.2 6.0 5.8 12 9.67 445 9.72 932 0.27 068 9.94 513 48 3 9.3 9.0 8.7 13 9.67 468 9.72 963 0.27 037 9.94 506 47 4 12.4 12.0 11.6 14 9.67 492 9.72 993 0.27 007 9.94 499 46 5 15.5 15.0 14.5 23 30 7 f. 18.6 18.0 17.4 15 9.67 515 24 23 24 23 9.73 023 31 30 30 30 0.26 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13.8 13.2 33 9.67 936 9.73 567 0.26 433 9.94 369 27 7 16.8 16.1 15.4 34 9.67 959 9.73 597 0.26 403 9.94 362 26 8 19.2 18.4 17.6 23 30 7 9 21.6 20.7 18.8 35 9.67 982 24 23 23 23 9.73 627 30 30 30 30 0.26 373 9.94 355 6 7 7 7 25 36 9.68 006 9.73 657 0.26 343 9.94 349 24 37 9.68 029 9.73 687 0.26 313 9.94 342 23 38 9.68 052 9.73 717 0.26 283 9.94 335 22 39 9.68 075 9.73 747 0.26 253 9.94 328 21 23 30 7 40 9.68 098 23 23 23 23 9.73 777 30 30 30 30 0.26 223 9.94 321 7 7 7 7 20 41 9.68 121 9.73 807 0.26 193 9.94 314 19 42 9.68 144 9.73 837 0.26 163 9.94 307 18 43 9.68 167 9.73 867 0.26 133 9.94 300 17 44 9.68 190 9.73 897 0.26 103 9.94 293 16 7 6 23 30 7 45 9.68 213 24 23 23 22 9.73 927 30 30 30 30 0.26 073 9.94 286 7 6 7 7 15 1 0.7 0.6 46 9.68 237 9.73 957 0.26 043 9.94 279 14 2 1.4 1.2 47 9.68 260 9.73 987 0.26 013 9.94 273 13 3 2.1 1.8 48 9.68 283 9.74 017 0.25 983 9.94 266 12 4 2.8 2.4 49 9.68 305 9.74 047 0.25 953 9.94 259 11 5 3.5 3.0 23 30 7 6 4.2 3.6 50 9.68 328 23 23 23 23 9.74 077 30 30 29 30 0.25 923 9.94 252 7 7 7 7 IO 7 4.9 4.2 51 9.68 351 9.74 107 0.25 893 9.94 245 9 8 5.6 4.8 52 9.68 374 9.74 137 0.25 863 9.94 238 8 9 6.3 5.4 53 9.68 397 9.74 166 0.25 834 9.94 231 7 54 9.68 420 9.74 196 0.25 804 9.94 224 ' 6 23 30 7 55 9.68 443 23 23 23 22 9.74 226 30 30 30 29 0.25 774 9.94 217 7 7 7 7 5 56 9.68 466 9.74 256 0.25 744 9.94 210 4 57 9.68 489 9.74 286 0.25 714 9.94 203 3 58 9.68 512 9.74 316 0.25 684 9.94 196 2 59 9.68 534 9.74 345 0.25 655 9.94 189 1 23 30 7 60 9.68 557 9.74 375 0.25 625 9.94 182 O LCos d LCot cd LTan LSin d / PP 61 Logarithms of Trigonometric Functions II] 29 Logarithms of Trigonometric Functions 57 / LSin d LTan cd LCot LCos d PP o 9.68 557 23 23 22 23 9.74 375 30 30 30 29 0.25 625 9.94 182 7 7 7 7 eo 1 9.68 580 9.74 405 0.25 595 9.94 175 59 2 9.68 603 9.74 43g 0.25 565 9.94 168 58 3 9.68 625 9.74 465 0.25 535 9.94 161 57 4 9.68 648 9.74 494 0.25 506 9.94 154 56 23 30 7 5 9.68 671 23 22 23 23 9.74 524 30 29 30 30 0.25 476 9.94 147 7 7 7 7 55 6 9.68 694 9.74 554 0.25 446 9.94 140 54 7 9.68 716 9.74 583 0.25 417 9.94 133 53 8 9.68 739 9.74 613 0.25 387 9.94 126 52 9 9.68 762 9.74 643 0.25 357 9.94 119 51 22 30 7 io 9.68 784 23 22 23 23 9.74 673 29 30 30 29 0.25 327 9.94 112 7 7 8 7 50 11 9.68 807 9.74 702 0.25 298 9.94 105 49 12 9.68 829 9.74 732 0.25 268 9.94 098 48 13 9.68 852 9.74 762 0.25 238 9.94 090 47 14 9.68 873 9.74 791 0.25 209 9.94 083 46 22 30 7 30 29 23 15 9.68 897 23 22 23 22 9.74 821 30 29 30 29 0.25 179 9.94 076 9.94 069 7 7 7 7 45 16 9.68 920 9.74 851 0.25 149 44 1 3.0 2.9 2.3 17 9.68 942 9.74 880 0.25 120 9.94 062 43 2 6.0 5.8 4.6 18 9.68 965 9.74 910 0.25 090 9.94 055 42 3 9.0 8.7 6.9 19 9.68 987 9.74 939 0.25 061 9.94 048 41 4 12.0 11.6 9.2 23 30 7 5 15.0 14.5 11.5 20 9.69 010 22 23 22 23 9.74 969 29 30 30 29 0.25 031 9.94 041 7 7 7 8 40 6 18.0 17.4 13.8 21 9.69 032 9.74 998 0.25 002 9.94 034 39 7 21.0 20.3 16.1 22 9.69 055 9.75 028 0.24 972 9.94 027 38 8 24.0 23.2 18.4 23 9.69 077 9.75 058 0.24 942 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9.69 501 9.75 t317 0.24 383 9.93 884 18 4 8.8 3.2 2.8 43 9.69 523 9.75 647 0.24 353 9.93 876 17 5 11.0 4.0 3.5 44 9.69 545 9.75 676 0.24 324 9.93 869 16 6 13.2 4.8 4.2 22 29 7 7 15.4 5.6 4.9 45 9.69 567 22 22 22 22 9.75 705 30 29 29 29 0.24 295 9.93 862 7 8 7 7 15 S 17.6 6.4 5.6 46 9.69 589 9.75 735 0.24 265 9.93 855 14 9 19.8 7.2 6.3 47 9.69 611 9.75 764 0.24 236 9.93 847 13 48 9.69 633 9.75 793 0.24 207 9.93 840 12 49 9.69 655 9.75 822 0.24 178 9.93 833 11 22 30 7 50 9.69 677 22 22 22 22 9.75 852 29 29 29 30 0.24 148 9.93 826 7 8 7 7 IO 51 9.69 699 9.75 881 0.24 119 9.93 819 9 52 9.69 721 9.75 910 0.24 090 9.93 811 8 53 9.69 743 9.75 939 0.24 061 9.93 804 7 54 9.69 765 9.75 969 0.24 031 9.93 797 6 22 29 8 55 9.69 787 22 22 22 22 9.75 998 29 29 30 29 0.24 002 9.93 789 7 7 7 8 5 56 9.69 809 9.76 027 0.23 973 9.93 782 4 57 9.69 831 9.76 056 0.23 944 9.93 775 3 58 9.69 853 9.76 086 0.23 914 9.93 768 2 59 9.69 875 9.76 115 0.23 885 9.93 760 1 22 29 7 <> 9.69 897 9.76 144 0.23 856 9.93 753 O LCos d LCot cd LTan LSin d / PP 60 Logarithms of Trigonometric Functions 58 30 Logarithms of Trigonometric Functions [II / L Sin d LTan cd LCot LCos d PP o 9.69 897 22 22 22 21 9.76 144 29 29 29 30 0.23 856 9.93 753 7 8 7 7 60 1 9.69 919 9.76 173 0.23 827 9.93 746 59 2 9.69 941 9.76 202 0.23 798 9.93 738 58 3 9.. 69 963 9.76 231 0.23 769 9.93 731 57 4 9.69 984 9.76 261 0.23 739 9.93 724 56 22 29 7 5 9.70 006 22 22 22 21 9.76 290 29 29 29 29 0.23 710 9.93 717 8 7 7 8 55 6 9.70 028 9.76 319 0.23 681 9.93 709 54 7 9.70 050 9.76 348 0.23 652 9.93 702 53 8 9.70 072 9.76 377 0.23 623 9.93 695 52 9 9.70 093 9.76 406 0.23 594 9.93 687 51 30 29 28 22 29 7 io 9.70 115 22 22 21 22 9.76 435 29 29 29 29 0.23 565 9.93 680 7 8 7 8 50 1 3.0 2.9 2.8 11 9.70 137 9.76 464 0.23 536 9.93 673 49 2 6.0 5.8 5.6 12 9.70 159 9.76 493 0.23 507 9.93 665 48 3 9.0 8.7 8.4 13 9.70 180 9.76 522 0.23 478 9.93 658 47 4 12.0 11.6 11.2 14 9.70 202 9.76 551 0.23 449 9.93 650 46 5 15.0 14.5 14.0 22 29 7 (i 18.0 17.4 16.8 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8 7 8 55 6 9.71 310 9.78 049 0.21 951 9.93 261 54 7 9.71 331 9.78 077 0.21 923 9.93 253 53 8 9.71 352 9.78 106 0.21 894 9.93 246 52 9 9.71 373 9.78 135 0.21 865 9.93 238 51 29 28 20 28 8 io 9.71 393 21 21 21 21 9.78 163 29 28 29 28 0.21 837 9.93 230 7 8 8 7 50 1 2.9 2.8 11 9.71 414 9.78 192 0.21 808 9.93 223 49 2 5.8 5.6 12 9.71 435 9.78 220 0.21 780 9.93 215 48 3 8.7 8.4 13 9.71 456 9.78 249 0.21 751 9.93 207 47 4 11.6 11.2 14 9.71 477 9.78 277 0.21 723 9.93 200 46 5 14.5 14.0 21 29 8 6 17.4 16.8 15 9.71 498 21 20 21 21 9.78 306 28 29 28 28 0.21 694 9.93 192 8 7 8 8 45 7 20.3 19.6 16 9.71 519 9.78 334 0.21 666 9.93 184 44 8 23.2 22.4 17 9.71 539 9.78 363 0.21 637 9.93 177 43 9 26.1 25.2 18 9.71 560 9.78 391 0.21 609 9.93 169 42 19 9.71 581 9.78 419 0.21 581 9.93 161 41 21 29 7 20 9.71 602 20 21 21 21 9.78 448 28 29 28 29 0.21 552 9.93 154 8 8 7 8 40 21 9.71 622 9.78 476 0.21 524 9.93 146 39 22 9.71 643 9.78 505 0.21 495 9.93 138 38 23 9.71 664 9.78 533 0.21 467 9.93 131 37 24 9.71 685 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9.92 991 19 42 9.72 055 9.79 072 0.20 928 9.92 983 18 43 9.72 075 9.79 100 0.20 900 9.92 976 17 44 9.72 096 9.79 128 0.20 872 9.92 968 16 8 7 20 28 8 45 9.72 116 21 20 20 21 9.79 156 29 28 28 28 0.20 844 9.92 960 8 8 8 7 15 1 0.8 0.7 46 9.72 137 9.79 185 0.20 815 9.92 952 14 2 1.6 1.4 47 9.72 157 9.79 213 0.20 787 9.92 944 13 3 2.4 2.1 48 9.72 177 9.79 241 0.20 759 9.92 936 12 4 3.2 2.8 49 9.72 198 9.79 269 0.20 731 9.92 929 11 5 4.0 3.5 20 28 8 6 4.8 4.2 50 9.72 218 20 21 20 20 9.79 297 29 28 28 28 0.20 703 9.92 921 8 8 8 8 IO 7 5.6 4.9 51 9.72 238 9.79 326 0.20 674 9.92 913 9 8 6.4 5.6 52 9.72 259 9.79 354 0.20 646 9.92 905 8 9 7.2 6.3 53 9.72 279 9.79 382 0.20 618 9.92 897 7 54 9.72 299 9.79 410 0.20 590 9.92 889 6 21 28 8 55 9.72 320 20 20 21 20 9.79 438 28 29 28 28 0.20 562 9.92 881 7 8 8 8 5 56 9.72 340 9.79 466 0.20 534 9.92 874 4 57 9.72 360 9.79 495 0.20 505 9.92 866 3 58 9.72 381 9.79 523 0.20 477 9.92 858 2 59 9.72 401 9.79 551 0.20 449 9.92 850 1 20 28 8 60 9.72 421 9.79 579 0.20 421 9.92 842 O LCos d LCot cd LTan LSin d / PP 58 Logarithms of Trigonometric Functions 60 32 Logarithms of Trigonometric Functions [II / LSin d LTan cd LCot LCos d PP o 9.72 421 20 20 21 20 9.79 579 28 28 28 28 0.20 421 9.92 842 8 8 8 8 60 1 9.72 441 9.79 607 0.20 393 9.92 834 59 2 9.72 461 9.79 635 0.20 365 9.92 826 58 3 9.72 482 9.79 663 0.20 337 9.92 818 57 4 9.72 502 9.79 691 0.20 309 9.92 810 56 20 28 7 5 9.72 522 20 20 20 20 9.79 719 28 29 28 28 0.20 281 9.92 803 8 8 8 8 55 6 9.72 542 9.79 747 0.20 253 9.92 795 54 7 9.72 562 9.79 776 0.20 224 9.92 787 53 8 9.72 582 9.79 804 0.20 196 9.92 779 52 9 9.72 602 9.79 832 0.20 168 9.92 771 51 29 28 27 20 28 8 io 9.72 622 21 20 20 20 9.79 860 28 28 28 28 0.20 140 9.92 763 8 8 8 8 50 1 2.9 2.8 2.7 11 9.72 643 9.79 888 0.20 112 9.92 755 49 2 5.8 5.6 5.4 12 9.72 663 9.79 916 0.20 084 9.92 747 48 3 8.7 8.4 8.1 13 9.72 683 9.79 944 0.20 056 9.92 739 47 4 11.6 11.2 10.8 14 9.72 703 9.79 972 0.20 028 9.92 731 46 5 14.5 14.0 13.5 20 28 8 6 17.4 16.8 16.2 15 9.72 723 20 20 20 20 9.80 000 28 28 28 28 0.20 000 9.92 723 8 8 8 8 45 7 20.3 19.6 18.9 16 9.72 743 9.80 028 0.19 972 9.92 715 44 8 23.2 22.4 21.6 17 9.72 763 9.80 056 0.19 944 9.92 707 43 9 26.1 25.2 24.3 18 9.72 783 9.80 084 0.19 916 9.92 699 42 19 9.72 803 9.80 112 0.19 888 9.92 691 41 ^ 20 28 .8 20 9.72 823 20 20 20 19 9.80 140 28 27 28 28 0.19 860 9.92 683 8 8 8 8 40 21 9.72 843 9.80 168 0.19 832 9.92 675 39 22 9.72 863 9.80 195 0.19 805 9.92 667 38 23 9.72 883 9.80 223 0.19 777 9.92 659 37 24 9.72 902 9.80 251 0.19 749 9.92 651 36 20 28 8 25 9.72 922 20 20 20 20 9.80 279 28 28 28 28 0.19 721 9.92 643 8 8 8 8 35 26 9.72 942 9.80 307 0.19 693 9.92 635 34 21 20 19 27 9.72 962 9.80 335 0.19 665 9.92 627 33 28 9.72 982 9.80 363 0.19 637 9.92 619 32 1 2.1 2.0 1.9 29 9.73 002 9.80 391 0.19 609 9.92 611 31 2 4.2 4.0 3.8 20 28 8 3 6.3 6.0 5.7 SO 9.73 022 19 20 20 20 9.80 419 28 27 28 28 0.19 581 9.92 603 8 8 8 8 30 4 8.4 8.0 7.6 31 9.73 041 9.80 447 0.19 553 9.92 595 29 5 10.5 10.0 9.5 32 9.73 061 9.80 474 0.19 526 9.92 587 28 6 12.6 12.0 11.4 33 9.73 081 9.80 502 0.19 498 9.92 579 27 7 14.7 14.0 13.3 34 9.73 101 9.80 530 0.19 470 9.92 571 26 8 16.8 16.0 15.2 20 28 8 9 18.9 18.0 17.1 35 9.73 121 19 20 20 20 9.80 558 28 28 28 27 0.19 442 9.92 563 8 9 8 8 25 36 9.73 140 9.80 586 0.19 414 9.92 555 24 37 9.73 160 9.80 614 0.19 386 9.92 546 23 38 9.73 180 9.80 642 0.19 358 9.92 538 22 39 9.73 200 9.80 669 0.19 331 9.92 530 21 19 28 8 40 9.73 219 20 20 19 20 9.80 697 28 28 28 27 0.19 303 9.92 522 8 8 8 8 20 41 9.73 239 9.80 725 0.19 275 9.92 514 19 42 9.73 259 9.80 753 0.19 247 9.92 506 18 43 9.73 278 9.80 781 0.19 219 9.92 498 17 44 9.73 298 9.80 808 0.19 192 9.92 490 16 9 8 7 20 28 8 45 9.73 318 19 20 20 19 9.80 836 28 28 27 28 0.19 164 9.92 482 9 8 8 8 15 1 0.9 0.8 0.7 46 9.73 337 9.80 864 0.19 136 9.92 473 14 2 1.8 1.6 1.4 47 9.73 357 9.80 892 0.19 108 9.92 465 13 3 2.7 2.4 2.1 48 9.73 377 9.80 919 0.19 081 9.92 457 12 4 3.6 3.2 2.8 49 9.73 396 9.80 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319 9.87 446 30 4 6.0 5.6 31 9.82 141 9.94 706 0.05 294 9.87 434 29 5 7.5 7.0 32 9.82 155 9.94 732 0.05 268 9.87 423 28 6 9.0 8.4 33 9.82 169 9.94 757 0.05 243 9.87 412 27 7 10.5 9.8 34 9.82 184 9.94 783 0.05 217 9.87 401 26 8 12.0 11.2 14 25 9 13.5 12.6 35 9.82 198 14 14 14 15 9.94 808 26 25 25 26 0.05 192 9.87 390 25 36 9.82 212 9.94 834 0.05 166 9.87 378 24 37 9.82 226 9.94 859 0.05 141 9.87 367 23 38 9.82 240 9.94 884 0.05 116 9.87 356 22 39 9.82 255 9.94 910 0.05 090 9.87 345 21 14 25 40 9.82 269 14 14 14 15 9.94 935 26 25 26 25 0.05 065 9.87 334 20 41 9.82 283 9.94 961 0.05 039 9.87 322 19 42 9.82 297 9.94 986 0.05 014 9.87 311 18 43 9.82 311 9.95 012 0.04 988 9.87 300 17 44- 9.82 326 9.95 037 0.04 963 9.87 288 16 12 11 14 25 45 9.82 340 14 14 14 14 9.95 062 26 25 26 25 0.04 938 9.87 277 15 1 1.2 1.1 46 9.82 354 9.95 088 0.04 912 9.87 266 14 2 2.4 2.2 47 9.82 368 9.95 113 0.04 887 9.87 255 13 3 3.6 3.3 48 9.82 382 9.95 139 0.04 861 9.87 243 12 4 4.8 4.4 49 9.82 396 9.95 164 0.04 836 9.87 232 11 5 6.0 5.5 14 26 6 7.2 6.6 50 9.82 410 14 15 14 14 9.95 190 25 25 26 25 0.04 810 9.87 221 IO 7 8.4 7.7 51 9.82 424 9.95 215 0.04 785 9.87 209 9 8 9.6 8.8 52 9.82 439 9.95 240 0.04 760 9.87 198 8 9 10.8 9.9 53 9.82 453 9.95 266 0.04 734 9.87 187 7 54 9.82 467 9.95 291 0.04 709 9.87 175 6 14 26 55 9.82 481 14 14 14 14 9.95 317 25 26 25 25 0.04 683 9.87 164 5 56 9.82 495 9.95 342 0.04 658 9.87 153 4 57 9. "82 509 9.95 368 0.04 632 9.87 141 3 58 9.82 523 9.95 393 0.04 607 9.87 130 2 59 9.82 537 9.95 418 0.04 582 9.87 119 1 14 26 12 60 9.82 551 9.95 444 0.04 556 9.87 107 O LCos d LCot cd LTan LSin d r PP 48 Logarithms of Trigonometric Functions 70 42 Logarithms of Trigonometric Functions 1 LSin * LTan cd LCot LCos d PP o 9.82 551 14 14 14 14 9.95 444 0'\ 04 556 9.87 107 n n 12 11 60 1 9.82 565 9.95 469 40 26 25 25 0.04 531 9.87 096 59 2 9.82 579 9.95 495 0.04 505 9.87 08 58 3 9.82 593 9.95 520 0.04 4S() 9.87 073 57 4 9.82 607 9.95 545 0.04 455 9.87 062 56 14 26 12 5 9.82 621 14 14 14 14 9.95 571 25 26 25 25 0.04 429 9.87 050 11 11 12 11 55 6 9.82 635 9.95 596 0.04 404 9.87 039 54 7 9.82 649 9.95 622 0.04 378 9.87 028 53 8 9.82 663 9.95 647 0.04 353 9.87 016 52 9 9.82 677 9.95 672 0.04 328 9.87 005 51 26 25 14 26 12 io 9.82 691 14 14 14 14 9.95 698 25 25 26 25 0.04 302 9.86 993 11 12 11 12 50 1 2.6 2.5 11 9.82 705 9.95 723 0.04 277 9.86 982 49 2 5.2 5.0 12 9.82 719 9.95 748 0.04 252 9.86 970 48 3 7.8 7.5 13 9.82 733 9.95 774 0.04 226 9.86 959 47 4 10.4 10.0 14 9.82 747 9.95 799 0.04 201 9.86 947 46 5 13.0 12.5 14 26 11 6 15.6 15.0 15 9.82 761 14 13 14 14 9.95 825 25 25 26 25 0.04 175 9.86 936 12 11 11 12 45 7 18.2 17.5 16 9.82 775 9.95 850 0.04 150 9.86 924 44 8 20.8 20.0 17 9.82 788 9.95 875 0.04 125 9.86 913 43 9 23.4 22.5 18 9.82 802 9.95 901 0.04 099 9.86 902 42 19 9.82 816 9.95 926 0.04 074 9.86 890 41 14 26 11 20 9.82 830 14 14 14 13 9.95 952 25 25 26 25 0.04 048 9.86 879 12 12 11 12 40 21 9.82 844 9.95 977 0.04 023 9.86 867 39 22 9.82 858 9.96 002 0.03 998 9.86 855 38 23 9.82 872 9.96 028 0.03 972 9.86 844 37 24 9.82 885 9.96 053 0.03 947 9.86 832 36 14 25 11 25 9.82 899 14 14 14 14 9.96 078 26 25 26 25 0.03 922 9.86 821 12 11 12 11 35 26 9.82 913 9.96 104 0.03 896 9.86 809 34 14 13 27 9.82 927 9.96 129 0.03 871 9.86 798 33 28 9.82 941 9.96 155 0.03 845 9.86 786 32 1 1.4 1.3 29 9.82 955 9.96 180 0.03 820 9.86 775 31 2 2.8 2.6 13 25 12 3 4.2 3.9 30 9.82 968 14 14 14 13 9.96 205 26 25 25 26 0.03 795 9.86 763 11 12 12 11 30 4 5.6 5.2 31 9.82 982 9.96 231 0.03 769 9.86 752 29 5 7.0 6.5 32 9.82 996 9.96 256 0.03 744 9.86 740 28 6 8.4 7.8 33 9.83 010 9.96 281 0.03 719 9.86 728 27 7 9.8 9.1 34 9/83 023 9.96 307 0.03 693 9.86 717 26 8 11.2 10.4 14 25 12 9 12.6 11.7 35 9.83 037 14 14 13 14 9.96 332 25 26 25 25 0.03 668 9.86 705 11 12 12 11 25 36 9.83 051 9.96 357 0.03 643 9.86 694 24 37 9.83 065 9.96 383 0.03 617 9.86 682 23 38 9.83 078 9.96 408 0.03 592 9.86 670 22 39 9.83 092 9.96 433 0.03 567 9.86 659 21 14 26 12 40 9.83 106 14 13 14 14 9.96 459 25 26 25 25 0.03 541 9.86 647 12 11 12 12 20 41 9.83 120 9.96 484 0.03 516 9.86 635 19 42 9.83 133 9.96 510 0.03 490 9.86 624 18 43 9.83 147 9.96 535 0.03 465 9.86 612 17 44 9.83 161 9.96 560 0.03 440 9.86 600 16 12 11 13 26 11 45 9.83 174 14 14 13 14 9.96 586 25 25 26 25 0.03 414 9.86 589 12 12 11 12 15 1 12 1.1 46 9.83 188 9.96 611 0.03 389 9.86 577 14 2 2.4 2.2 47 9.83 202 9.96 636 0.03 364 9.86 565 13 3 3.6 3.3 48 9.83 215 9.96 662 0.03 338 9.86 554 12 4 4.8 4.4 49 9.83 229 9.96 687 0.03 313 9.86 542 11 5 6.0 5.5 13 25 12 6 7.2 6.6 5(> 9.83 242 14 14 13 14 9.96 712 26 25 25 26 0.03 288 9.86 530 12 11 12 12 IO 7 8.4 7.7 51 9.83 256 9.96 738 0.03 262 9.86 518 9 8 9.6 8.8 52 9. S3 270 9.96 763 0.03 237 9.86 507 8 9 10.8 9.9 53 9.83 283 9.96 7SS 0.03 212 9.86 495 7 54 9.83 297 9.96 814 0.03 186 9.86 483 6 13 25 11 55 9.83 310 14 14 13 14 9.96 839 25 26 25 25 0.03 101 9.86 472 12 12 12 11 5 56 9.83 324 9.96 864 0.03 136 9. 86 460 4 57 9. S3 338 9.96 890 0.03 110 9.86 448 3 58 9.83 351 9.96 915 0.03 085 9.86 436 2 59 9. S3 365 9.96 940 0.03 060 9.S6 425 1 13 26 12 60 9.83 378 9.96 966 0.03' 03V 9.86 413 L Cos d LCot cd LTan LSin d / PP 47 Logarithms of Trigonometric Functions II] 43 Logarithms of Trigonometric Functions 71 / LSin d LTan cd LCot LCos d PP o 9.83 378 14 13 14 13 9.96 966 25 25 26 25 0.03 034 9.86 413 12 12 12 11 GO 1 9.83 392 9.96 991 0.03 009 9.86 401 59 2 9.83 405 9.97 016 0.02 984 9.86 389 58 3 9.83 419 9.97 042 0.02 958 9.86 377 57 4 9.83 432 9.97 067 0.02 933 9.86 366 56 14 25 12 5 9.83 446 13 14 13 14 9.97 092 26 25 25 25 0.02 908 9.86 354 12 12 12 12 55 6 9.83 459 9.97 118 0.02 882 9.86 342 54 7 9.83 473 9.97 143 0.02 857 9.86 330 53 8 9.83 486 9.97 168 0.02 832 9.86 318 52 9 9.83 500 9.97 193 0.02 807 9.86 306 51 26 25 13 26 11 io 9.83 513 14 13 14 13 9.97 219 25 25 26 25 0.02 781 9.86 295 12 12 12 12 50 1 2.6 2.5 11 9.83 527 9.97 244 0.02 756 9.86 283 49 2 5.2 5.0 12 9.83 540 9.97 269 0.02 731 9.86 271 48 3 7.8 7.5 13 9.83 554 9.97 295 0.02 705 9.86 259 47 4 10.4 10.0 14 9.83 567 9.97 320 0.02 680 9.86 247 46 5 13.0 12.5 14 25 12 6 15.6 15.0 15 9.83 581 13 14 13 13 9.97 345 26 25 25 26 0.02 655 9.86 235 12 12 11 12- 45 7 18.2 17.5 16 9.83 594 9.97 371 0.02 629 9.86 223 44 8 20.8 20.0 17 9.83 608 9.97 396 0.02 604 9.86 211 43 9 23.4 22.5 18 9.83 621 9.97 421 0.02 579 9.86 200 42 19 9.83 634 9.97 447 0.02 553 9.86 188 41 14 25 12 20 9.83 648 13 13 14 13 9.97 472 25 26 25 25 0.02 528 9.86 176 12 12 12 12 40 21 9.83 661 9.97 497 0.02 503 9.86 164 39 22 9.83 674 9.97 523 0.02 477 9.86 152 38 23 9.83 688 9.97 548 0.02 452 9.86 140 37 24 9.83 701 9.97 573 0.02 427 9.86 128 36 14 25 12 25 9.83 715 i3 13 14 13 9.97 598 26 25 25 26 0.02 402 9.86 116 j 19 9.86 104 to 35 26 9.83 728 9.97 624 0.02 376 34 14 IS 27 9.83 741 9.97 649 0.02 351 9.86 092 12 12 33 28 9.83 755 9.97 674 0.02 326 9.86 080 32 1 1.4 1.3 29 9.83 768 9.97 700 0.02 300 9.86 068 31 2 2.8 2.6 13 25 12 3 4.2 3.9 30 9.83 781 14 13 13 13 9.97 725 25 26 25 25 0.02 275 9.86 056 12 1 o 30 4 5.6 5.2 31 9.83 795 9.97 750 0.02 250 9.86 044 29 5 7.0 6.5 32 9.83 808 9.97 776 0.02 224 9.86 032 ! to 9.86 020 }o 9.86 008 LZ 28 6 8.4 7.8 33 9.83 821 9.97 801 0.02 199 27 7 9.8 9.1 34 9.83 834 9.97 826 0.02 174 26 8 11.2 10.4 14 25 12 9 12.6 11.7 35 9.83 848 13 13 13 14 9.97 851 26 25 25 26 0.02 149 9.85 996 | 19 9.85 984 1 jo 9.85 972 jo 9.85 960 ! to 25 36 9.83 861 9.97 877 0.02 123 24 37 9.83 874 9.97 902 0.02 098 23 38 9.83 887 9.97 927 0.02 073 22 39 9.83 901 9.97 953 0.02 047 9.85 948 1 21 13 25 12 40 9.83 914 13 13 14 13 9.97 978 25 26 25 25 0.02 022 9.85 936 12 12 12 12 20 41 9.83 927 9.98 003 0.01 997 9.85 924 19 42 9.83 940 9.98 029 0.01 971 9.85 912 18 43 9.83 954 9.98 054 0.01 946 9.85 900 17 44 9.83 967 9.98 079 0.01 921 9.85 888 16 12 11 13 25 12 45 9.83 980 13 13 14 13 9.98 104 26 25 25 26 0.01 896 9.85 876 12 13 12 12 15 1 1.2 1.1 46 9.83 993 9.98 130 0.01 870 9.85 864 14 2 2.4 2.2 47 9.84 006 9.98 155 0.01 845 9.85 851 13 3 3.6 3.3 48 9.84 020 9.98 180 0.01 820 9.85 839 12 4 4.8 4.4 49 9.84 033 9.98 206 0.01 794 9.85 827 11 5 6.0 5.5 13 25 12 6 7.2 6.6 50 9.84 046 13 13 13 13 9.98 231 25 25 26 25 0.01 769 9.85 815 12 12 12 13 IO 7 8.4 7.7 51 9.84 059 9.98 256 0.01 744 9.85 803 9 8 9.6 8.8 52 9.84 072 9.98 281 0.01 719 9.85 791 8 9 10.8 9.9 53 9.84 OSS 9.98 307 0.01 693 9.85 779 7 54 9.84 098 9.98 332 0.01 668 9.85 766 6 14 25 12 55 9.84 112 13 13 13 13 9.98 357 26 25 25 25 0.01 643 9.85 754 12 12 12 12 5 56 9.84 125 9.98 383 0.01 617 9.85 742 4 57 9.84 138 9.98 408 0.01 592 9.85 730 3 58 9.84 151 9.98 433 0.01 567 9.85 718 2 59 9.84 164 9.98 458 0.01 542 9.85 706 1 13 26 13 60 9.84 177 9.98 484 O'.Ol 516 9.85 693 O LCos d LCot cd LTan LSin ! d t PP 46 Logarithms of Trigonometric Functions 72 44 Logarithms of Trigonometric Functions [II r LSin d LTan cd LCot LCos d PP 9.84 177 13 13 13 13 9.98 484 25 25 26 25 0.01 516 9.85 693 12 12 12 12 60 1 9.84 190 9.98 509 0.01 491 9.85 681 59 2 9.84 203 9.98 534 0.01 466 9.85 669 58 3 9.84 216 9.98 560 0.01 440 9.85 657 57 4 9.84 229 9.98 585 0.01 415 9.85 645 56 13 25 13 5 9.84 242 13 14 13 13 9.98 610 25 26 25 25 0.01 390 9.85 632 12 12 12 13 55 6 9.84 255 9.98 635 0.01 365 9.85 620 54 7 9.84 269 9.98 661 0.01 339 9.85 608 53 8 9.84 282 9.98 686 0.01 314 9.85 596 52 9 9.84 295 9.98 711 0.01 289 9.85 583 51 13 26 12 io 9.84 308 13 13 13 13 9.98 737 25 25 25 26 0.01 263 9.85 571 12 12 13 12 50 11 9.84 321 9.98 762 0.01 238 9.85 559 49 12 9.84 334 9.98 787 0.01 213 9.85 547 48 13 9.84 347 9.98 812 0.01 188 9.85 534 47 14 9.84 360 9.98 838 0.01 162 9.85 522 46 13 25 12 26 25 14 15 9.84 373 12 13 13 13 9.98 863 25 25 26 25 0.01 137 9.85 510 13 12 12 13 45 16 9.84 385 9.98 888 0.01 112 9.85 497 44 1 2.6 2.5 1.4 17 9.84 398 9.98 913 0.01 087 9.85 485 43 2 5.2 5.0 2.8 18 9.84 411 9.98 939 0.01 061 9.85 473 42 3 7.8 7.5 4.2 19 9.84 424 9.98 964 0.01 036 9.85 460 41 4 10.4 10.0 5.6 13 25 12 5 13.0 12.5 7.0 20 9.84 437 13 13 13 13 9.98 989 26 25 25 25 0.01 Oil 9.85 448 12 13 12 12 40 6 15.6 15.0 8.4 21 9.84 450 9.99 015 0.00 985 9.85 436 39 7 18.2 17.5 9.8 22 9.84 463 9.99 040 0.00 960 9.85 423 38 8 20.8 20.0 11.2 23 9.84 476 9.99 065 0.00 935 9.85 411 37 9 23.4 22.5 12.6 24 9.84 489 9.99 090 0.00 910 9.85 399 36 13 26 13 25 9.84 502 13 13 12 13 9.99 116 25 25 25 26 0.00 884 9.85 386 12 13 12 12 35 26 9.84 515 9.99 141 0.00 859 9.85 374 34 27 9.84 528 9.99 166 0.00 834 9.85 361 33 28 9.84 540 9.99 191 0.00 809 9.85 349 32 29 9.84 553 9.99 217 0.00 783 9.85 337 31 13 25 13 30 9.84 566 13 13 13 13 9.99 242 25 26 25 25 0.00 758 9.85 324 12 13 12 13 30 31 9.84 579 9.99 267 0.00 733 9.85 312 29 32 9.84 592 9.99 293 0.00 707 9.85 299 28 33 9.84 605 9.99 318 0.00 682 9.85 287 27 34 9.84 618 9.99 343 0.00 657 9.85 274 26 12 25 12 35 9.84 630 13 13 13 13 9.99 368 26 25 25 25 0.00 632 9.85 262 12 13 12 13 25 36 9.84 643 9.99 394 0.00 606 9.85 250 24 37 9.84 656 9.99 419 0.00 581 9.85 237 23 38 9.84 669 9.99 444 0.00 556 9.85 225 22 13 12 39 9.84 682 9.99 469 0.00 531 9.85 212 21 12 26 12 1 1.3 1.2 40 9.84 694 13 13 13 12 9.99 495 25 25 25 26 0.00 505 9.85 200 13 12 13 12 2 2 2.6 2.4 41 9.84 707 9.99 520 0.00 480 9.85 187 19 3 3.9 3.6 42 9.84 720 9.99 545 0.00 455 9.85 175 18 4 5.2 4.8 43 9.84 733 9.99 570 0.00 430 9.85 162 17 5 6.5 6.0 44 9.84 745 9.99 596 0.00 404 9.85 150 16 6 7.8 7.2 13 25 13 7 9.1 8.4 45 9.84 758 13 13 12 13 9.99 621 25 26 25 25 0.00 379 9.85 137 12 13 12 13 15 8 10.4 9.6 46 9.84 771 9.99 646 0.00 354 9.85 125 14 9 11.7 10.8 47 9.84 784 9.99 672 0.00 328 9.85 112 13 48 9.84 796 9.99 697 0.00 303 9.85 100 12 49 9.84 809 9.99 722 0.00 278 9.85 087 11 13 25 13 5 9.84 822 13 12 13 13 9.99 747 26 25 25 25 0.00 253 9.85 074 12 13 12 13 IO 51 9.84 835 9.99 773 0.00 227 9.85 062 9 52 9.84 847 9.99 798 0.00 202 9.85 049 8 53 9.84 860 9.99 823 0.00 177 9.85 037 7 54 9.84 873 9.99 848 0.00 152 9.85 024 6 12 26 12 55 9.84 885 13 13 12 13 9.99 874 25 25 25 26 0.00 126 9.85 012 13 13 12 13 5 56 9.84 898 9.99 899 0.00 101 9.84 999 4 57 9.84 911 9.99 924 0.00 076 9.84 986 3 58 9.84 923 9.99 949 0.00 051 9.84 974 2 59 9.84 936 9.99 975 0.00 025 9.84 961 1 13 25 12 60 9.84 949 0.00 000 0.00 000 9.84 949 LCos d LCot cd LTan LSin d / PP 45 Logarithms of Trigonometric Functions TABLE HI Values of the Natural Trigonometric Functions for Every Minute from to 90 to Five Decimal Places Ill] O 6 Natural Functions l c 75 ' N Sin N Tan N Cot NCOS .000001. ooooo 1 oc 1 . 0000 o 1 020 020 3437 . 7 000 59 2 058 058 1718.9 000 58 3 0S7 087 1145.9 000 57 4 116 116 859 . 44 000 56 5 .00145 .00145 687. 55 1.0000 55 6 175 175 572.96 000 54 7 2041 204 491.11 000 53 8 233 233 429.72 000 52 9 262 262 381.97 000 51 io .00291 .00291343.77 1 . 0000 50 11 320 320 312.52 .99999 49 12 349 349 286 . 48 -999 48 13 378 378 264.44 999 47 14 407 407 245.55 999 46 15 . 00436 .00436 229.18 . 99999 45 16 465 465 214.86 999 44 17 495 495 202 . 22 999 43 18 524 524 190 . 98 999 42 19 553 553 180.93 998 41 20 . 00582 . 00582 171.89 . 99998 40 21 611 611J163.70 998 39 22 640 6401156.26 998 38 23 669 669 149.47 998 37 24 698 698 143. 24 998 36 25 . 00727 .00727 137.51 .99997 35 26 7561 756 132.22 997 34 27 785 785J127.32 997 33 28 814 815 122.77 997 32 29 844 844 118.54 996 31 30 . 00873 .00873' 114. 59 . 99996 30 31 902 902 110.89 996 29 32 931 931 107.43 996 28 33 960 960 104.17 995 27 34 . 00989 .00989 101.11 995 26 35 .01018 .01018'98.218 . 99995 25 36 047 047 95 . 489 995 24 37 076 076 92.908 994 23 38 105 105 90.463 994 22 39 134 133 88.144 994 21 40 .01164 .01164 85.940 . 99993 20 41 193 193 83 . 844 993 19 42 222 222 81.847 993 18 43 251 25179.943 992 17 44 280. 280 78.126 992 16 45 .01309'. 01809 76.390 .99991 15 46 338 338 74.729 991 14 47 367 367 73.139 991 13 48 396 396 71.615 990 12 49 425 425,70.153 990 11 50 . 01454 .01455 68.750 . 99989 10 51 483 484 67 . 402 989 9 52 513 513 66.105 989 8 53 542 542 64.858 988 7 54 571 571;63.657 988 6 55 .01600 .01600 62.499 . 99987 5 56 629 629 61.383 987 4 57 658 658 60.306 986 3 58 687 687159.266 986 2 59 716 71658.261 985 1 60 . 01745 .01746 57.290 . 99985 O NCos N Cot N Tan NSin / ' N Sin'NTanJN Cot 1 N Cos o .01745 .01746 57.290 .99985 60 1 774| 775 56.351 984 59 2 8031 804 55.442 984 58 3 832 833 54.561 983 57 4 862 862 53 . 709 983 56 5 .01891 .01891 52 . 882 . 99982 55 6 920 920 52.081 982 54 7 949 949 51.303 981 53 8 .01978 .01978 50.549 980 52 9 . 02007 .02007 49.816 980 51 IO . 02036 . 02036149. 104 . 99979 50 11 065 066 48.412 979 49 12 094 095! 47. 740 978 48 13 123 124 47.085 977 47 14 152 153 46 . 449 977 46 15 .02181 .02182 45 . 829 . 99976 45 16 211! 211 45 . 226 976 44 17 240 240 44 . 639 975 43 18 269 269 44.066 974 42 19 298 298 43 . 508 974 41 20 . 02327 . 02328 42.964 . 99973 40 21 356 357 42 . 433 972 39 22 385 386 41.916 972 38 23 414 415 41.41 971 37 24 443 444 40.917 970 36 25 . 02472 I 02473 40 . 436 . 99969 35 26 501 502 39 . 965 969 34 27 530 531 39 . 506 968 33 28 560 560 39 . 057 967 32 29 589 589 38.618 966 31 30 .02618 .02619 38.188 . 99966 30 31 647 648 37 . 769 965 29 32 676 677 37.358 964 28 33 705 706 36 . 956 963 27 34 734 735 36 . 563 963 26 35 . 02763 .02764 36.178 . 99962 25 36 792 793 35.801 961 24 37 821 822 35.431 960 23 38 850 851 35.070 959 22 39 879 881 34,715 959 21 40 . 02908 .02910 34.368 . 99958 20 41 938 939 34 . 027 957 19 42 967 968 33 . 694 956 18 43 . 02996 . 02997 33 . 366 955 17 44 .03025 . 03026 33.045 954 16 45 . 03054 . 03055 32 . 730 ,99953 15 46 OSS! 084 ll^ 114 32.421 952 14 47 ' 32.118 952 13 48 141 143 31.821 951 12 49 170 172 31.528 950 11 50 .03199 .03201 31.242 . 99949 IO 51 228 230 30 . 960 948 9 52 257 259 30.683 947 8 53 286 288 30.412 946 7 54 316 317 30.145 945 6 55 . 03345 .03346 29.882 . 99944 5 56 374 376 29 . 624 943 4 57 403 405 29.371 942 3 58 432 434 29.122 941 2 59 461 463 28.877 940 1 60 . 03490 .03492 28.636 . 99939 O NCOS N Cot NTan NSin i 89 Natural Functions 88 c 76 2 Natural Functions 3 C [III / N Sin NTan N Cot N Cos o .03490 . 03492 28.636 .99939 GO 1 519 521 .399 938 59 2 548 550 28.166 937 58 3 577 579 27 . 937 936 57 4 606 609 .712 935 56 5 . 03635 .03638 27 . 490 . 99934 55 6 664 667 .271 933 54 7 693 696 27.057 932 53 8 723 725 26.845 931 52 9 752 754 .637 930 51 io .03781 . 03783 26.432 . 99929 50 11 810 812 .230 927 49 12 839 842 26.031 926 48 13 868 871 25 . 835 925 47 14 897 900 .642 924 46 15 . 03926 . 03929 25.452 . 99923 45 16 955 958 .264 922 44 17 . 03984 . 03987 25.080 921 43 18 . 04013 .04016 24.898 919 42 19 042 046 .719 918 41 20 . 04071 .04075 24.542 . 99917 40 21 100 104 .368 916 39 22 129 133 .196 915 38 23 159 162 24.026 913 37 24 188 191 23.859 912 36 25 .04217 . 04220 23 . 695 .99911 35 26 246 250 .532 910 34 27 275 279 .372 909 33 28 304 308 .214 907 32 29 333 337 23 . 058 906 31 30 .04362 . 04366 22.904 . 99905 30 31 391 395 .752 904 29 32 420 424 .602 902 28 33 449 454 .454 901 27 34 478 483 .308 900 26 35 .04507 .04512 22.164 . 99898 25 36 536 541 22.022 897 24 37 565 570 21.881 896 23 38 594 599 .743 894 22 39 623 628 .606 893 21 40 .04653 . 04658 21.470 . 99892 20 41 682 687 .337 890 19 42 711 716 .205 889 18 43 740 745 21.075 888 17 44 769 774 20.946 886 16 45 . 04798 . 04803 20.819 .99885 15 46 827 833 .693 883 14 47 856 862 .569 882 13 48 885 891 .446 881 12 49 914 920 .325 879 11 50 .04943 . 04949 20.206 .99878 IO 51 .04972 .04978 20.087 876 9 52 .05001 .05007 19.970 875 8 53 030 037 .855 873 7 54 059 066 .740 872 6 55 . 05088 .05095 19.627 . 99870 5 56 117 124 .516 869 4 57 146 153 .405 867 3 58 175 182 .296 866 2 59 205 212 .188 864 1 60 . 05234 . 05241 19.081 . 99863 O NCOS N Cot NTan NSin / / N Sin NTan N Cot N Cos O . 05234 . 05241 19.081 . 99863 OO 1 263 270 18.976 861 59 2 292 299 .871 860 58 3 321 328 .768 858 57 4 350 357 .666 857 56 5 . 05379 . 05387 18.564 . 99855 55 6 408 416 .464 854 54 7 437 445 .366 852 53 8 466 474 .268 851 52 9 495 503 .171 849 51 IO . 05524 . 05533 18.075 . 99847 50 11 553 562 17 . 980 846 49 12 582 591 .886 844 48 13 611 620 .793 842 47 14 640 649 .702 841 46 15 . 05669 . 05678 17.611 . 99839 45 16 698 708 .521 838 44 17 727 737 .431 836 43 18 756 766 .343 834 42 19 785 795 .256 833 41 20 . 05814 . 05824 17.169 .99831 40 21 844 854 17 . 084 829 39 22 873 883 16 . 999 827 38 23 902 912 .915 826 37 24 931 941 .832 824 36 25 . 05960 . 05970 16.750 . 99822 35 26 . 05989 . 05999 .668 821 34 27 .06018 . 06029 .587 819 33 28 047 058 .507 817 32 29 076 087 .428 815 31 30 .06105 .06116 16.350 .99813 30 31 134 145 .272 812 29 32 163 175 .195 810 28 33 192 204 119 80S - ST- " zzf 233 16 . 043 806 35 . 06250 . 06262 15 . 969 . 99804 25 36 279 291 .895 803 24 37 308 321 .821 801 23 38 337 350 .748 799 22 39 366 379 .676 797 21 40 . 06395 . 06408 15 . 605 . 99795 20 41 424 438 .534 793 19 42 453 467 .464 792 18 43 482 496 .394 790 17 44 511 525 .325 788 16 45 . 06540 . 06554 15 . 257 . 99786 15 46 569 584 .189 784 14 47 598 613 .122 782 13 48 627 642 15 . 056 780 12 49 656 671 14 . 990 778 11 50 . 06685 .06700 14.924 . 99776 IO 51 714 730 .860 774 9 52 743 759 .795 772 8 53 773 788 .732 770 7 54 802 817 .669 768 6 55 .06831 .06847 14.606 . 99766 5 56 860 876 .544 764 4 57 889 905 .482 762 3 58 918 934 .421 760 2 59 947 963 .361 758 1 OO . 06976 . 06993 14.301 . 99756 O NCOS N Cot NTan NSin ' 87 Natural Functions 86 Ill] 4 Natural Functions 5 ' N Sin NTan N Cot N Cos o .06976 .06993 14.301 . 99756 60 1 . 07005 . 07022 .241 754 59 2 034 051 .182 752 58 3 063 080 .124 750 57 4 092 110 .065 748 56 5 .07121 .07139 14.008 .99746 55 6 150 168 13.951 744 54 7 179 197 .894 742 53 8 208 227 .838 740 52 9 237 256 .782 738 51 io .07266 . 07285 13.727 . 99736 50 11 295 314 .672 734 49 12 324 344 .617 731 48 13 353 373 .563 729 47 14 382 402 .510 727 46 15 .07411 .07431 13.457 . 99725 45 16 440 461 .404 723 44 17 469 490 .352 721 43 18 498 519 .300 719 42 19 527 548 .248 716 41 20 . 07556 . 07578 13.197 .99714 40 21 585 607 .146 712 39 22 614 636 .096 710 38 23 643 665 13 . 046 708 37 24 672 695 12 . 996 705 36 25 . 07701 . 07724 12 . 947 . 99703 35 26 730 753 .898 701 34 27 759 782 .850 699 33 28 788 812 .801 696 32 29 817 841 .754 694 31 30 . 07846 . 07870 12.706 .99692 30 31 875 899 .659 689 29 32 904 929 .612 687 28 33 933 958 .566 685 27 34 962 . 07987 .520 683 26 35 . 07991 .08017 12.474 .99680 25 36 . 08020 046 .429 678 24 37 049 075 .384 676 23 38 078 104 .339 673 22 39 107 134 .295 671 21 40 .08136 .08163 12.251 . 99668 20 41 165 192 .207 666 19 42 194 221 .163 664 18 43 223 251 .120 661 17 44 252 280 .077 659 16 45 .08281 .08309 12 . 035 . 99657 15 46 310 339 11.992 654 14 47 339 368 .950 652 13 48 368 397 .909 649 12 49 397 427 .867 647 11 50 .08426 .08456 11.826 . 99644 IO 51 455 485 .785 642 9 52 484 514 .745 639 8 53 513 544 .705 637 7 54 542 573 .664 635 6 55 .08571 . 08602 11.625 . 99632 5 56 600 632 .585 630 4 57 629 661 .546 627 3 58 658 690 .507 625 2 59 687 720 .468 622 1 60 .08716 . 08749 11.430 . 99619 O NCOS N CotjNTan NSin ' / N Sin NTan N Cotl N Cos o .08716 . 08749 11.430 .99619 60 1 745 778 .392 617 59 2 774 807 .354 614 58 3 803 837 .316 612 57 4 831 866 .279 609 56 5 . 08860 . 08895 11.242 . 99607 55 6 889 925 .205 604 54 7 918 954 .168 602 53 8 947 . 08983 .132 599 52 9 .08976 .09013 .095 596 51 IO . 09005 .09042 11.059 . 99594 50 11 034 071 11.024 591 49 12 063 101 10 . 988 588 48 13 092 130 .953 586 47 14 121 159 -.918 583 46 15 .09150 . 09189 10.883 . 99580 45 16 179 218 .848 578 44 17 208 247 .814 575 43 18 237 277 .780 572 42 19 266 306 .746 570 41 20 .09295 . 09335 10.712 . 99567 40 21 324 365 .678 564 39 22 353 394 .645 562 38 23 382 423 .612 559 37 24 411 453 .579 556 36 25 .09440 .09482 10.546 .99553 35 26 469 511 .514 551 34 27 498 541 .481 548 33 28 527 570 .449 545 32 29 556 600 .417 542 31 30 . 09585 . 09629 10.385 . 99540 30 31 614 658 .354 537 29 32 642 688 .322 534 28 33 671 717 .291 531 27 34 700 746 .260 528 26 35 . 09729 .09776 10.229 . 99526 25 36 758 805 .199 523 24 37 787 834 .168 520 23 38 816 864 .138 517 22 39 845 893 .108 514 21 40 . 09874 . 09923 10.078 .99511 20 41 903 952 .048 508 19 42 932 .09981 10.019 506 18 43 961 .10011 9.9893 503 17 44 . 09990 040 .9601 500 16 45 . 10019 .10069 9.9310 . 99497 15 46 048 099 .9021 494 14 47 077 128 .8734 491 13 48 106 158 .8448 488 12 49 135 187 .8164 485 11 50 . 10164 . 10216 9.7882 .99482 IO 51 192 246 .7601 479 9 52 221 275 .7322 476 8 53 250 305 .7044 473 7 54 279 334 .6768 470 6 55 . 10308 . 10363 9 . 6493 .99467 5 56 337 393 .6220 464 4 57 366 422 .5949 461 3 58 395 452 .5679 458 2 59 424 481 .5411 455 1 60 .10453 . 10510 9.5144 .99452 O NCOS N Cot NTan NSin ' 85 Natural Functions 84 c 6 Natural Functions 7 [III ' N Sin NTan N Cot N Cos o .10453 .10510 9.5144 . 99452 60 1 482 540 .4878 449 59 2 511 569 .4614 446 58 3 540 599 .4352 443 57 4 569 628 .4090 440 56 5 . 10597 . 10657 9.3831 . 99437 55 6 626 687 .3572 434 54 7 655 716 .3315 431 53 8 684 746 .3060 428 52 9 713 775 .2806 424 51 io . 10742 . 10805 9.2553 .99421 50 11 771 834 .2302 418 49 12 800 863 .2052 415 48 13 829 893 .1803 412 47 14 858 922 .1555 409 46^ 15 . 10887 . 10952 9.1309 . 99406 15 16 916 . 10981 .1065 402 44 17 945 .11011 .0821 399 43 18 . 10973 040 .0579 396 42 19 .11002 070 .0338 393 41 20 .11031 . 11099 9 . 0098 . 99390 40 21 060 128 8.9860 386 39 22 089 158 .9623 383 38 23 118 187 .9387 380 37 24 147 217 .9152 377 36 25 .11176 . 11246 8.8919 . 99374 35 26 205 276 .8686 370 34 27 234 305 .8455 367 33 28 263 335 .8225 364 32 29 291 364 .7996 360 31 30 .11320 .11394 8.7769 . 99357 30 31 349 423 .7542 354 29 32 378 452 .7317 351 28 33 407 482 .7093 347 27 34 436 511 .6870 344 26 35 .11465 .11541 8.6648 . 99341 25 36 494 570 .6427 337 24 37 523 600 .6208 334 23 38 552 629 .5989 331 22 39 580 659 .5772 327 21 40 . 11609 .11688 8 . 5555 . 99324 20 41 638 718 .5340 320 19 42 667 747 .5126 317 18 43 696 777 .4913 314 17 44 725 806 .4701 310 16 45 .11754 . 11836 8.4490 . 99307 15 46 783 865 .4280 303 14 47 812 895 .4071 300 13 48 840 924 .3863 297 12 49 869 954 .3656 293 11 50 .11898 .11983 8.3450 . 99290 IO 51 927 . 12013 .3245 286 9 52 956 042 .3041 2S3 8 63 .11985 072 .2838 279 7 54 . 12014 101 .2636 276 6 55 . 12043 .12131 8.2434 . 99272 5 56 071 160 .2234 269 4 57 100 190 .2035 265 3 58 129 219 .1837 262 2 59 158 249 .1640 258 1 GO .12187 . 12278 8.1443 . 99255 O NCosJN Cot NTan NSin / i N Sin'NTanN Cot 1 N Cos o . 12187 .12278 8.1443 .99255 eo 1 216 308 .1248 251 59 2 245 338 .1054 248 58 3 274 367 .0860 244 57 4 302 397 .0667 240 -56 5 .12331 . 12426 8 . 0476 . 99237 55 6 360 456 .0285 233 54 7 389 485 8.0095 230 53 8 418 515 7.9906 226 52 9 447 544 .9718 222 51 IO . 12476 . 12574 7 . 9530 .99219 50 11 504 603 .9344 215 49 12 533 633 .9158 211 48 13 562 662 .8973 208 47 14 591 692 .8789 204 46 15 . 12620 . 12722 7 . 8606 . 99200 45 16 649 751 .8424 197 44 17 678 781 .8243 193 43 18 706 810 .8062 189 42 19 735 840 .7882 186 41 20 . 12764 . 12869 7 . 7704 .99182 40 21 793 899 .7525 178 39 22 822 929 .7348 175 38 23 851 958 .7171 171- 37 24 880 . 12988 .6996 167 36 25 . 12908 . 13017 7 . 6821 .99163 35 26 937 047 .6647 160 34 27 966 076 .6473 156 33 28 .12995 106 .6301 152 32 29 . 13024 136 .6129 148 31 30 . 13053 .13165 7 . 5958 .99144 30 31 081 195 .5787 141 29 32 110 224 .5618 137 28 33 139 254 .5449 133 27 34 168 284 .5281 129 26 35 . 13197 .13313 7.5113 .99125 25 36 226 343 .4947 122 24 37 254 372 .4781 118 23 38 283 402 .4615 114 22 39 312 432 .4451 110 21 40 . 13341 . 13461 7 . 4287 .99106 20 41 370 491 .4124 102 19 42 399 521 .3962 098 18 43 427 550 .3800 094 17 44 456 580 .3639 091 16 45 . 13485 . 13609 7.3479 . 99087 15 46 514 639 .3319 083 14 47 543 669 .3160 079 13 48 572 698 .3002 075 12 49 600 728 .2844 071 11 50 . 13629 . 13758 7 . 2687 . 99067 IO 51 658 787 .2531 063 9 52 687 817 . 2375 059 8 53 716 846 .2220 055 7 54 744 876 .2066 051 6 55 . 13773 . 13906 7.1912 . 99047 5 56 802 935 .1759 043 4 57 831 965 .1607 039 3 58 860 . 13995 .1455 035 2 59 889 . 14024 .1304 031 1 60 .13917 . 14054 7.1154 .99027 O NCOS N Cot NTan NSin ' 83 Natural Functions 82 c Ill] 8 Natural Functions 9 79 / N Sin N Tan N Cot N Cos t N SinNTan ! N Cot N Cos o .13917 . 14054 7.1154 .99027 OO O . 15643 . 15838 6.3138 . 98769 60 1 9461 084 .1004 023 59 1 672 868 .301S 764 59 2 .139731 113 .0855 019 58 2 701 898 .2901 750 58 3 . 14004 143 .0706 015 57 3 730 928 .2783 755 57 4 033 173 .0558 Oil 56 4 * 758 958 .2666 751 56 5 .14061 .14202 7.0410 .99006 55 5 . 15787 . 15988 6 . 2549 . 98746 55 6 090! 232 .0264. 99002 54 6 816 . 16017 .2432 741 54 7 119 262 7.0117 .98998 53 7 845 047 .2316 737 53 8 148 291 6.9972 994 52 8 873 077 .2200 - 732 52 9 177 321 .9827 990 51 9 902 107 .2085 728 51 10 . 14205 . 14351 6.9682 . 98986 50 IO . 15931 .16137 6.1970 . 98723 50 11 234 381 . 9538 982 49 11 959 167 . 1S56 718 49 12 263 410 . 9395 978 48 12 . 15988 196 .1742 714 48 13 292 440 . 9252 973 47 13 .16017 226 .1628 709 47 14 320 470 .9110 969 46 14 046 256 . 1515 704 46 15 . 14349 .14499 6. 8969 .98965 45 15 . 16074 . 16286 6.1402 . 98700 45 16 378 529 .8828 961 44 16 103 316 . 1290 695 44 17 407 559 .8687 957 43 17 132 346 .1178 690 43 18 436 588 .8548 953 42 18 160 376 .1066 686 42 19 464 618 .8408 948 41 19 189 405 .0955 681 41 20 . 14493 . 14648 6.8269 .98944 40 20 .16218 . 16435 6 . 0844 . 98676 40 21 522 678 .8131 940 39 21 246 465 .0734 671 39 22 551 707 .7994 936 38 22 275 495 .0624 667 38 23 580 737 . 7856 931 37 23 304 525 .0514 662 37 24 608 767 .7720 927 36 24 333 555 .0405 657 36 25 . 14637 . 14796 6 . 7584 . 98923 35 25 . 16361 . 16585 6 . 0296 . 98652 35 26 666 826 .7448 919 34 26 390 615 .0188 648 34 27 695 856 .7313 914 33 27 419 645 6.0080 643 33 28 723 886 .7179 910 32 28 447 674 5 . 9972 638 32 29 752 915 .7045 906 31 29 476 704 .9865 633 31 30 . 14781 . 14945 6.6912 . 98902 30 30 . 16505 . 16734 5 . 9758 . 98629 30 31 - 810 . 14975 .6779 897 29 31 533 764 .9651 624 29 32 838 . 15005 .6646 893 28 32 562 794 .9545 619 28 33 867 034 .6514 889 27 33 591 824 .9439 614 27 34 896 064 .6383 884 26 34 620 854 .9333 609 26 35 . 14925 . 15094 6.6252 . 98880 25 35 . 16648 . 16884 5 . 9228 . 98604 25 36 954 124 .6122 876 24 36 677 914 .9124 600 24 37 . 14982 153 .5992 871 23 37 706 944 .9019 595 23 38 .15011 183 .5863 867 22 38 734 . 16974 .8915 590 22 39 040 213 .5734 863 21 39 763 . 17004 .8811 585 21 JO . 15069 . 15243 6 . 5606 . 98858 20 40 . 16792 . 17033 5 . 8708 . 98580 20 41 097 272 .5478 854 19 41 820 063 .8605 575 19 42 126 302 .5350 849 18 42 849 093 .8502 570 18 43 155 332 .5223 845 17 43 878 123 .8400 565 17 44 184 362 .5097 841 16 44 906 153 .8298 561 16 45 . 15212 .15391 6.4971 . 98836 15 45 . 16935 .17183 5.8197 . 98556 15 46 241 421 .4846 832 14 46 964 213 .8095 551 14 47 270 451 .4721 827 13 47 . 16992 243 .7994 546 13 48 299 481 .4596 823 12 48 .17021 273 .7894 541 12 49 327 511 .4472 818 11 49 050 303 . 7794 536 11 50 . 15356 . 15540 6.4348 .98814 io 50 . 17078 . 17333 5.7694 .98531 IO 51 085 570 .4225 809 9 51 107 363 . 7594 526 9 52 414 600 .4103 805 8 52 136 393 .7495 521 8 53 442 630 .3980 800 7 53 164 423 .7396 516 7 54 471 660 .3859 796 6 54 193 453 .7297 511 6 55 . 15500 . 15689 6.3737 .98791 5 55 . 17222 . 17483 5.7199 . 98506 5 56 529 719 .3617 787 4 56 250 513 .7101 501 4 57 557 749 .3496 782 3 57 279 543 .7004 496 3 58 586 779 .3376 778 2 58 308 573 .6906 491 2 59 615 809 .3257 773 1 59 336 603 .6809 486 1 60 . 15643 .15838 6.3138 .98769 O 60 . 17365 . 17633 5.6713 .98481 O NCosN CotNTan N Sin r NCOS N Cot NTan NSin ' 81 Natural Functions 80 c 80 10 Natural Functions ll c ' N Sin NTanN Cot N Cos o . 17365 .17633 5.6713 .98481 60 1 393 663 .6617 476 59 2 422 693 .6521 471 58 3 451 723 .6425 466 57 4 479 753 .6329 461 56 5 . 17508 . 17783 5.6234 . 98455 55 6 537 813 .6140 450 54 7 565 843 .6045 445 53 8 594 873 .5951 440 52 9 623 903 .5857 435 51 io . 17651 . 17933 5 . 5764 . 98430 50 11 680 963 .5671 425 49 12 708 . 17993 .5578 420 48 13 737 . 18023 .5485 414 47 14 766 053 .5393 409 46 15 . 17794 . 18083 5.5301 . 98404 45 16 823 113 .5209 399 44 17 852 143 .5118 394 43 18 880 173 .5026 389 42 19 909 203 .4936 383 41 20 . 17937 . 18233 5.4845 . 98378 40 21 966 263 .4755 373 39 22 . 17995 293 .4665 368 38 23 . 18023 323 .4575 362 37 24 052 353 .4486 357 36 25 . 18081 . 18384 5.4397 . 98352 35 26 109 414 .4308 347 34 27 138 444 .4219 341 33 28 166 474 .4131 336 32 29 195 504 .4043 331 31 30 . 18224 . 18534 5.3955 . 98325 30 31 252 564 .3868 320 29 32 281 594 .3781 315 28 33 309 624 .3694 310 27 34 338 654 .3607 304 26 35 . 18367 . 18684 5.3521 . 98299 25 36 395 714 .3435 294 24 37 424 745 .3349 288 23 38 452 775 .3263 283 22 39 481 8O5 .3178 277 21 40 . 18509 . 18835 5 . 3093 . 98272 20 41 538 865 .3008 267 19 42 567 895 .2924 261 18 43 595 925 .2839 256 17 44 624 955 .2755 250 16 45 . 18652 . 18986 5.2672 . 98245 15 46 681 . 19016 .2588 240 14 47 710 046 .2505 234 13 48 738 076 .2422 229 12 49 767 106 .2339 223 11 50 . 18795 .19136 5.2257 .98218 IO 51 824 166 .2174 212 9 52 852 197 .2092 207 8 53 881 227 .2011 201 7 54 910 257 .1929 196 6 55 . 18938 . 19287 5.1848 .98190 5 56 967 317 .1767 185 4 57 . 18995 347 .1686 179 3 58 . 19024 378 .1606 174 2 59 052 408 .1526 168 1 60 . 19081 . 19438 5.1446 .98163 O NCOS N Cot N Tan NSin / o 1 2 3 4 5 6 7 8 9 IO 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 N Sin 19081 109 138 167 195 19224 252 281 309 338 19366 395 423 452 481 19509 538 566 595 623 . 19652 680 709 737 766 . 19794 823 851 880 908 . 19937 965 . 19994 . 20022 051 .20079 108 136 165 193 . 20222 250 279 307 336 . 20364 393 421 450 478 .20507 535 563 592 620 . 20649 677 706 734 763 . 20791 NTanN Cot NCos 19438 468 498 529 559 19589 619 649 680 710 19740 770 801 831 861 19891 921 952 19982 20012 20042 073 103 133 164 20194 224 254 285 315 20345 376 406 436 466 20497 527 557 588 618 . 20648 679 709 739 770 .20800 830 861 891 921 . 20952 . 20982 .21013 043 073 5.1446 .1366 .1286 .1207 .1128 5.1049 0970 0892 0814 0736 5 . 0658 0581 0504 0427 0350 5.0273 0197 0121 5 . 0045 4.9969 4.9894 .9819 .9744 .9669 .9594 4.9520 .9446 .9372 .9298 .9225 4.9152 9078 9006 8933 8860 4.8788 .8716 .8644 .8573 .8501 4 . 8430 .8359 .8288 .8218 .8147 4.8077 .8007 .7937 .7867 .7798 4.7729 .7659 .7591 .7522 .7453 N Cos 21104 4.7385 134 164 195 225 21256 .7317 .7249 .7181 .7114 4.7046 N Cot N Tan 98163 157 152 146 140 98135 129 124 118 112 98107 101 096 090 084 98079 073 067 061 056 98050 044 039 033 027 98021 016 010 98004 97998 97992 987 981 975 969 97963 958 952 946 940 . 97934 928 922 916 910 . 97905 899 893 887 881 .97875 869 863 857 851 .97845 839 833 827 821 .97815 NSin 79 Natural Functions 78 c Ill] 12 Natural Functions 13 t N Sin N Tan N Cot NCOS .20791 .21256 4.7046 .97815 60 1 820 286 .6979 809 59 2 848 316 .6912 803 58 3 877 347 .6845 797 57 4 905 377 .6779 791 56 5 .20933 .21408 4.6712 . 97784 55 6 962 438 .6646 778 54 7 . 20990 469 .6580 - 772 53 8 .21019 499 .6514 766 52 9 047 529 .6448 760 51 io .21076 .21560 4 . 6382 . 97754 50 11 104 590 .6317 748 49 12 132 621 .6252 742 48 13 161 651 .6187 735 47 14 189 682 .6122 729 46 15 .21218 .21712 4.6057 .97723 45 16 246 743 .5993 717 44 17 275 773 .5928 - 711 43 18 303 804 .5864 705 42 19 331 834 .5800 698 41 20 .21360 .21864 4.5736 . 97692 40 21 388 895 .5673 686 39 22 417 925 .5609 680 38 23 445 956 .5546 673 37 24 474 .21986 .5483 667 36 25 .21502 .22017 4.5420 .97661 35 26 530 047 .5357 655 34 27 559 078 .5294 648 33 28 587 108 .5232 642 32 29 616 139 .5169 636 31 30 . 21644 .22169 4.5107 . 97630 30 31 672 200 .5045 623 29 32 701 231 .4983 617 28 33 729 261 .4922 611 27 34 758 292 .4860 604 26 35 .21786 . 22322 4.4799 . 97598 25 36 814 353 .4737 592 24 37 843 383 .4676 585 23 38 871 414 .4615 579 22 39 899 444 .4555 573 21 40 .21928 .22475 4.4494 . 97566 20 41 956 505 .4434 560 19 42 .21985 536 .4373 553 18 43 .22013 567 .4313 547 17 44 041 597 .4253 541 16 45 . 22070 . 22628 4.4194 .97534 15 46 098 658 .4134 528 14 47 126 689 .4075 521 13 48 155 719 .4015 515 12 49 183 750 .3956 508 11 50 .22212 .22781 4.3897 .97502 10 51 240 811 . 3838 496 9 52 268 842 .3779 489 8 53 297 872 .3721 483 7 54 325 903 .3662 476 6 55 .22353 .22934 4.3604 .97470 5 56 382 i 964 .3546 463 4 57 410 .22995 .3488 457 3 58 438 .23026 .3430 450 2 59 467 1 056 .3372 444 1 60 . 22495 ! . 23087 4 . 33 1 5 . 97437 O NCosN Cot N Tan N Sin / ' N Sin N Tan N Cot N Cos O .22495 . 23087 4.3315 .97437 GO 1 523 117 .3257 430 59 2 552 148 .3200 424 58 3 580 179 .3143 417 57 4 608 209 .3086 411 56 5 .22637 .23240 4.3029 . 97404 55 6 665 271 .2972 398 54 7 693 301 .2916 391 53 8 722 332 .2859 384 52 9 750 363 .2803 378 51 10 . 22778 .23393 4.2747 .97371 50 11 807 424 .2691 365 49 12 835 455 .2635 358 48 13 863 485 .2580 351 47 14 892 516 .2524 34g 46 15 . 22920 . 23547 4.2468 .97338 45 16 948 578 .2413 331 44 17 . 22977 608 .2358 325 43 18 . 23005 639 .2303 318 42 19 033 670 .2248 311 41 20 . 23062 .23700 4.2193 .97304 40 21 090 731 .2139 298 39 22 118 762 .2084 291 38 23 146 793 .2030 284 37 24 175 823 .1976 278 36 25 . 23203 . 23854 4.1922 .97271 35 26 231 885 .1868 264 34 27 260 916 .1814 257 33 28 288 946 .1760 251 32 29 316 .23977 .1706 244 31 30 . 23345 . 24008 4.1653 .97237 30 31 373 039 .1600 230 29 32 401 069 .1547 223 28 33 429 100 .1493 217 27 34 458 131 .1441 210 26 35 . 23486 .24162 4.1388 . 97203 25 36 514 193 .1335 196 24 37 542 223 .1282 189 23 38 571 254 .1230 182 22 39 599 285 .1178 176 21 40 .23627 .24316 4.1126 .97169 20 41 656 347 .1074 162 19 42 684 377 .1022 155 18 43 712 408 .0970 148 17 44 740 439 .0918 141 16 45 . 23769 .24470 4.0867 .97134 15 46 797 501 .0815 127 14 47 825 532 .0764 120 13 48 853 562 .0713 113 12 49 882 593 .0662 106 11 50 .23910 . 24624 4.0611 .97100 IO 51 938 655 .0560 093 9 52 966 686 .0509 086 8 53 . 23995 717 .0459 079 7 54 .24023 747 .0408 072 6 55 .24051 . 24778 4.0358 . 97065 5 56 079 809 .0308 058 4 57 108 840 .0257 051 3 58 136 871 .0207 044 2 59 164 902 .0158 037 1 60 .24192 .24933 4.0108 .97030 O NCOS N Cot NTan NSin i 77 Natural Functions 76 82 14 Natural Functions 15 [III / N Sin NTanN Cot N Cos o .24192 . 24933 4.0108 . 97030 oo 1 220 964 .0058 023 59 2 249 . 24995 4.0009 015 58 3 277 . 25026 3 . 9959 008 57 4 305 056 . 9910. . 97001 56 5 .24333 . 25087 3.9861 .96994 55 6 362 118 .9812 987 54 7 390 149 .9763 980 53 8 418 180 .9714 973 52 9 446 211 .9665 966 51 io .24474 . 25242 3.9617 . 96959 50 11 503 273 .9568 952 49 12 531 304 .9520 945 48 13 559 335 .9471 937 47 14 587 366 .9423 930 46 15 . 24615 . 25397 3 . 9375 . 96923 45 16 644 428 .9327 916 44 17 672 459 .9279 909 43 18 700 490 .9232 902 42 19 728 521 .9184 894 41 20 . 24756 . 25552 3.9136 . 96887 40 21 784 583 .9089 880 39 22 813 614 .9042 873 38 23 841 645 .8995 866 37 24 869 676 .8947 858 36 25 .24897 . 25707 3.8900 .96851 35 26 925 738 .8854 844 34 27 954 769 .8807 837 33 28 . 24982 800 .8760 829 32 29 . 25010 831 .8714 822 31 30 . 25038 . 25862 3 . 8667 .96815 30 31 066 893 .8621 807 29 32 094 924 .8575 800 28 33 122 955 .8528 793 27 34 151 . 25986 .8482 786 26 35 .25179 .26017 3 . 8436 . 96778 25 36 207 048 .8391 771 24 37 235 079 .8345 764 23 38 263 110 .8299 756 22 39 291 141 .8254 749 21 40 .25320 .26172 3 . 8208 . 96742 20 41 348 203 .8163 734 19 42 376 235 .8118 727 18 43 404 266 .8073 719 17 44 432 297 .8028 712 16 45 . 25460 . 26328 3.7983 . 96705 15 46 488 359 .7938 697 14 47 516 390 .7893 690 13 48 54B 421 .7848 682 12 49 573 452 .7804 675 11 50 . 25601 . 26483 3 . 7760 . 96667 IO 51 629 515 .7715 660 9 52 657 546 .7671 653 8 53 685 577 .7627 645 7 54 713 608 .7583 638 6 55 .25741 . 26639 3.7539 . 96630 5 56 769 670 . 7495 623 4 57 798 701 .7451 615 3 58 826 733, .7408 608 2 59 854 764 .7364 600 1 60 .25882 . 26795 3.7321 . 96593 O NCos N Cot NTan NSin ' / N Sin NTanN Cot 1 N Cos o .25882 .26795 3.7321 . 96593 60 1 910 826 .7277 585 59 2 938 857 .7234 578 58 3 966 888 .7191 570 57 4 . 25994 920 .7148 562 56 5 . 26022 .26951 3.7105 . 96555 55 6 050 .26982 .7062 547 54 7 079 .27013 .7019 540 53 8 107 044 .6976 532 52 9 135 076 .6933 524 51 IO .26163 .27107 3.6891 . 96517 50 11 191 138 .6848 509 49 12 219 169 .6806 502 48 13 247 201 .6764 494 47 14 275 232 .6722 486 46 15 .26303 . 27263 3.6680 . 96479 45 16 331 294 .6638 471 44 17 359 326 .6596 463 43 18 387 357 .6554 456 42 19 415 388 .6512 448 41 20 .26443 .27419 3 . 6470 . 90440 40 21 471 451 .6429 433 39 22 500 482 . 6387 425 38 23 528 513 .6346 417 37 24 556 545 .6305 410 36 25 . 26584 . 27576 3 . 6264 . 96402 35 26 612 607 .6222 394 34 27 640 638 .6181 386 33 28 668 670 .6140 379 32 29 696 701 .6100 371 31 30 .26724 .27732 3 . 6059 . 96363 30 31 752 764 .6018 355 29 32 780 795 .5978 347 28 33 808 826 .5937 340 27 34 836 858 .5897 332 26 35 . 26864 . 27889 3 . 5856 . 96324 25 36 892 921 .5816 316 24 37 920 952 .5776 308 23 38 948 . 27983 . 5736 301 22 39 .26976 .28015 .5696 293 21 40 .27004 . 28046 3 . 5656 . 96285 20 41 032 077 .5616 277 19 42 060 109 .5576 269 18 43 088 140 .5536 261 17 44 116 172 .5497 253 16 45 .27144 . 28203 3 . 5457 . 96246 15 46 172 234 .5418 238 14 47 200 266 . 5379 230 13 48 228 297 .5339 222 12 49 256 329 .5300 214 11 50 . 27284 .28360 3.5261 . 96206 IO 51 312 391 .5222 198 9 52 340 368 423 .5183 190 8 53 454 .61 H 182 7 54 396 486 .5105 174 6 55 . 27424 .28517 3 . 5067 .96166 5 56 452 549 .5028 158 4 57 480 580 .4989 150 3 58 508 612 .4951 142 2 59 536 643 .4912 134 1 OO .27564 . 28675 3.4874 .96126 O NCOS N Cot NTan NSin f 75 Natural Functions 74 c 16 Natural Functions 17 c 83 / N Sin N Tan N Cot NCOS o .27564 .28675 3.4874 .96126 60 1 592 706 .4836 118 59 2 620 738 .479S 110 58 3 648 769 .4760 102 57 4 676 801 .4722 094 56 5 .27704 . 28832 ! 3. 4684 . 96086 55 6 731 864 .4646 078 54 7 759 895 .4608 070 53 8 787 927 .4570 062 52 9 815 958 .4533 054 51 io . 27843 . 28990 3.4495 .96046 50 11 871 .29021 . 4458 037 49 12 899 053 .4420 029 48 13 927 084 .4383 021 47 14 955 116 .4346 013 46 15 . 27983 . 29147 3 . 4308 . 96005 45 16 .28011 179 .4271 . 95997 44 17 039 210 .4234 989 43 18 067 242 .4197 981 42 19 095 274 .4160 972 41 20 .28123 . 29305 3.4124 . 95964 40 21 150 337 .4087 956 39 22 178 368 .4050 948 38 23 206 400 .4014 940 37 24 234 432 .3977 931 36 25 . 28262 . 29463 3.3941 . 95923 35 26 290 495 .3904 915 34 27 318 526 .3868 907 33 28 346 558 .3832 898 32 29 374 590 .3796 890 31 30 . 28402 .29621 3.3759 . 95882 30 31 429 653 .3723 874 -29 32 457 685 .3687 865 28 33 485 716 . 3652 857 27 34 513 748 .3616 849 26 35 . 28541 . 29780 3 . 3580 .95841 25 36 569 811 .3544 832 24 37 597 843 .3509 824 23 38 625 875 .3473 816 22 39 652 906 .3438 807 21 40 . 28680 . 29938 3 . 3402 . 95799 20 41 708 . 29970 . 3367 791 19 42 736 .30001 .3332 782 18 43 764 033 .3297 774 17 44 792 065 .3261 766 16 45 . 28820 . 30097 3.3226 .95757 15 46 847 128 .3191 749 14 47 875 160 .3156 740 13 48 903 192 .3122 732 12 49 931 224 .3087 724 11 50 . 28959 . 30255 3 . 3052 .95715 IO 51 . 28987 287 .3017 707 9 52 .29015 319 .2983 698 8 53 042 351 .2948 690 7 54 070 382 .2914 681 6 55 . 29098 .30414 3.2879 . 95673 5 56 126 446 .2845 664 4 57 154 478 .2811 656 3 58 182 509 .2777 647 2 59 209 541 .2743 639 1 60 .29237 . 30573 3 . 2709 . 95630 O N Cos N Cot N Tan N Sin / / N Sin NTan N Cot N Cos 60 O . 29237 . 30573 3 . 2709 . 95630 1 265 605 .2675 622 59 2 293 637 .2641 613 58 3 321 669 .2607 605 57 4 348 700 .2573 596 56 5 . 29376 . 30732 3 . 2539 . 95588 55 6 404 764 .2506 579 54 7 432 796 .2472 571 53 8 460 828 .2438 562 52 9 487 860 .2405 554 51 IO .29515 .30891 3.2371 . 95545 50 11 543 923 .2338 536 49 12 571 955 .2305 528 48 13 599 . 30987 .2272 519 47 14 626 .31019 .2238 511 46 15 . 29654 .31051 3 . 2205 . 95502 45 16 682 083 .2172 493 44 17 710 115 .2139 485 43 18 737 147 .2106 476 42 19 765 178 .2073 467 41 20 . 29793 .31210 3.2041 . 95459 40 21 821 242 .2008 450 39 22 849 274 .1975 441 38 23 876 306 .1943 433 37 24 904 338 .1910 424 36 25 . 29932 .31370 3 . 1878 .95415 35 26 960 402 .1845 407 34 27 . 29987 434 .1813 398 33 28 .30015 466 .1780 389 32 29 043 498 .1748 380 31 30 . 3007 1 .31530 3.1716 . 95372 30 31 098 562 .1684 363 29 32 126 594 .1652 354 28 33 154 626 .1620 345 27 34 182 658 .1588 337 26 35 . 30209 .31690 3.1556 . 95328 25 36 237 722 .1524 319 24 37 265 754 .1492 310 23 38 292 786 .1460 301 22 39 320 818 .1429 293 21 40 . 30348 .31850 3.1397 . 95284 20 41 376 882 .1366 275 19 42 403 914 .1334 266 18 43 431 946 .1303 257 17 44 459 .31978 ^271 248 16 45 . 30486 .32010 3.1240 .95240 15 46 514 042 .1209 231 14 47 542 074 .1178 222 13 48 570 106 .1146 213 12 49 597 139 .1115 204 11 50 .30625 .32171 3 . 1084 .95195 IO 51 653 203 .1053 186 9 52 680 235 .1022 177 8 53 708 267 .0991 168 7 54 736 299 .0961 159 6 55 . 30763 .32331 3.0930 .95150 5 56 791 363 .0899 142 4 57 819 396 .0868 133 3 58 846 428 .0838 124 2 59 874 460 .0807 115 1 60 . 30902 NCos . 32492 3 . 0777 .95106 O N Cot NTan NSin / 73 Natural Functions 72 c 84 18 Natural Functions 19 / N SinNTanN Cot N Cos / N Sin NTan N Cot N Cos o . 30902 .32492 3.0777 . 95106 60 o . 32557 . 34433 2.9042 .94552 oo 1 929 524 .0746 097 59 1 584 465 .9015 542 59 2 957 556 .0716 088 58 2 612 498 .8987 533 58 3 . 30985 588 .0686 079 57 3 639 530 .8960 523 57 4 .31012 621 .0655 070 56 4 667 563 .8933 514 56 5 .31040 . 32653 3 . 0625 .95061 55 5 . 32694 . 34596 2.8905 . 94504 55 6 068 685 .0595 052 44 6 722 628 .8878 495 54 7 095 717 .0565 043 53 7 749 661 .8851 485 53 8 123 749 .0535 033 52 8 777 693 .8824 476 52 9 151 782 .0505 024 51 9 804 720 .8797 466 51 io .31178 .32814 3.0475 .95015 50 IO . 32832 . 34758 2.8770 . 94457 50 11 206 846 .0445 . 95006 49 11 859 791 .8743 447 49 12 233 878 .0415 . 94997 48 12 887 824 .8716 438 48 13 261 911 .0385 988 47 13 914 856 .8689 428 47 14 289 943 .0356 979 46 14 942 889 .8662 418 46 15 .31316 . 32975 3 . 0326 . 94970 45 15 .32969 .34922 2.8636 . 94409 45 16 344 . 33007 .0296 961 44 16 . 32997 954 .8609 399 44 17 372 040 .0267 952 43 17 .33024 . 34987 .8582 390 43 18 399 072 .0237 943 42 18 051 . 35020 .8556 380 42 19 427 104 .0208 933 41 19 079 052 .8529 370 41 2 .31454 .33136 3.0178 . 94924 40 20 .33106 .35085 2 . 8502 . 94361 40 21 482 169 .0149 915 39 21 134 118 .8476 351 39 22 510 201 .0120 906 38 22 161 150 .8449 342 38 23 ,537 233 .0090 897 37 23 189 183 .8423 332 37 24 565 266 .0061 888 36 24 216 216 .8397 322 36 25 .31593 . 33298 3 . 0032 . 94878 35 25 . 33244 .35248 2.8370 .94313 35 26 620 330 3 . 0003 869 34 26 271 281 .8344 303 34 27 648 363 2 . 9974 860 33 27 298 314 .8318 293 33 28 675 395 .9945 851 32 28 326 346 .8291 284 32 29 703 427 .9916 842 31 29 353 379 .8265 274 31 30 .31730 . 33460 2 . 9887 . 94832 30 30 .33381 .35412 2 . 8239 . 94264 30 31 758 492 .9858 823 29 31 408 445 .8213 254 29 32 786 524 .9829 814 28 32 436 477 .8187 245 28 33 813 557 .9800 805 27 33 463 510 .8161 235 27 34 841 589 .9772 795 26 34 490 543 .8135 225 26 35 .31868 .33621 2.9743 . 94786 25 35 .33518 . 35576 2.8109 .94215 25 36 896 654 .9714 777 24 36 545 608 .8083 206 24 37 923 686 .9686 768 23 37 573 641 .8057 196 23 38 951 718 .9657 758 22 38 600 674 .8032 186 22 39 .31979 751 .9629 749 21 39 627 707 .8006 176 21 40 . 32006 .33783 2 . 9600 . 94740 20 40 . 33655 . 35740 2 . 7980 .94167 20 41 034 816 .9572 730 19 41 682 772 .7955 157 19 42 061 848 .9544 721 18 42 710 805 .7929 147 18 43 089 881 .9515 712 17 43 737 838 .7903 137 17 44 116 913 .9487 702 16 44 764 871 .7878 127 16 45 .32144 . 33945 2 . 9459 . 94693 15 45 . 33792 . 35904 2 . 7852 .94118 15 46 171 . 33978 .9431 684 14 46 819 937 .7827 108 14 47 199 .34010 .9403 674 13 47 846 . 35969 .7801 098 13 48 227 043 .9375 665 12 48 874 . 36002 .7776 088 12 49 254 075 .9347 656 11 49 901 035 .7751 078 11 50 . 32282 .34108 2.9319 . 94646 IO 50 . 33929 . 36068 2.7725 . 94068 IO 51 309 140 .9291 637 9 51 956 101 .7700 058 9 52 337 173 .9263 627 8 52 . 33983 134 .7675 049 8 53 364 205 .9235 618 7 53 .34011 167 .7650 039 7 54 392 238 .9208 609 6 54 038 199 .7625 029 6 55 .32419 .34270 2.9180 . 94599 5 55 . 34065 .36232 2 . 7600 .94019 5 56 447 303 .9152 590 4 56 093 265 . 7575 . 94009 4 57 474 335 .9125 580 3 57 120 298 .7550 .03999 3 58 502 368 .901)7 571 2 58 147 331 .7525 989 2 59 529 400 .9070 561 1 59 175 364 .7500 979 1 eo . 32557 . 34433 2.9042 . 94552 O 60 .34202 . 36397 2.7475 . 93969 O N Cos N Cot NTan NSin ' N Cos N Cot NTan NSin 71 Natural Functions 70 c 20 Natural Functions 21 c 85 / N Sin N Tan N Cot N Cos o . 34202 . 36397 2.7475 . 93969 60 1 229 430 .7450 959 59 2 257 463 .7425 - 949 58 3 284 496 .7400 939 57 4 311 529 .7376 929 56 5 .34339 . 36562 2.7351 .93919 55 6 366 595 .7326 909 54 7 393 628 .7302 899 53 8 421 661 .7277 889 52 9 448 694 .7253 879 51 10 .34475 .36727 2.7228 . 93869 50 11 503 760 .7204 859 49 12 530 793 .7179 849 48 13 557 826 .7155 839 47 14 584 859 .7130 829 46 15 .34612 .36892 2.7106 .93819 45 16 639 925 .7082 809 44 17 666 958 .7058 799 43 18 694 .36991 .7034 789 42 19 721 .37024 .7009 779 41 20 .34748 . 37057 2.6985 .93769 40 21 775 090 .6961 759 39 22 803 123 .6937 748 38 23 830 157 .6913 738 37 24 857 190 .6889 728 36 25 . 34884 .37223 2.6865 .93718 35 26 9121 256 .6841 708 34 27 939 289 .6818 698 33 28 966 322 .6794 688 32 29 .34993 355 .6770 677 31 30 .35021 '.37388 2.6746 . 93667 30 31 048 422 .6723 657 29 32 075! 455 .6699 647 28 33 102 488 .6675 637 27 34 130 521 .6652 626 26 35 .35157 . 37554 2.6628 .93616 25 36 184 588 .6605 606 24 37 211 621 .6581 596 23 38 239 654 .6558 585 22 39 266 687 j .6534 575 21 40 .35293 .3772o'2.6511 . 93565 20 41 320 754 : 6488 555 19 42 347 787 .6464 544 18 43 375 820 .6441 534 17 44 402 853 .6418 524 16 45 .35429 .37887 2.6395 .93514 15 46 456 920 .6371 503 14 47 484 953 .6348 493 13 48 511 .37986 .6325 483 12 49 538 38020 .6302 472 11 50 .35565 .38053 2.6279 . 93462 io 51 592 086 .6256 452 9 52 619 120 .6233 441 8 53 647 153 .6210 431 7 54 674 186 .6187 420 6 55 .35701 .38220 2.6165 .93410 5 56 728 253 .6142 400 4 57 755 286 .6119 389 3 58 782 320 .6096 379 2 59 810 353 .6074 368 1 60 . 35837 . 38386 2.6051 . 93358 O NCOS N Cot NTan NSin / / N Sin NTan N Cot NCos o . 35837 .38386 2.6051 . 93358 60 1 864 420 .6028 348 59 2 891 453 .6006 337 58 3 918 487 .5983 327 57 4 945 520 .5961 316 56 5 . 35973 . 38553 2.5938 . 93306" 55 6 . 36000 587 .5916 295 54 7 027 620 .5893 285 53 8 054 654 .5871 274 52 9 081 687 .5848 264 51 IO .36108 .38721 2 . 5826 . 93253 50 11 135 754 .5804 243 49 12 . 162 787 .5782 232 48 13 190 821 .5759 222 47 14 217 854 .5737 211 46 15 .36244 .38888 2.5715 . 93201 45 16 271 921 .5693 190 44 17 298 955 .5671 180 43 18 325 . 38988 .5649 169 42 19 352 .39022 .5627 159 41 20 .36379 . 39055 2.5605 .93148 40 21 406 089 .5583 137 39 22 434 122 .5561 127 38 23 461 156 .5539 116 37 24 488 190 .5517 106 36 25 .36515 .39223 2.5495 . 93095 35 26 542 257 .5473 084 34 27 569 290 .5452 074 33 28 596 324 .5430 063 32 29 623 357 .5408 052 31 30 .36650 .39391 2.5386 . 93042 30 31 677 425 .5365 031 29 32 704 458 .5343 020 28 33 731 492 .5322 . 93010 27 34 758 526 .5300 . 92999 26 35 . 36785 .39559 2.5279 .92988 25 36 812 593 .5257 978 24 37 839 626 .5236 967 23 38 867 660 .5214 956 22 39 894 694 .5193 945 21 40 .36921 . 39727 2.5172 . 92935 20 41 948 761 .5150 924 19 42 .36975 795 .5129 913 18 43 . 37002 829 .5108 902 17 44 029 862 .5086 892 16 45 . 37056 . 39896 2.5065 .92881 15 46 083 930 .5044 870 14 47 110 963 .5023 859 13 48 137 . 39997 .5002 849 12 49 164 .40031 .4981 838 11 50 .37191 . 40065 2.4960 . 92827 IO 51 218 098 .4939 816 9 52 245 132 .4918 805 8 53 272 166 .4897 794 7 54 299 200 .4876 784 6 55 .37326 . 40234 2.4855 . 92773 5 56 353 267 .4834 762 4 57 380 301 .4813 751 3 58 407 335 .4792 740 2 59 434 369 .4772 729 1 60 . 37461 .40403 2.4751 .92718 O NCOS N Cot NTan NSin t 69 Natural Functions 68 c 86 22 Natural Functions 23 c ' N Sin N Tan N Cot N Cos o .37461 .40403 2.4751 .92718 60 1 488 436 .4730 707 59 2 515 470 .4709 697 58 3 542 504 .4689 686 57 4 569 538 .4668 675 56 5 .37595 .40572 2.4648 . 92664 55 6 622 606 .4627 653 54 7 649 640 .4606 642 53 8 676 674 .4586 631 52 9 703 707 . 4566 620 51 io . 37730 .40741 2.45451.92609 50 11 757 775 .4525 598 49 12 784 809 .4504 587 48 13 811 843 .4484 576 47 14 838 877 .4464 565 46 15 .37865 .40911 2.4443 . 92554 45 16 892 945 .4423 543 44 17 919 . 40979 .4403 532 43 18 946 .41013 .4383 521 42 19 973 047 .4362 510 41 SO . 37999 .41081 2.4342 . 92499 40 21 . 38026 115 .4322 488 39 22 053 149 .4302 477 38 23 080 183 .4282 466 37 24 107 217 .4262 455 36 25 .38134 .41251 2.4242 . 92444 35 26 161 285 .4222 432 34 27 188 319 .4202 421 33 28 215 353 .4182 410 32 29 241 387 .4162 399 31 30 .38268 .41421 2.4142 . 92388 30 31 295 455 .4122 377 29 32 322 490 .4102 366 28 33 349 524 .4083 355 27 34 376 558 .4063 343 26 35 .38403 .41592 2.4043 .92332 25 36 430 626 .4023 321 24 37 456 660 .4004 310 23 38 483 694 .3984 299 22 39 510 728 .3964 287 21 40 . 38537 . 41763 2.3945 .92276 SO 41 564 797 . 3925 265 19 42 591 831 . 3906 254 18 43 617 865 .3886 243 17 44 644 899 .3867 231 16 45 .38671 .41933 2 . 3847 . 92220 15 46 698 .41968 . 3828 209 14 47 725 .42002 .3808 198 13 48 752 036 .3789 186 12 49 778 070 .3770 175 11 50 . 38805 .42105 2 . 3750 .92164 IO 51 832 139 .3731 152 9 52 859 173 .3712 141 8 53 886 207 . 3693 130 7 54 912 242 .3673 119 6 55 . 38939 .42276 2.3654 .92107 5 56 966 310 . 363B 096 4 57 . 38993 345 .3616 085 3 58 . 39020 379 . 3597 073 2 59 046 413 .3578 062 1 60 . 39073 .42447 2.3559 . 92050 O NCOS N Cot NTan NSin / / N Sin N Tan N Cot| N Cos O .39073 .42447 2.3559 .92050 60 1 100 482 3539 039 59 2 127 516 . 3520 028 58 3 153 551 .3501 016 57 4 180 585 .3483 . 92005 56 5 .392071.42619 2.3464 .91994 55 6 234 654 .3445 982 54 7 260 688 .3426 971 53 8 287 722 .3407 959 52 9 314 757 .3388 948 51 10 . 39341 .42791 2.3369 .91936 50 11 367 826 .3351 925 49 12 394 860 .3332 914 48 13 421 894 .3313 902 47 14 448 929 .3294 891 46 15 . 39474 .42963 2.3276 .91879 45 16 501 .42998 . 3257 868 44 17 528 . 43032 .3238 856 43 18 555 067 .3220 845 42 19 581 101 .3201 833 41 SO . 39608 .43136 2.3183 .91822 40 21 635 170 .3164 810 39 22 661 205 .3146 799 38 23 688 239 .3127 787 37 24 715 274 .3109 775 36 S5 .39741 . 43308 2 . 3090 .91764 35 26 768 343 .3072 752 34 27 795 378 .3053 741 33 28 822 412 .3035 729 32 29 848 447 .3017 718 31 30 .39875 .43481 2.2998 .91706 30 31 902 516 .2980 694 29 32 928 550 .2962 683 28 33 955 585 .2944 671 27 34 . 39982 620 .2925 660 26 35 . 40008 .43654 2.2907 .91648 35 36 035 689 .2889 636 24 37 062 724 .2871 625 23 38 088 758 .2853 613 22 39 115 793 .2835 601 21 40 .40141 .43828'2.2817 .91590 SO 41 168 862 .2799 578 19 42 195 897 .2781 566 18 43 221 932 . 2763 555 17 44 248 . 43966 2745 543 16 45 .40275 .44001 2.2727 .91531 15 46 301 036 .2709 519 14 47 328 071 .2691 508 13 48 355 105 .2673 496 12 49 381 140 .2655 484 11 50 . 40408 .44175 2.2637 .91472 10 51 434 210 .2620 461 9 52 461 244 .2602 449 8 53 488 279 . 2584 437 7 54 514 314 .2566 425 6 55 .40541 .44349 2.2549 .91414 5 56 507 384 .2531 402 4 57 594 418 .2513 390 3 58 621 453 . 2496 378 2 59 647 488 .2478 366 1 o .40674 .44523 2 . 2460 .91355 O N Cos N Cot NTan NSin / 67 Natural Functions 66 c Ill] 24 Natural Functions 25 c 87 / N Sin N Tan N Cot N Cos o .406741.44523 2.2460 .91355 60 1 700 558 .2443 343 59 2 727 593 .2425 331 58 3 753 627 .2408- 319 57 4 780; 662 .2390 307 56 5 .40806 '.44697 2 . 2373 .91295 55 6 833; 732 . 2355 283 54 7 860 767 . 2338 272 53 8 886 802 .2320 260 52 9 913 837 .2303 248 51 io .40939 .44872 2 . 2286 .91236 50 11 966: 907| .2268 224 49 12 .40992 942 .2251 212 48 13 .41019 .44977 .2234 200 47 14 045.45012 .2216 188 46 15 .41072 .45047 2.2199 .91176 45 16 098 082 .2182 164 44 17 125 117 .2165 152 43 18 151 152 .2148 140 42 19 178 187 .2130 128 41 20 .41204 .45222 2.2113 .91116 40 21 231 257 .2096 104 39 22 257 292 .2079 092 38 23 284 327 .2062 080 37 24 310 362 .2045 068 36 25 .41337 .45397 2.2028 . 91056 35 26 363 432 .2011 044 34 27 390 467 .1994 032 33 28 416 502 .1977 020 32 29 443 1 538 .1960 . 91008 31 30 .41469 .45573 2.1943 . 90996 30 31 496 60S .1926 984 29 32 522 643 .1909 972 28 33 549 678| .1892 960 27 34 575 713 .1876j 948 26 35 .41602 .45748 2. 1859 i. 90936 25 36 628 784 .18421 924 24 37 655 819 .1825 911 23 38 681 854 . 1 808 899 22 39 707 889 .1792 887 21 40 .41734 .45924 2.1775 . 90875 20 41 760 960 .1758 863 19 42 787 .45995 .1742 851 18 43 813 .46030 .1725 839 17 44 840 ; 065 .1708 826 16 45 .41866 .46101 2.1692 .90814 15 46 892 136 .1675 802 14 47 919 171 .1659 790 13 48 945 206 .1642 778 12 49 972 242 .1625 766 11 50 .41998 .46277 2.1609 . 90753 IO 51 .42024 312 .1592: 741 9 52 051 348 .1576 729 8 53 077 383 .1560 717 7 54 104 418 .1543! 704 6 55 .42130 .46454 2.1527 .90692 5 56 156 489 .1510 680 4 57 183 525 .1494 668 3 58 209 560 .1478 655 2 59 235 595 .1461 643 1 60 .42262 .46631 2.1445 .90631 O NCosN Cot N Tan; N Sin ' / N Sin 1 N Tan N Cot N Cos o . 42262 .46631 ;2.1445 .90631 60 1 288 666 .1429 618 59 2 315 702 .1413 606 58 3 341 737 .1396 594 57 4 367 772 .1380 582 56 5 .42394 .46808 2.1364 . 90569 55 6 420 843 . 1348 557 54 7 446 879 .1332 545 53 8 473 914 .1315 532 52 9 499 950 .1299 520 51 IO .42525 .46985 2 . 1283 . 90507 50 11 552 .47021 .1267 495 49 12 578 056 .1251 483 48 13 604 092 .1235 470 47 14 631 128 .1219 458 46 15 .42657 .47163 2.1203 . 90446 45 16 683 199 .1187 433 44 17 709 234 .1171 421 43 18 736 270 .1155 408 42 19 762 305 .1139 396 41 20 .42788 .47341 2.1123 .90383 40 21 815 377 .1107 371 39 22 841 412 .1092 358 38 23 867 448 .1076 346 37 24 894 483 .1060 334 36 25 .42920 .47519 2.1044 .90321 35 26 946 555 .1028 309 34 27 972 590 .1013 296 33 28 . 42999 626 .0997 284 32 29 . 43025 662 .0981 271 31 30 .43051 . 47698 2.0965 .0950 . 90259 30 31 077 733 246 29 32 104 769 .0934 233 28 33 130 805 .0918 221 27 34 156 840 j .0903 208 26 35 .43182 .47876 2.0887 .90196 25 36 209 912 .0872 183 24 37 235 948j .0856 171 23 38 261 .47984 .0840 158 22 39 287 .48019: .0825, 146 21 40 .43313 .48055 2.0809 .90133 20 41 340 091 .0794 120 19 42 366 127 .0778! 108 18 43 392 163| .0763 095 17 44 418 198 .0748 082 16 45 .43445 .48234 2. 0732 .90070 15 46 471 270 .0717 057 14 47 497 306 .0701 045 13 48 523 342 .0686 032 12 49 549 378 .0671 019 11 50 .43575 .48414 2.0655 .90007 IO 51 602 450 .0640.89994 9 52 628 486 .0625 981 8 53 654 521 .0609 968 7 54 680 557 . 0594 956 6 55 . 43706 .48593 2.0579 . 89943 5 56 733 629 i .0564 930 4 57 759 665 . 0549 918 3 58 785 701 .0533 905 2 59 811 737 .0518 892 1 60 .43837 .48773 2.0503 .89879 O NCOS N Cot N Tan NSin / 65 Natural Functions 64 c 88 26 Natural Functions 2T .[III ' N Sin N Tan N Cot NCos o . 43837 . 48773 2.0503 .89879 60 1 863 809 .0488 867 59 2 889 845 .0473 854 58 3 916 881 .0458 841 57 4 942 917 .0443 828 56 5 . 43968 .48953 2 . 0428 .89816 55 6 . 43994 .48989 .0413 803 54 7 .44020 .49026 .0398 790 53 8 046 062 .0383 777 52 9 072 098 .0368 764 51 io .44098 .49134 2.0353 . 89752 50 11 124 170 .0338 739 49 12 151 206 .0323 726 48 13 177 242 .0308 713 47 14 203 278 .0293 700 46 15 .44229 .49315 2.0278 .89687 45 16 255 351 .0263 674 44 17 281 387 .0248 662 43 18 307 423 .0233 649 42 19 333 459 .0219 636 41 20 .44359 .49495 2.0204 . 89623 40 21 385 532 .0189 610 39 22 411 568 .0174 597 38 23 437 604 .0160 584 37 24 464 640 .0145 571 36 25 .44490 . 49677 2.0130 . 89558 35 26 516 713 .0115 545 34 27 542 749 .0101 532 33 28 568 786 .0086 519 32 29 594 822 .0072 506 31 30 .44620 .49858 2 . 0057 .89493 30 31 646 894 .0042 480 29 32 672 931 .0028 467 28 33 698 .49967 2 . 0013 454 27 34 724 .50004 1 . 9999 441 26 35 .44750 . 50040 1 . 9984 .89428 25 36 776 076 .9970 415 24 37 802 113 .9955 402 23 38 828 149 .9941 389 22 39 854 185 .9926 376 21 40 .44880 . 50222 1.9912 .89363 20 41 906 258 .9897 350 19 42 932 295 .9883 337 18 43 958 331 .9868 324 17 44 .44984 368 .9854 311 16 45 .45010 .50404 1.9840 . 89298 15 46 036 441 .9825 285 14 47 062 477 .9811 272 13 48 088 514 .9797 259 12 49 114 550 .9782 245 11 50 .45140 . 50587 1 . 9768 . 89232 10 51 166 623 .9754 219 9 52 192 660 .9740 206 8 53 218 696 .9725 193 7 54 243 733 .9711 180 6 55 .45269 . 50769 1 . 9697 .89167 5 56 295 806 .9683 153 4 57 321 843 .9669 140 3 58 347 879 .9654 127 2 59 373 916 .9640 114 1 60 .45399 .50953 1 . 9626 .89101 O NCOS N Cot NTan NSin / / N Sin NTan N Cot N Cos o .45399 .50953 1.9626 .89101 60 1 425 .50989 .9612 087 59 2 451 .51026 .9598 074 58 3 477 063 .9584 061 57 4 503 099 .9570 048 56 5 . 45529 .51136 1.9556 .89035 55 6 554 173 .9542 021 54 7 580 209 .9528 . 89008 53 8 606 246 .9514 . 88995 52 9 632 283 .9500 981 51 10 45658 .51319 1 . 9486 . 88968 50 11 684 356 .9472 955 49 12 710 393 .9458 942 48 13 736 403 .9444 928 47 14 762 467 .9430 915 46 15 . 45787 .51503 1.9416 . 88902 45 16 813 540 .9402 888 44 17 839 577 .9388 875 43 18 865 614 .9375 862 42 19 891 651 .9361 848 41 20 . 45917 .51688 1 . 9347 . 88835 40 21 942 724 .9333 822 39 22 968 761 .9319 808 38 23 .45994 798 .9306 795 37 24 . 46020 835 .9292 782 36 25 .46046 .51872 1 . 9278 . 88768 35 26 072 909 .9265 755 34 27 097 946 .9251 741 33 28 123 .51983 .9237 728 32 29 149 . 52020 .9223 715 31 30 .46175 . 52057 1.9210 . 88701 30 31 201 094 .9196 688 29 32 226 131 .9183 674 28 33 252 168 .9169 661 27 34 278 203 .9155 647 26 35 .46304 . 52242 1.9142 .88634 25 36 330 279 .9128 620 24 37 355 316 .9115 607 23 38 381 353 .9101 593 22 39 407 390 .9088 580 21 40 .46433 . 52427 1.9074 .88566 20 41 458 464 .9061 553 19 42 484 501 .9047 539 18 43 510 538 .9034 526 17 44 536 575 .9020 512 16 45 .46561 .52613 1.9007 . 88499 15 46 587 650 .8993 485 14 47 613 687 .8980 472 13 48 639 724 .8967 458 12 49 664 761 .8953 445 11 50 .46690 . 52798 1 . 8940 .88431 IO 51 716 836 .8927 417 9 52 742 873 .8913 404 8 53 767 910 .8900 390 7 54 793 947 .8887 377 6 55 .46819 . 52985 1.8873 . 88363 5 56 844 . 53022 .8860 349 4 57 870 059 .8847 336 3 58 896 096 .8834 322 2 59 921 134 .8820 308 1 60 ,46947 .53171 1 . 8807 .88295 O NCOS N Cot NTan NSin 63 Natural Functions 62 c 28 Natural Functions 29 89 / N Sin NTanjN Cot N Cos o . 46947 .53171 1.8807 .88295 60 1 973 208 .8794 281 59 2 .46999 246| .8781 267 58 3 . 47024 283 .8768 1 254 57 4 050, 320 .875^ 240 56 5 .47076 .53358 1.8741 .88226 55 6 101 395 .8728 213 54 7 127 432 .8715 199 53 8 153 470 .8702 185 52 9 178, 507 .8689, 172 51 io . 47204 . 53545 1 . 8676 . 88158 50 11 229 582 .8663 144 49 12 255 620 .8650 130 48 13 281 6571 .8637; 117 47 14 306 694 .8624 j 103 46 15 .47332 .53732 1.8611 .88089 45 16 358 769 .8598 075 44 17 383 807 .8585 062 43 18 409 844 .8572 048 42 19 434: 882! .8559 034 41 20 . 47460 . 53920 1 . 8546 ' . 88020 40 21 486 957 .8533 .88006 39 22 511 .53995 .8520 . 87993 38 23 537 .54032 .8507 979 37 24 562 . 070 .8495 965 36 25 .47588 '.54107 1.8482 . 87951 35 26 614 145 .8469 937 34 27 639 183 .8456 923 33 28 665 220 .8443 909 32 29 690 258 .8430 896 31 30 .47716 .54296 1.8418 . 87882 30 31 741 333 .8405 868 29 32 767 371 .8392 854 28 33 793 409 .8379 840 27 34 818 446 .8367 826 26 35 .47844 .54484 1 . 8354 .87812 25 36 869j 522 .8341 798 24 37 895 560 .8329 784 23 38 920| 597 .8316 770 22 39 946' 635 .8303 756 21 40 .479711.54673 1 . 8291 .87743 20 41 .47997; 711 .8278 729 19 42 .48022 748 .8265 715 18 43 048 786 .8253 701 17 44 073 824 .8240 687 16 45 .48099 .54862 1.8228 . 87673 15 46 124 900 .8215 659 14 47 150 938 .8202 645 13 48 175 .54975 .8190 631 12 49 201 55013 .8177 617 11 50 .48226 55051 1.8165 . 87603 IO 51 252 089, .8152 589 9 52 277 127 .8140 575 8 53 303 165! .8127 561 7 54 328! 203 .8115 546 6 55 .48354 .55241 1.8103 . 87532 5 56 379 279 .8090 518 4 57 405 317 .8078J 504 3 58 430 355 .8065; 490 2 59 456 393 .8053 476 1 60 .48481 .55431 1.8040 .87462 O NCos;N CotjNTan NSin t 1 N Sin N Tan N Cot NCos O .48481 .55431 1.8040 .87462 OO 1 506 469 .8028 448 59 2 532 507 .8016 434 58 3 557 545 .8003 420 57 4 583 583 .7991 406 56 5 . 48608 .55621 1.7979 . 87391 55 6 634 659 .7966 377 54 7 659 697 .7954 1 363 53 8 684 736 .7942 349 52 9 710 774 .7930 335 51 10 .48735 .55812 1.7917 .87321 50 11 761 85O .7905! 306 49 12 786 888 . 7893 292 48 13 811 926 .7881, 278 47 14 837 . 55964 .7868 264 46 15 .48862 .56003 1.7856 '.87250 45 16 888 041! .7844 235 44 17 913 079 .7832 221 43 18 938 1171 .7820 207 42 19 964 156 .7808 193 41 20 . 48989 .5619411.7796 .87178 40 21 . 49014 232 .7783 164 39 22 040 270 .7771 150 38 23 065 309 .7759 136 37 24 090 347 .7747 121 36 25 .49116 . 56385 1.7735 .87107 35 26 141 424 .7723 093 34 27 166 462 .7711 079 33 28 192 501 .7699 064 32 29 217 539 .7687 050 31 30 . 49242 . 56577 1.7675 . 87036 30 31 268 616 .7663 021 29 32 293 654 .7651 . 87007 28 33 318 693 .7639 .86993 27 34 344 731 .7627 978 26 35 .49369 .56769 1.7615 . 86964 25 36 394 808 . 7603 949 24 37 419 846 .7591 935 23 38 445 885 .7579 921 22 39 470 923 .7567 906 21 40 . 49495 . 56962 1.7556 . 86892 20 41 521 . 57000 .7544 878 19 42 546 039 .7532 863 18 43 571 078 .7520 849 17 44 596 116 .7508 834 16 45 .49622 .57155 1.7496 . 86820 15 46 647 193 . 7485 805 14 47 672 232 .7473 791 13 48 697 271 .7461 777 12 49 723 309 .7449 762 11 50 . 49748 . 57348 1.7437 .86748 IO 51 773 386 .7426 733 9 52 798 425 .7414 719 8 53 824 464 .7402 704 7 54 849 503 .7391 690 6 55 . 49874 .57541 1.7379 . 86675 5 56 899 580 .7367 661 4 57 924 619 . 7355 646 3 58 950 657 .7344 632 2 59 .49975 696 .7332 617 1 60 .50000 .57735 1.7321 '.86603 O N Cos N Cot N Tan N Sin ' 61 Natural Functions 60 c 90 30 Natural Functions 31/ 1 N Sin N Tan N Cot N Cos o .50000 .57735 ' 1.7321 .86603 OO 1 025 774 .73091 588 59 2 050 813 .7297 573 58 3 076 851 .7286 559 57 4 101 890 .7274 544 56 5 .50126 . 57929 1 . 7262 . 86530 55 6 151 .57968 .7251! 515 54 7 176 . 58007 i .7239 501 53 8 201 0461 .7228 486 52 9 227 OS5 .7216 471 51 io . 50252 .58124 1.7205 . 86457 50 11 277 162 .7193 442 49 12 302 201 .7182| 427 48 13 327 240 .7170 413 47 14 352 279 .7159 398 4 ? 15 . 50377 .58318 1.7147 . 86384 45 16 403 357 .7136 369 44 17 428 396 .7124 354 43 18 453 435 .7113 340 42 19 478 474 .7102 325 41 20 . 50503 .58513 1 . 7090 .86310 40 21 528 552 .7079 295 39 22 553 591 .7067 281 38 23 578 631 .7056 266 37 24 603 670 .7045 251 36 25 . 50628 .58709 1 . 7033 . 86237 35 26 654 748 .7022 222 34 27 679 787 .7011 207 33 28 704 826 . 6999 192 32 29 729 865 .6988 178 31 30 . 50754 . 58905 1 . 6977 .86163 30 31 779 944 .6965' 148 29 32 804 . 58983 .6954J 133 28 33 829 .59022 .6943! 119 27 34 854 061 .6932 104 26 35 . 50879' . 59101 1 . 6920 . 86089 25 36 904 140 . 6909 074 24 37 929 179 .6898! 059 23 38 954 218 .6887 045 22 39 . 50979 258 .6875 030 21 40 .51004 . 59297 1 . 6864 .86015 20 41 029 336 .6853 .86000 19 42 054 376 .6842 .85985 18 43 079 415 .6831 970 17 44 104 454 . 6820 956 16 15 .51129 .59494 1.6808 .85941 15 46 154 533 . 13797 926 14 47 179 573 .6786 911 13 48 204 612 .6775 896 12 49 229 651 .6764 881 11 50 .51254 .59691 1.6753 . 85866 IO 51 279 730 .6742 851 9 52 304 770 .6731 836 8 53 329 809 .6720 S21 7 54 354 849 .6709 806 6 55 .51379 .59888 1.6698 .85792 5 56 404 928! .6687 777 4 57 429 .599(17 .6676 762 3 58 454 . (10007 . 666g 747 2 59 479 040 . (1054 732 1 GO UloOl .60086 J. 66431. 85717 O N Cos N Cot N Tan' N Sin ' ' N Sin N Tan N Cot N Cos O .51504 .60086 1.6643 .S57 17 60 1 529 126 .6632 702 59 2 554 165 .6621 687 58 3 579 205 .6610 672 57 4 604 24 .6599 657 56 5 .51628 .60284 1.6588 . 85642 55 6 653 324 .6577 627 54 7 678! 364 . 6566 612 53 8 703 403 . 6555 597 52 9 728 443 .6545 582 51 IO .51753 . 60483 1 . 6534 . 85567 50 11 778 522 .6523 551 49 12 803 562 .6512 536 48 13 828 602 .6501 521 47 14 852 642 .6490 506 46 15 .51877 .60681 1.6479 .85491 45 16 902! 721 .6469 476 44 17 927! "761 .6458 461 43 18 952 801 .6447 . 446 42 19 .51977 841 .6436 431 41 20 .52002|.60881 1.6426 .85416 40 21 026 921! .6415 401 39 22 051 1.60960! .6404 385 38 23 076 .61000! .6393 370 37 24 101 040 .6383 355 36 25 .52126 .61080 1.6372 . 85340 35 26 151 120j .6361 325 34 27 175 1601 .6351 310 33 28 200 200! -6340 294 32 29 225 240 .6329 279 31 30 .52250 .61280 1.6319 . 85264 30 31 275 320 .6308 249 29 32 299 360 .6297 234 28 33 324 400 .6287 218 27 34 349 440 .6276 203 26 35 .52374 .61480 1.6265 .85188 25 36 399 1 520 .0255 173 24 37 423 561 ! .6244 157 23 38 448 601! .6234 142 22 39. 473 641 .6223 127 21 40 . 52498 .61681 1.6212 .85112 20 41 522 721 .6202 096 19 42 547 761 .6191 081 18 43 572 801 .6181 066 17 44 597 842 .6170 051 16 45 .52621 61882 1.6160 . 85035 15 46 646 922 .6149 020 14 47 671 .61962 .6139 .85005 13 48 696 .62003 .6128 . 84989 12 49 720 043 .6118 974 11 50 . 52745 ft0f )ftV A 1ii u 84959 IO 51 770 ^mm^Bfr 043 9 52 794 m^^^^m 928 8 53 819 TI^^!Tf76 913 7 54 844 Z45 . 6066 897 6 55 52869 .62283 1 . 6055 . 84882 5 56 893 325 .6045 866 4 57 918 366 .6034 3 58 943 406 . 6024 S36 2 59 967 446 .6014 820 1 60 . 52992 .62487 1 .;< 103 .84805 L .,._ NCos N Cot NTan N Sin / 59 Natural Functions 58 c 32 Natural Functions 33 c 91 / N Sin N Tan N Cot 1 N Cos o . 52992 . 62487 1 . 6003 .84805 60 1 .53017 527 .5993 789 59 2 041 568 .5983 774 58 3 066 608 .5972 759 57 4 091 649 .5962 743 56 5 .53115 . 62689 1 . 5952 . 84728 55 6 140 730 .5941 712 54 7 164 770 .5931 697 53 8 189 811 .5921 681 52 9 214 852 .5911 666 51 10 . 53238 . 62892 1 . 5900 .84650 50 11 263 i 933 .5890 635 49 12 288 .62973 .5880 619 48 13 312 .63014 .5869 604 47 14 337 055 .5859 588 46 15 .53361 . 63095 1.5849 .84573 45 16 386 136 .5839 557 44 17 411 177 .5829 542 43 18 435 217 .5818! 526 42 19 460 258 .5808 511 41 20 . 53484 . 63299 1.5798 .84495 40 21 509 340 .5788 480 39 22 534 380 .5778 464 38 23 558 421 .5768 448 37 24 583 462 .5757 433 36 25 . 53607 . 63503 1.5747 .84417 35 26 632 544 .5737 402 34 27 656 584 .5727 386 33 28 681 625 .5717 370 32 29 705 666 .5707j 355 31 30 .53730 .63707 1.5697 .84339 30 31 754 748 . 56S7 324 29 32 779 789 . 5677 308 28 33 804 830 .5667 292 27 34 828 871 .5657, 277 26 35 . 53853 .63912 1.5647 .84261 25 36 877 953 . 5637 245 24 37 902 .63994 .5627 230 23 38 926 .64035 .5617 214 22 39 951 076 .5607, 198 21 40 .53975 .64117 1.5597 .84182 20 41 . 54000 158 . 5587 167 19 42 024 199 .5577 151 18 43 049 240 .5567 135 17 44 073 281 .5557 120 16 45 .54097 .64322 1.5547 .84104 15 46 122 363 . 5537 088 14 47 146! 404 . 5527 072 13 48 171 446 .5517 057 12 49 195. 487 .5507 041 11 50 .54220 .64528 1.5497 .84025 IO 51 244 56ft tfK 84009 9 52 269 6 H 83994 8 53 293 eoS^WKs* 978 7 54 317 693 .54158 962 6 55 .54342 .64734 1.5448 .83946 5 56 3661 775 .5438 930 4 57 391. 817 .5428 915 3 58 415 858 .5418 899 2 59 440 899 . 6408 883 1 OO .54464 .64941 1.5399 .83867 1 O N Cos N Cot N Tan N Sin ' f N SinNTanN Cot N Cos O .54464 .64941 1.5399 .83867 60 1 488i.64982| .5389 851 59 2 513 . 65024 ! .5379 835 58 3 537 O65 .5369 819 57 4 561 106 .5359 804 56 5 .54586 . 65148 1.5350 .83788 55 6 610 189 .5340 772 54 7 635 231 .5330 756 53 8 659 272 .5320 740 52 9 683 314 .5311 724 51 10 .54708 . 65355 1.5301 .83708 50 11 732 397 .5291' 692 49 12 756 438 .5282: 676 48 13 781 480 .5272 660 47 14 805 521 .5262j 645 46 15 . 54829 . 65563 1.5253 .83629 45 16 854 604( .5243! 613 44 17 878 646 .5233 597 43 18 902 688 .5224 581 42 19 927 729 .5214J 565 41 20 .54951 .65771 1.5204 '.83549 40 21 975 813 .5195 533 39 22 .54999! 854 .5185 517 38 23 . 55024 i 896 .5175 501 37 24 048 93S .5166: 485 36 25 .55072 '.65980 1.5156 .83469 35 26 097 .66021 .5147 453 34 27 121 063 .5137 437 33 23 145 105 .5127 421 32 29 169 147 .5118! 405 31 30 .55194 .66189 1.5108 .83389 30 31 218 230 .5099! 373 29 32 242 272 . 5089 356 28 33 266 314 .5080 340 27 34 291 r 356 .5070 324 26 35 .55315 .66398 1.5061 .83308 25 36 339 ; 440 .5051 292 24 37 363 482 .5042 276 23 38 388! 524 . 5032 260 22 39 412 , 566 .5023 244 21 40 . 55436 . 66608 1.5013 .83228 20 41 460 650 .5004 212 19 42 484 692 . 4994 195 18 43 509 734 .4985 179 17 44 533 776 .4975 163 16 45 .55557 .66818 1 . 4966 .83147 15 46 581 860 .4957 131 14 47 605! 902 .4947 115 13 48 630 944 .4938 098 12 49 654,. 66986 .4928 082 11 SO .55678 '.67028 1.4919 . 83066 IO 51 702 071 .4910 050 9 52 72fi 113 .4900 034 8 53 750 155 .4891 017 7 54 775 197 .4882 .83001 6 55 .55799 .67239 1 . 4872 . 82985 5 56 823 2S2 .4863 969 4 57 847! 324 .4854 953 3 58 87 1 366 . 4844 936 2 59 895 409 .4835 920 1 60 .559 19 .67451 1.4826 .82904 O NCosN Cot N Tan NSin / 57 Natural Functions 56 c 92 34 Natural Functions 35 c / N Sin NTan'N Cot N Cos o .55919 .67451 1.4826 .82904 60 1 943 493 .4816 ! 887 59 2 968 536 .48071 871 58 3 . 55992 578 .4798 855 57 4 .56016 620 .4788 839 56 5 . 56040 . 67663 1.4779 . 82822 55 6 064 705 .4770 806 54 7 088 748 .4761 790 53 8 112 790 .4751 773 52 9 136 832 .4742 757 51 io .56160 .67875 1 . 4733 .82741 50 11 184 917 .4724 724 49 12 208 . 67960 .4715 708 48 13 232 . 68002 .4705 692 47 14 256 045 .4696 675 46 15 . 56280 . 68088 1.4687 . 82659 45 16 305 130 .4678 643 44 17 329 173 .4669 626 43 18 353 215 .4659 610 42 19 377 258 .4650 593 41 20 . 56401 . 68301 1.4641 . 82577 40 21 425 343 .4632 561 39 22 449 386 .4623 544 38 23 473 429 .4614 528 37 24 497 471 .4605 511 36 25 .56521 . 68514 1.4596 . 82495 35 26 545 557 .4586 478 34 27 569 600 .4577 462 33 28 593 642 .4568 446 32 29 617 685 .4559 429 31 30 .56641 . 68728 1 . 4550 .82413 30 31 665 771 .4541 396 29 32 689 814 .4532 380 28 33 713 857 .4523 363 27 34 736 900 .4514 347 26 35 . 56760 . 68942 1.4505 .82330 25 36 784 . 68985 .4496 314 24 37 808 . 69028 .4487 297 23 38 832 071 .4478 281 22 39 856 114 .4469 264 21 40 . 56880 .69157 1.4460 . 82248 20 41 904 200 .4451 231 19 42 928 243 .4442 214 18 43 952 286 .4433 198 17 44 . 56976 329 .4424 181 16 45 . 57000 . 69372 1.4415 .82165 15 46 024 416 .4406 148 14 47 047 459 .4397 132 13 48 071 502 .4388 115 12 49 095 545 .4379 098 11 50 .57119 . 69588 1.4370 . 82082 IO 51 143 631 .4361 065 9 52 167 675 .4352 048 8 53 191 718 .4344 032 7 54 215 761 .4335 .82015 6 55 . 57238 .69804 1.4326 .81999 5 56 262 847 .4317 982 4 57 286 891 .4308 965 3 58 310 934 .4299 949 2 59 334 . 69977 .4290 932 1 60 .57358 .70021 1.4281 .81915 O NCOS N Cot NTan NSin i i N Sin NTan N Cot N Cos o . 57358 .70021 1.4281 .81915 OO 1 381 064 .4273 899 59 2 405 107 .4264 882 58 3 429 151 .4255 865 57 4 453 194 .4246 848 56 5 . 57477 . 70238 1.4237 .81832 55 6 501 281 .4229 815 54 7 524 32g .4220 798 53 8 548 368 .4211 782 52 9 572 412 .4202 765 51 IO . 57596 .70455 1.4193 .81748 50 11 619 499 .4185 731 49 12 643 542 .4176 714 48 13 667 586 .4167 698 47 14 691 629 .4158 681 46 15 .57715 . 70673 1.4150 . 81664 45 16 738 717 .4141 647 44 17 762 760 .4132 631 43 18 786 804 .4124 614 42 19 810 848 .4115 597 41 20 . 57833 . 70891 1 . 4106 . 81580 40 21 857 935 .4097 563 39 22 881 . 70979 .4089 546 38 23 904 .71023 .4080 530 37 24 928 066 .4071 513 36 25 . 57952 .71110 1 . 4063 . 81496 35 26 976 154 .4054 479 34 27 . 57999 198 .4045 462 33 28 . 58023 242 .4037 445 32 29 047 285 .4028 428 31 30 . 58070 .71329 1.4019 .81412 30 31 094 373 .4011 395 29 32 118 417 .4002 378 28 33 141 461 .3994 361 27 34 165 505 .3985 344 26 35 .58189 .71549 1 . 3976 .81327 25 36 212 593 .3968 310 24 37 236 637 .3959 293 23 38 260 681 .3951 276 22 39 283 725 .3942 259 21 40 . 58307 .71769 1 . 3934 .81242 20 41 330 813 .3925 225 19 42 354 857 .3916 208 18 43 378 901 .3908 191 17 44 401 946 .3899 174 16 45 . 58425 .71990 1.3891 .81157 15 46 449 .72034 .3882 140 14 47 472 078 .3874 123 13 48 496 122 .3865 106 12 49 519 167 .3857 089 11 50 . 58543 .72211 1 . 3848 .81072 IO 51 567 255 .3840 055 9 52 590 299 .3831 038 8 53 614 344 .3823 021 7 54 637 388 .3814 .81004 6 55 .58661 . 72432 1 . 3806 . 80987 5 56 684 477 .3798 970 4 57 708 521 .3789 953 3 58 731 565 .3781 936 2 59 755 610 .3772 919 1 60 . 58779 . 72654 1.3764 . 80902 O NCos N Cot NTan NSin / 55 Natural Functions 54 36 Natural Functions 37 c 93 / N Sin NTan'N Cot N Cos / N Sin NTan'N Cot NCos o . 58779 . 72654 1 . 3764 .80902 60 o . 60182 .75355 1.3270 . 79864 60 1 802 699 .3755 885 59 1 205 401 .3262 846 59 2 826 743 .3747 867 58 2 228 447 .3254 829 58 3 849 788 .3739 850 57 3 251 492 .3246 811 57 4 873 832 .3730 833 56 4 274 538 .3238 793 56 5 .58896 . 72877 1.3722 .80816 55 5 . 60298 . 75584 1.3230 . 79776 55 6 920 921 .3713 799 54 6 321 629 .3222 758 54 7 943 . 72966 .3705 782 53 7 344 675 .3214 741 53 8 967 .73010 .3697 765 52 8 367 721 .3206 723 52 9 . 58990 055 .3688 748 51 9 390 767 .3198 706 51 io .59014 .73100 1.3680 .80730 50 10 .60414 .75812 1.3190 .79688 50 11 037 144 .3672 713 49 11 437 858 .3182 671 49 12 061 189 .3663 696 48 12 460 904 .3175 653 48 13 084 234 .3655 679 47 13 483 950 .3167 635 47 14 108 278 .3647 662 46 14 506 . 75996 .3159 618 46 15 .59131 . 73323 1.3638 .80644 45 15 . 60529 .76042 1.3151 .79600 45 16 154 368 .3630 627 44 16 553 088 .3143 583 44 17 178 413 .3622 610 43 17 576 134 .3135 565 43 18 201 457 .3613 593 42 18 599 180 .3127 547 42 19 225 502 .3605 576 41 19 622 226 .3119 530 41 20 . 59248 . 73547 1.3597 .80558 40 20 . 60645 . 76272 1.3111 .79512 40 21 272 592 .3588 541 39 21 668 318 .3103 494 39 22 295 637 .3580 524 38 22 691 364 .3095 477 38 23 318 681 .3572 507 37 23 714 410 .3087 459 37 24 342 726 .3564 489 36 24 738 456 .3079 441 36 25 . 59365 .73771 1.3555 .80472 35 25 .60761 . 76502 1.3072 . 79424 35 26 389 816 .3547 455 34 26 784 548 .3064 406 34 27 412 861 .3539 438 33 27 807 594 .3056 388 33 28 436 906 .3531 420 32 28 830 640 .3048 371 32 29 459 951 .3522 403 31 29 853 686 .3040 353 31 30 . 59482 . 73996 1.3514 . 80386 30 30 .60876 . 76733 1 . 3032 .79335 30 31 506 .74041 .3506 368 29 31 899 779 .3024 318 29 32 529 086 .3498 351 28 32 922 825 .3017 300 28 33 552 131 .3490 334 27 33 945 871 .3009 282 27 34 576 176 .3481 316 26 34 968 918 .3001 264 26 35 . 59599 .74221 1.3473 . 80299 25 35 . 60991 . 76964 1 . 2993 . 79247 25 36 622 267 .3465 282 24 36 .61015 .77010 .2985 229 24 37 646 312 .3457 264 23 37 038 057 .2977 211 23 38 669 357 .3449 247 22 38 061 103 . 2970 193 22 39 693 402 .3440 230 21 39 084 149 .2962 176 21 4 .59716 .74447 1.3432 .80212 20 40 .61107 .77196 1 . 2954 .79158 20 41 739 492 .3424 195 19 41 130 242 .2946 140 19 42 763 538 .3416 178 18 42 153 289 .2938 122 18 43 786 583 .3408 160 17 43 176 335 .2931 105 17 44 809 628 .3400 143 16 44 199 382 .2923 087 16 45 . 59832 .74674 1.3392 .80125 15 45 .61222 .77428 1.2915 . 79069 15 46 856 719 .3384 108 14 46 245 475 .2907 051 14 47 879 764 .3375 091 13 47 268 521 .2900 033 13 48 902 810 .3367 073 12 48 291 568 .2892 .79016 12 49 926 855 .3359 056 11 49 314 615 .2884 . 78998 11 50 . 59949 . 74900 1.3351 .80038 IO 50 .61337 .77661 1 . 2876 .78980 IO 51 972 946 .3343 021 9 51 360 708 .2869 962 9 52 . 59995 .74991 .3335 .80003 8 52 383 754 .2861 944 8 R- .60019 . 75037 .3327 . 79986 7 53 406 801 .2853 926 7 042 082 .3319 968 6 54 429 848 .2846 908 6 55 .60065 .75128 1.3311 .79951 5 55 .61451 .77895 1 . 2838 .78891 5 56 089 173 .3303 934 4 56 474 941 .2830 873 4 57 112 219 .3295 916 3 57 497 .77988 .2822 855 3 58 135 264 .3287 899 2 58 520 .78035 .2815 837 2 59 158 310 .3278 881 1 59 543 082 .2807 819 1 60 . 60182 .75355 1.3270 . 79864 O 60 .61566 . 78129 1 . 2799 . 78801 O NCosJN Cot NTan NSin ' NCOS N Cot NTan NSin / 53 Natural Functions 52 94 38 Natural Functions 39 d ' N Sin N Tan N Cot N Cos o .61566 .78129 1.2799 .78801 60 1 589 175 .27921 783 59 2 612 222 .2784 765 58 3 635 269 .2776 747 57 4 658 316 . 2769 729 56 5 .61681 . 78363 1 . 2761 .78711 55 6 704 410 .2753 694 54 7 726 457 .2746 676 53 8 749 504 .2738 658 52 9 772 551 .2731 640 51 io .61795 . 78598 1 . 2723 . 78622 SO 11 818 645 .2715 604 49 12 841 692 .2708 586 48 13 864 739 .2700 568 47 14 887 786 .2693 550 46 15 . 61909 . 78834 1 . 2685 . 78532 45 16 932 881 .2677 514 44 17 955 928 .2670 496 43 18 .61978 . 78975 .2662 478 42 19 . 62001 .79022 .2655 460 41 *20 . 62024 . 79070 1 . 2647 .78442 40 21 046 117 .2640 424 39 22 069 164 .2632 405 38 23 092 212 .2624 387 37 24 115 259 .2617 369 36 25 .62138 . 79306 1 . 2609 . 78351 35 26 160 354 .2602 333 34 27 183 401 .2594 315 33 28 206 449 .2587 297 32 29 229 496 .2579 279 31 30 .62251 .79544 1 . 2572 .78261 30 31 274 591 . 2564 243 29 32 297 639 .2557 225 28 33 320 686 .2549 206 27 34 342 734 .2542 188 26 35 . 62365 .79781 1 . 2534 .78170 25 36 388 829 .2527 152 24 37 411 877 .2519 134 23 38 433 924 .2512 116 22 39 456 .79972 .2504 098 21 40 . 62479 . 80020 1 . 2497 .78079 20 41 502 067 .2489 061 19 42 524 115 .2482 043 18 43 547 163 .2475 025 17 44 570 211 .2467 . 78007 16 45 . 62592 . 80258 1 . 2460 . 77988 15 46 615 306 .2452 970 14 47 638 354 .2445 952 13 48 860 402 . 2437 934 12 49 683 4g0 .2430 916 11 50 . 62706 . 80498 1 . 2423 . 77897 IO 51 728 546 .2415 879 9 52 751 594 .2408 861 8 53 774 642 .2401 843 7 54 796 690 .2393 824 6 55 .62819 . 80738 1 . 2386 . 77806 5 56 842 786 . 2378 788 4 57 864 834 .2:571 769 3 58 887 882 . 2:'64 751 2 59 909 930 .2356 733 1 60 . 62932 .80978 ! 1.2349 .77715 O NCOS N Cot N Tan N Sin ' ' N Sin N Tan N Cot N Cos o .62932 .80978 1 . 2349 .77715 60 1 955 .81027 .2342 696 59 2 .62977 075 .2334 678 58 3 .63000 123 .2327 660 57 4 022 171 .2320 641 56 5 .63045 .81220 1.2312 . 77623 55 6 068 268 .2305 605 54 7 090 316 .2298 586 53 8 113 364 .2290 568 52 9 135 413 .2283 550 51 IO .63158 .81461 1 . 2276 .77531 50 11 180 510 .2268 513 49 12 203 558 .2261 494 48 13 225 606 .2254 476 47 14 248 655 .2247 458 46 15 .63271 . 81703 1 . 2239 . 77439 45 16 293 752 .2232 421 44 17 316 800 .2225 402 43 18 338 849 .2218 384 42 19 361 898 .2210 366 41 20 . 63383 .81946 1 . 2203 . 77347 40 21 406 .81995 .2196 329 39 22 428 . 82044 .2189 310 38 23 451 092 .2181 292 37 24 473 141 .2174 273 36 25 . 63496 .82190 1.2167 .77255 35 26 518 238 .2160 236 34 27 540 287 .2153 218 33 28 563 336 .2145 199 32 29 585 385 .2138 181 31 30 . 63608 . 82434 1.2131 .77162 30 31 630 483 .2124 144 29 32 653 531 .2117 125 28 33 675 580 .2109 107 27 34 698 629 .2102 088 26 35 .63720 . 82678 1.2095 . 77070 25 36 742 727 .2088 051 24 37 765 776 .2081 033 23 38 787 825 .2074 .77014 22 39 810 874 .2066 . 76996 21 40 . 63832 . 82923 1 . 2059 . 76977 20 41 854 .82972 .2052 959 19 42 877 . 83022 .2045 940 is 43 899 071 .2038 921 17 44 922 120 .2031 903 16 45 . 63944 .83169 1 . 2024 . 76884 15 46 966 218 .2017 866 14 47 . 63989 268 .2009 847 13 48 .64011 317 .2002 828 12 49 033 366 .1995 810 11 50 . 64056 .83415 1.1988 .76791 IO 51 078 465 .1981 772 9 52 100 514 . 1974 754 8 53 123 564 .1967 7X5 7 54 145 613 .1960 717 6 55 .64167 .83662 1 . 1953 . 76698 5 56 190 712 .1946 679 4 57 212 761 . L939 661 3 58 234 811 .1932 642 2 59 256 860 .1925 623 1 60 . 64279 .83910 1.1918 . 76604 O NCOS N Cot NTan NSin / 51 Natural Functions 50 c 40 6 Natural Functions 41 ( 1 N Sin NTan N Cot NCos .64279 .83910 1.1918 . 76604 60 1 301 . 83960 .1910 586 59 2 323 . 84009 .1903 567 58 3 346 059 .1896 548 57 4 368 108 .1889 530 56 5 . 64390 .84158 1 . 1882 .76511 55 6 412 208 .1875 492 54 7 435 258 .1868 473 53 8 457 307 .1861 455 52 9 479 357 .1854 436 51 10 .64501 . 84407 1.1847 .76417 50 11 524 457 .1840 398 49 12 546 507 .1833 380 48 13 568 ! 556 .1826 361 47 14 590 606 .1819 342 46 15 .64612. 84656 1.1812 .76323 45 16 635 706 .1806 304 44 17 657 756 . 1799 286 43 18 679 806 .1792 267 42 19 701 856 .1785 248 41 20 .64723 .84906 1.1778 . 76229 40 21 746 .84956 .1771 210 39 22 768 .85006 .1764 192 38 23 790 057 . 1757 173 37 24 812 107 .1750 154 36 25 .64834 .85157 1 . 1743 .76135 35 26 856 207 .1736 116 34 27 878 257 .1729 097 33 28 901 308 .1722 078 32 29 923 358 .1715 059 31 30 .64945 .85408 1 . 1708 .76041 30 31 967; 458 .1702 022 29 32 . 64989 ' 509 .1695-76003 28 33 .65011; 559 .1688.75984 27 34 033, 609 .1681 965 26 35 .65055 .85660 1.1674 . 75946 25 36 077 710 .1667 927 24 37 100! 761 .1660 908 23 38 122 811 .1653 889 22 39 144 862, .1647 870 21 40 .65166 .85912 1.1640 .75851 20 41 188 .85963 .1633 832 19 42 210 .86014 .1626 813 18 43 282 064 .1619 794 17 44 254 115 .1612 775 16 45 .65276 .86166 1 . 1606 . 75756 15 46 298! - 216 . 1599 738 14 47 320 267 . 1592 719 13 48 342| 318 .1585 700 12 49 364 368 .1578 680 11 50 .65386 .86419 1.1571 .75661 io 51 408 470 .1565 642 9 52 430 521 .1558 623 8 53 452 572 .1551 604 7 54 474 623 .1544 585 6 55 . 65496 .86674 1.1538 . 75566 5 56 518 725 .1531 547 4 57 540 776 .1524 528 3 58 562 827 .1517 509 2 59 584 878 .1510 490 1 60 . 65606 . 86929 1 . 1504 .75471 O NCosN Cot N Tan N Sin i r N Sin NTan N Cot N Cos O . 65606 . 86929 1.1504 .75471 60 1 628 . 86980 .1497 452 59 2 650 .87031 .1490 433 58 3 672 082 .1483 414 57 4 694 133 .1477 395 56 5 .65716 .87184 1 . 1470 . 75375 55 6 738 236 .1463 356 54 7 759 287 .1456 337 53 8 781 338 .1450 318 52 9 803 389 .1443 299 51 IO . 65825 .87441 1 . 1436 . 75280 50 11 847 492 .1430 261 49 12 869 543 . 1423 241 48 13 891 595 .1416 222 47 14 913 646 .1410 203 46 15 .65935 . 87698 1 . 1403 .75184 45 16 956 749 .1396 165 44 17 . 65978 801 .1389 146 43 18 . 66000 852 .1383 126 42 19 022 904 .1376 107 41 20 .66044 . 87955 1.1369 .75088 40 21 066 .88007 .1363 069 ' 39 22 088 059 .1356 050 38 23 109 110 .1349 030 37 24 131 162 .1343 .75011 36 25 .66153 .88214- 1.1336 . 74992 35 26 175 265 .1329 973 34 27 197 317 .1323 953 33 28 218 369 . 1316 934 32 29 240 421 .1310 915 31 30 . 66262 . 88473 1 . 1303 . 74896 30 31 284 524 .1296 876 29 32 306 576 .1290 857 28 33 327 628 .1283 838 27 34 349 680 .1276 818 26 35 .66371 . 88732 1.1270 . 74799 25 36 393 784 .1263 780 24 37 414 836 .1257 760 23 38 436 888 .1250 741 22 39 458 940 .1243 722 21 40 . 66480 . 88992 1 . 1237 . 74703 20 41 501 . 89045 .1230 683 1.9 42 523 097 . 1224 664 18 43 545 149 .1217 644 17 44 566 201 .1211 625 16 45 . 66588 . 89253 1.1204 . 74606 15 46 610 306 .1197 586 14 47 632 358 .1191 567 13 48 653 410 1184 548 12 49 675 463 .1178 528 11 50 . 66697 .89515 1.1171 . 74509 IO 51 718 567 .1165 489 9 52 740 620 .1158 470 8 53 762 672 .1152 451 7 54 783 725 .1145 431 6 55 . 66805 . 89777 1.1139 .74412 5 56 827 830 .1132 392 4 57 848 883 .1126 373 3 58 870 935 .1119 353 2 59 891 . 89988 .1113 334 1 GO .66913 .90040 1.1106 .74314 O NCosN CotNTan ! N Sin / 49 Natural Functions 48 c 96 42 6 Natural Functions 43 t N Sin NTanN Cot N Cos N Sin NTan N Cot N Cos o .66913 . 90040 1.1106 .74314 60 o 68200 .93252 1 . 0724 .73135 60 1 935 093 .1100 295 59 1 221 308 .0717 116 59 2 956 146 .1093 276 58 . 2 242 360 .0711 096 58 3 978 199 .1087 256 57 3 264 415 .0705 076 57 4 . 66999 251 .1080 237 56 4 285 469 .0699 056 56 5 .67021 . 90304 1.1074 . 74217 55 5 68306 . 93524 1 . 0692 . 73036 55 6 043 357 .1067 198 54 6 327 578 .0686 .73016 54 7 064 410 . 1061 178 53 7 349 633 .0680 . 72996 53 8 086 463 .1054 159 52 8 370 688 .0674 976 52 9 107 516 .1048 139 51 9 391 742 .0668 957 51 io .67129 . 90569 1.1041 . 74120 50 IO .68412 . 93797 1.0661 . 72937 50 11 151 621 .1035 100 49 11 434 852 .0655 917 49 12 172 674 .1028 080 48 12 455 906 .0649 897 48 13 194 727 .1022 061 47 13 476 .93961 .0643 877 47 14 215 781 .1016 041 46 14 497 .94016 .0637 857 46 15 . 67237 . 90834 1 . 1009 . 74022 45 15 .68518 .94071 1 . 0630 . 72837 45 16 258 887 .1003 .74002 44 16 539 125 .0624 817 44 17 280 940 .0996 . 73983 43 17 561 180 .0618 797 43 18 301 . 90993 .0990 963 42 18 582 235 .0612 777 42 19 323 .91046 .0983 944 41 19 603 290 .0606 757 41 20 . 67344 . 91099 1 . 0977 . 73924 40 20 . 68624 .94345 1 . 0599 . 72737 40 21 366 153 .0971 904 39 21 645 400 .0593 717 39 22 387 206 .0964 885 38 22 666 455 .0587 697 38 23 409 259 .0958 865 37 23 688 510 .0581 677 37 24 430 313 .0951 846 36 24 709 565 .0575 657 36 25 . 67452 .91366 1 . 0945 .73826 35 25 . 68730 . 94620 1 . 0569 .72637 35 ' 26 473 419 .0939 806 34 26 751 676 .0562 617 34 i 27 495 473 .0932 787 33 27 772 731 .0556 597 33 28 516 526 .0926 767 32 28 793 786 .0550 577 32 29 538 580 .0919 747 31 29 814 841 .0544 557 31 30 .67559 .91633 1 . 0913 .73728 30 30 . 68835 . 94896 1 . 0538 . 72537 30 31 580 687 .0907 708 29 31 8571.94952 .0532 517 29 32 602 740 . 0900 688 28 32 878 .95007 .0526 497 28 33 623 794 .0894 669 27 33 899 062 .0519 477 27 34 645 847 .0888 649 26 34 920 - 118 .0513 457 26 35 . 67666 .91901 1.0881 .73629 25 35 . 68941 .95173 1.0507 . 72437 25 36 688 .91955 .0875 610 24 36 962 229 .0501 417 24 37 709 . 92008 .0869 590 23 37 . 68983 284 .0495 397 23 38 730 062 .0862 570 22 38 .69004 340 .0489 377 22 39 752 116 .0856 551 21 39 025 395 .0483 357 21 40 . 67773 .92170 1 . 0850 .73531 20 40 . 69046 .95451 1.0477 .72337 20 41 795 224 .0843 511 19 41 067 506 .0470 317 19 42 816 277 .0837 491 18 42 088 562 .0464 297 18 43 837 331 .0831 472 17 43 109 618 .0458 277 17 44 859 385 .0824 452 16 44 130 673 .0452 257 16 45 . 67880 . 92439 1.0818 .73432 15 45 .69151 . 95729 1 . 0446 . 72236 15 46 901 493 .0812 413 14 46 172 785 .0440 216 14 47 923 547 .0805 393 13 47 193 841 .0434 196 13 48 944 601 .0799 373 12 48 214 897 .0428 176 12 49 965 655 .0793 353 11 49 235 . 95952 .0422 156 11 50 . 67987 . 92709 1 . 0786 .73333 IO 50 . 69256 . 96008 1.0416 .72136 IO 51 . 68008 763 .0780 314 9 51 277 064 .0410 116 9 52 029 817 .0774 294 8 52 298 120 .0404 095 8 53 051 872 .0768 274 7 53 319 176 .0398 075 I 54 072 926 .0761 254 6 54 340 232 .0392 055 6 55 . 68093 . 92980 1 . 0755 . 73234 5 55 . 69361 . 96288 1.0385 .72035 5 56 us . 93034 .0749 215 4 56 382 344 .0379 .72015 4 57 136 088 .0742 195 3 57 403 400 .0373 .71995 3 58 157 143 .0736 175 2 58 424 457 .0367 974 2 59 17S 197 .0730 155 1 59 44S 513 .0361 954 1 60 .6820C . 93252 1 . 0724 .73135 O 60 . 69466 . 96569 1 . 0355 7 . 1934 O N Cos N Col NTan NSin / NCos N Cot NTan NSin / 47 Natural Functions 46 44 Natural Functions 97 ' N Sin N Tan! N Cot N Cos o .69466 .96569 1.0355 .71934 60 1 487 625 . 0349 914 59 2 508 681 .0343 894 58 3 529 738 .0337 873 57 4 549 794 .0331 853 56 5 . 69570 . 96850 1 . 0325 .71833 55 6 591 907 .0319 813 54 7 612 . 96963 .0313 792 53 8 633 . 97020 .0307 772 52 9 654 076 .0301 752 51 io .69675 .97133 1 . 0295 .71732 50 11 696 189 .0289 711 49 12 717 246 .0283 691 48 13 737 302 .0277 671 47 14 758 359 .0271 650 46 15 . 69779 .97416 1 . 0265 .71630 45 16 800 472 .0259 610 44 17 821 529 .0253 590 43 18 842 586 .0247 569 42 19 862 643 .0241 549 41 20 . 69883 . 97700 1 . 0235 .71529 40 21 904 756 .0230 508 39 22 925 813 .0224 488 38 23 946 870 .0218 468 37 24 966 927 .0212 447 36 25 . 69987 .97984 1 . 0206 .71427 35 26 .70008 .98041 .0200 407 34 27 029 098 .0194 386 33 28 049 155 .0188 366 32 29 070 213 .0182 345 31 30 . 70091 . 98270 1.0176 71325 30 31 112 327 .0170 305 29 32 132 384 .0164 284 28 33 153 441 .0158 264 27 34 174 499 .0152 243 26 35 .70195 . 98556 1.0147 .71223 25 36 215 613 .0141 203 24 37 236 671 .0135 182 23 38 257 728 .0129 162 22 39 277 786 .0123 141 21 to . 70298 . 98843 1.0117 .71121 20 41 319 90 i .0111 100 19 42 339 . 98958 .0105 080 18 43 360 .99016 . 0099 059 17 44 381 073 .00u4 039 16 4i .70401 .99131 1.0088 .71019 15 46 422 189 .0082 .70998 14 47 443 247 .0076 978 13 48 463 304 .0070 957 12 49 484 362 .0064 937 11 50 . 70505 . 99420 1.0058 .70916 IO 51 525 478 .00521 896 9 52 546 536 .0047: 875 8 53 567 594 .0041 855 7 54 587 652 .0035, 834 6 55 . 70608 .99710 1.0029^.70813 5 56 628 768 . 002:^ 793 4 57 649 826 .0017 772 3 58 670 884 .0012 752 2 59 690 . 99942 .0006 731 1 60 .70711 1^.0000 1.0000 .70711 O NCOS N Cot: N Tar l NSin / 45 Natural Functions TABLE IV Table of Powers and Roots V* v ^ ivi Table of Powers and Roots 101 No. Squares Cube, |Xt? Cube Roots No. Squares Cubes Square Roots Cube Roots 1 2 3 4 1 4 ,9 16 1 8 27 64 1.000 1.414 1.732 2.000 1.000 1.259 1.442 1.587 51 52 53 54 2,601 2,704 2,809 2,916 132,651 140,608 148,877 157,464 7.141 7.211 7.280 7.348 3 . 708 3.732 3.756 3.779 5 6 7 8 9 25 36 49 64 81 125 216 343 512 729 2.236 2.449 2.645 2.828 3.000 1.709 1.817 1.912 2.000 2.080 55 56 57 58 - 59 3,025 3,136 3,249 3,364 3,481 166,375 175,616 185,193 195,112 205,379 7.416 7.483 7.549 7.615 7.681 3.802 3.825 3.848 3.870 3.892 io 11 12 13 14 100 121 144 169 196 1,000 1,331 1,728 2,197 2,744 3.162 3.316 3.464 3.605 3.741 2.154 2.223 2.289 2.351 2.410 60 61 62 63 64 3,600 3,721 3,844 3,969 4,096 216,000 226,981 238,328 250,047 262,144 7.745 7~M0 7.874 7.937 8.000 3.914 3.936 3.957 3.979 4.000 15 16 17 - 18 19 225 256 . 289 324 361 3,375 4,096 4,913 5,832 6,859 3.872 4.000 4.123 4.242 4.358 2.466 2.519 2.571 2.620 2.668 65 66 67 68 69 4,225 4,356 4,489 4,624 4,761 274,625 287,496 300,763 314,432 328,509 8.062 8.124 8.185 8.246 8.306 4.020 4.041 4.061 4.081 4.101 20 21 22 23 24 400 441 484 529 576 8,000 9,261 10,648 12,167 13,824 4.472 4.582 4.690 4.795 4.898 2.714 2.758 2.S02 2.843 2.884 TO 71 72 73 . 74 4,900 5,041 5,184 5,329 5,476 343,000 357,911 373,248 389,017 405,224 8.366 8.426 8.485 8.544 8.602 4.121 4.140 4.160 4.179 4: 198 25 26 27 28 29 625 676 729 784 841 15,625 17,576 19,683 21,952 24,389 5.000 5.099 5.196 5.291 5.385 2.924' 2.962 3.000 3.036 3.072 75 76 77 78 79 5,625 5,776 5,929 6,084 6,241 421,875 438,976 456,533 474,552 493,039 8.660 8.717 8.774 8.831 8.888 4.217 4.235 4.254 4.272 4.290 30 31 32 33 34 900 961 1,024 1,089 1,156 27,000 29,791 32,768 35,937 39,304 5.477 5.567 5.656 5.744 5.830 3.107 3.141 3.174 3.207 3.239 SO 81 82 83 84 6,400 6,561 6,724 6,889 7,056 512,000 531,441 551,368 571,787 592,704 8.944 9.000 9.055 9.110 9.165 4.308 4.326 4.344 4.362 4.379 35 36 37 38 39 1,225 1,296 1,369 1,444 1,521 42,875 46,656 50,653 54,872 59,319 6.916 6.000 6.082 6.164 6.244 3.271 3.301 3.332 3.361 3.391 85 86 87 88 89 7,225 7,396 7,569 7,744 7,921 614,125 636,056 658.503 681,472 704,969 9.219 9.273 9.327 9.380 9.433 4.396 4.414 4.431 4.447 4.464 40 41 42 43 44 1,600 1,681 1,764 1,849 1,936 64,000 68.921 74,088 79,507 85,184 6.324 6.403 6.480 '6 . 557 6.633 3.419 3.448 3.476 | 3.503 3.530 90 91 92 93 94 8,100 8,281 8,464 8,649 8,836 729,000 753,571 778,688 804,357 830,584 9.486 9.539 9.591 9.643 9.695 4.481 4.497 4.514 4.530 4.546 45 46 47 48 49 2,025 2,116 2,209 2,304 2,401 91,125 97,336 103,823 110,592 117,649 6.708 6.782 6.855 6,928 IT 9.28 3.556 3.583 3.608 I 3.634 | 3.659 95 96 97 98 99 9,025 9,216 9,409 9,604 9,801 857,375 884,736 912.673 941,192 970,299 9.746 9.797 9.848 9.899 9.949 4.562 4.578 4.594 4.610 4.626 50 2,500 125,000 7.071 3.684 lOO 10,000 1,000,000 1Q 000 4.641 TABLE V Formulas Formulas 105 PLANE GEOMETRY 1. Length of circle = 2irr = 3. 141 59d 2. Area of circle =irr 2 3. Area of triangle = \bh = \ab sin C = \r(a-\-b+c) abc = ^ = V / s(s-a)(s-b)(s-c) 4. Area of parallelogram = bh 5. Area of square = a 2 6. Area of equilateral triangle = V3 7. Area of trapezoid = J/i(&i+& 2 ) = hm SOLID GEOMETRY 1 . Volume of prism = ba 2. Volume of pyramid = ^ba 3. Volume of right circular cylinder = irr 2 (\ 4. Total surface of right circular cylinder = 2irr (r+ A) 5. Lateral surface of right circular cylinder = 2irrd 6. Volume of right circular cone = 7rr 2 A 7. Lateral surface of right circular cone = 7rrs 8. Total surface of right circular cone = 7rr(r+s) 9. Surface of sphere = 47rr 2 4 10. Volume of sphere = ^jrr 3 series 1. Arithmetical progression : ft l = a-\-(nl)d; s = ~(a+l) 106 Formulas 2. Geometrical progression : , , a ar n . . . a l = ar n ~ x \ s= : if r< 1 and n->oo , s = - 1-r 1-r 3. Binomial theorem : {a+b) n = a n + n .a n -'b+ n{ \~} ) a n -W 1 1 A + n(n-^2 ) an . t6 , +ete The ftth term n(n-l)(n-2) .... ( n -/c+2) __, 4 , 1 ^_ 1 = L2.3. .. .k-1 a +b LOGARITHMS 1. log ab = log a+log b p. log r = log a log b 3. log a n = n log a . . */- log a 4. log V a = ^~ n 5. log 1 = logfciV 6. log a N = log 6 a 7. cologiV = log^=(10-logiV)-10 QUADRATIC EQUATION If az 2 +fcc+c = 0, a; = ^ . If 6 2 4ac = 0, the roots are real and equal. If b 2 4ac>0, the roots are real and unequal. If b 2 4ac<0, the roots are complex. Formulas 107 TRIGONOMETRIC FORMULAS aba 1. sin a = -, cos a = - , tan a = r, c c b c c b esc a = - , sec a = t , cot a = - a 1 6 a 2. sin 2 a + cos 2 a = l a cos a 6. cot a = - 3. sec 2 a = l+tan 2 a 1 4. csc 2 a=l + cot 2 a ' sec a = ^oTa sin a 5. tan a = 8. esc a cos a sin a 9. sin (a =*=)= sin a cos /3=*=cos a sin (3 10. cos (a =*= j8) = cos a cos /3 =f sin a sin (3 tan a^tan /?. 11. tan(a0) = 1 =f tan a tan /3 12. sin a+sin (3 = 2 sin J(a+j8) cos |(a /3) 13. sin a sin |S = 2 cos |(a-f /?) sin ^(a /3) 14. cos a+cos = 2 cos J( a +^) cos 2( a 0) 15. cos a cos = 2 sin J(a+/3) sin -(a jS) 16. sin a sin /3 = ^ cos (a ff) \ cos (a+/5) 17. cos a cos = \ cos (a j8)+J cos (a+/3) 18. sin a cos /? = J sin (a+/3)+J sin (a /3) 19. sin 2 a = 2 sin a cos a 20. cos 2 a = cos 2 a sin 2 a = 2cos 2 a-l = l-2sin 2 a 108 Formulas rti 2 tan a 29. cos (7r0) = cos 21. tan 2 a = - - r 1-tana 3() tan (tt == ^) = == tan ^ 22. sina=^ JE|*1* 3L sin (r**) ** 32. cos ( x) = cos z 23. cos^a^JH^ 33 - tan (-*) = -tans 34. esc ( x) = esc x 24. tanta=^ A / 1 ~ cosa ' 35. sec (-x)=secx 2 \l+cosa on , N 36. cot ( x) = cot x 25. sin f | J = cos (9 37. sin 30 = J v 38. sin 45 = 1/2 26. cos(j^0j - =f sin 39> gin 600 = 1^3 / x \ 40. cos 30 = \V 3 27. tan (-==0 = ^cot0 ^ - /- \2 / 41. cos 45 = 1/2 28. sin (tt0) = =Fsin 42. cos 60 = \ TRIANGLES AO a b c Aa a-b sin(A-) 4o. -: -r = n = ~Fi 40. = j ^ sin A sin B sin C c cos f C 44. a 2 = 6 2 +c 2 -26c cos A a+6 = cos (A-B) a+6 = tan (A+B) " c sin \ C a b tan (A B) Us = i (a+b+cy. y 48. sin \ A = ^ is ~ h) b ( c S ~ c) 49. cos \ A=^ 8 ~ a) . 50. tan jA-Jb-M'-Zc} 2 \ s(s-a) Formulas 109 If r = radius of inscribed circle : \ s 2 s-a 53. tani = -^. 54. tanC = ~. __ . , , . ~ c 2 sin A sin 5 55. Area = ao sin C =77 : 7= 2 sin C = l/s(s-a)(s-6)(s-c) 56. Diameter of circumscribed circle a b c sin A sin sin C TABLE VI Equivalents and Logarithms of Important Constants Equivalents and Logarithms of Important Constants 113 TCnmhpr Common .Number Logarithm 3.14 159 6 57.29 577 9 0.01 745 3 2.71 828 2 0.43 429 4 2.30 258 5 1 . 60 934 7 kilom. 0.91 440 2 meter 0.30 480 1 meter 25 . 40 005 mm. 39 . 37 inches 1.09 361 1 yard 3 . 28 083 3 feet 6080. 290 feet 0.62 137 Omile 7000 grains 453 . 59 242 8 grammes 28 . 34 953 grammes 31 . 10 348 grammes . 06 479 9 gramme 2.20 462 2 lbs. Av. 15.43 235 6 grains 1.05 668 U.S. quart 0.26 417 U.S. gal. 33.814 U.S. fluid oz. 0.94 636 liter 3.78 544 liters 0.02 957 3 liter 231 cu. inches 4.54 346 liters 36.34 77 liters 0.49 715 1.75 812 2 2.24 187 7 0.43 429 4 1.63 778 4 0.36 221 6 0.20 665 1.96 113 7 1.48 401 6 1.40 483 5 1.59 516 5 0.03 886 3 0.51 598 4 3.78 392 4 1.79 335 3.84 509 8 2.65 666 6 1.45 254 6 1.49 280 9 2.81 156 8 0.34 333 4 1.18 843 2 0.02 394 4 1.42 188 4 1.52 910 1.97 605 6 0.57 811 6 2.47 090 2.36 361 2 0.65 738 7 1.56 047 7 1 radian =i^ IT 1 degree =z-^. radians loO to (modulus of common loga- TO 1 mile 1 yard 1 foot 1 inch 1 nautical mile 1 kilometer 1 pound Av 1 ounce Av 1 ounce Troy 1 grain 1 kilogramme 1 gramme 1 liter 1 quart, U.S 1 gallon. U.S 1 fluid ounce. . . . 1 gallon U.S 1 British gallon 1 British bushel TABLE VII Reductions Degrees, Minutes, and Seconds Reduced to Radians 117 Radians ' Radians " Radians 1 2 3 4 0.01 745 33 0.03 490 66 0.05 235 99 0.06 981 32 1 2 3 4 0.00 029 09 0.00 058 18 0.00 087 27 0.00 116 36 1 2 3 4 0.00 000 48 0.00 000 97 0.00 001 45 0.00 001 94 5 6 7 8 9 0.08 726 65 0.10 471 98 0.12 217 30 0.13 962 63 0.15 707 96 5 6 7 8 9 0.00 145 44 0.00 174 53 0.00 203 62 0.00 232 71 0.00 261 80 5 6 7 8 9 0.00 002 42 0.00 002 91 0.00 003 39 0.00 003 88 0.00 004 36 io 0.17 453 29 IO 0.00 290 89 IO 0.00 004 85 20 0.34 906 59 15 0.00 436 33 15 0.00 007 27 30 0.52 359 88 20 0.00 581 78 20 0.00 009 70 40 0.69 813 17 25 0.00 727 22 25 0.00 012 12 50 0.87 266 46 30 0.00 872 66 30 0.00 014 54 60 1.04 719 76 35 0.01 018 11 35 0.00 016 97 70 1.22 173 05 40 0.01 163 55 40 0.00 019 39 80 1.39 626 34 50 0.01 454 44 50 0.00 024 24 90 1.57 079 63 60 0.01 745 33 60 0.00 029 09 Reduction of Minutes to Degrees 0' = = 0?000 !lO' = =0?166 20 = =0?333 30' = =0?500 40 = =0?666i5O' = =0?833 V .016 11' .183 21' .350 31' .516 41' .683 ! 51' .850 2' .033 12' .200 22' .366 32' .533 42' .700 52' .866 3' .050 13' .216 23' .383 33' .550 43' .716 53' .883 4' .066 14' .233 24' .400 34' .566 44' .733 54' .900 5' .083 15' .250 25' .416 35' .583 45' .750 55' .916 6' .100 16' .266 26' .433 36' .600! 46' .766 56' .933 7' .116 17' .283 27' .450 37' .616 47' .783 57' .950 8' .133 18' .300 28' .466 1 38' .633 48' .800 58' .966 9' .150 19' .316 29' 30' = .483 =0?500 39' .650 49' .816 59' 60' = .983 =1?000 Reduction of Seconds to Degrees 6" =0?00166 7" =0.00194 8" =000222 9" =0 ?0025C 10" =0?00277 15" =0.00416 20" =0.00555 30" =0?00833 35" =0!00970 40" =0.01111 45" =0.01250 50" =0?01388 118 Reduction of Degrees to Minutes and Seconds o?oo = 0' 0?3=18' 0?60 =36' 0?90 =54' .01 0'36" .31 18'36" .61 36'36" .91 54'36" .02 1/12" .32 19'12" .62 37'12" .92 55'12" .03 1'48" .33 19'48" . 63 37'48" .93 55'48" .04 2'24" . 34 20'24" .64 38'24" . 94 56'24" 0?05 3' 0?35 =21' 0?65 =39' 0?95 =57' .06 3'36" .36 21'36" .66 39'36" .96 57'36" .07" 4'12" .37 22'12" .67 40'12" .97 58'12" .08* 4'48" .38 22'48" . 68 40'48" . 98 58'48" .09 5'24" .39 23'24" .69 41'24" . 99 59'24" o?io - 6' 0?40 =24' 0?70 =42' 1?00 =60' .11 6'36" .41 24'36" .71 42'36" .12 7'12" .42 25'12" .72 43'12" .13 7'48" .43 25'48" .73 43'48" .14 8'24" .44 26'24" . 74 44'24" 0?15 - 9' 0?45 =27' 0?75 =45' .16 9'36" . 46 27'36" .76 45'36" o?ooo = or .17 10'12" .47 28'12" .77 46'12" .001 3 76 .18 10'48" .48 28'48" . 78 46'48" .002 7f2 .19 11 '24" .49 29'24" .79 47'24" .003 10*8 .004 14 * 4 0?20 = 12' 0?50 =30' 0?80 =48' .21 12'36" .51 30'36" .81 48'36" 0?005 =18f .22 13'12" .52 31'12" .82 49'12" .006 21 f6 .23 13'48" .53 31'48" . 83 49'48" .007 25 ".2 .24 14'24" .54 32'24" . 84 50'24" .008 28 f 8 .009 32 f4 O?2o = 15' 0?55 =33' 0?85 =51' .26 15'36" .56" 33'36" .86 51'36" 0?01 =36f .27 16'12" .57 34'12" .87 52'12" .28 16'48" . 58 34'48" .88 52'48" .29 17'24" .59 35'24" .89 53'24" 0?30 = 18' 0?60 =36' 0?90 =54' INDEX INDEX [References are to sections, not to pages] Abridged division 139 Abridged multiplication . . . 133 Addition theorem 195 Angle: between two curves 332 cosine of 32 Angles: in general 29 polyedral 320 sine of 32 spherical 332 tangent of 32 triedral 321 Area : of a cone 272 of a cylinder 272 of a frustum of a cone. . . 273 of a prism 269 of a pyramid 270 of a sphere 280 of oblique triangle 192 Arithmetical means 214 Arithmetical progression. . . 211 Axis of a cone 250 Bernoulli, James 5 Bernoulli, John 5 Bezout 72 Binomial theorem 206 Cauchy 72 Cavalieri's theorem 302 Characteristic 147 Circle, equation of 226 Cologarithm 188 Common logarithms 146 Complex fractions . 104 Complex numbers 89 Cone: altitude of 248 axis of 250 base of 248 circular 250 frustum of 267 lateral surface of 272 of revolution 250 right circular 250 sections of 268 similar. . . 275 Conical surface. . 248 directrix of 248 elements of 248 generatrix of 248 vertex of 248 Constant 2 Cosines, law of 181 Cramer 72 Cube.. 263 Cylinder: altitude of 251 base of 251 circular 251 elements of 251 lateral area of 269 of revolution 252 similar 274 volume of 298, 299 Denominator, rationalizing. 129 Determinant 75 solution by 76 Diameter of circumscribed circle 180 121 122 THIRD-YEAR MATHEMATICS [References are to sections, not to pages] Direct variation 11 Directrix: of a conical sur- face. . . , 248 of a cylindrical surface. . . 251 Discriminant 92 Division : abridged 139 synthetic 16 Dodecaedron 245 Elements of a progression. . 212 Ellipse, equation of 228 Equations: equivalent 73 exponential 162 formation of 93 homogeneous 236 inconsistent 73 irrational 132,238 linear 67,68,70 of quadratic form 87 quadratic, in one un- known 81, 83 solution by factoring .... 22 solution by graph 71 trigonometric. ... 88, 134, 202 with imaginary roots. ... 89 Evaluation of functions. . . 6, 21 Excess, spherical 340 Exponents: laws of 117 fractional 116 negative 115 zero 114 Exponential equations 162 Factor theorem 22 Factorial notation 208 Factoring 94-102 Fractional equations 238 Fractions, complex 104 Frustum: of a cone 267 of a pyramid 265 slant height of 266 Function 1,4 cubic 14 evaluation of 6 graph of linear 8 inverse 37 linear 7, 9 line representation of . . 39 of f--.) 60 62 of double an angle 199 of half an angle 200 of negative angles 57 quadratic 12, 13 trigonometric 32 Functional notation 5 Gauss 72 General quadratic equa- tion 225 Generatrix, of a conical sur- face 248 of a cylindrical surface. . . 251 Geometrical means 218 Geometrical progression ... 215 elements of 216 infinite 220 Graph: of cubic function. . 14 of linear function 8 of logarithmic function . . 148 of quadratic function. ... 13 of trigonometric func- tions 50 INDEX 123 [References are to sections, not to pages] Graphical solution: of a linear system 71 of a quadratic system in two variables 233 Hyperbola 229 Icosaedron 245 Imaginary numbers 90 Infinite geometrical series . . 220 Intercepts 10 Inverse functions 37 Inverse variation 24 Irrational equations. . . 132, 238 Irrational numbers 90 Joint variation 26 Lagrange 72 Laplace 72 Law of cosines 181 of sines 179 of tangents 183 Laws of exponents 117 Leibnitz 5, 72 L'Hopital 72 Linear equations in one un- known 67, 68 Linear function 7, 9 Line representation of func- tions 39 Logarithms 143 common 146 of a power 158 of a product 156 of a quotient 157 of a root 159 table of 152, 170 Lune 351 Mantissa 147 Means: arithmetical 214 geometrical . 218 Measure, radian 46 Measurement, precision of . 137 Mollweide's equations .... 184 Multiplication, abridged . . . 138 Napier 145 Notation, functional 5 Oblique triangle: area of.. 192 solution of 187 Octaedron 245 Parabola 13 Parallelopiped: oblique. . . 263 rectangular. 263 right 263 volume of 286, 291 Polar triangles 336 Polyedral angle 320 Polyedron 243 Polyedrons 245 regular 325 Polygon, spherical 323 Precision of measurement . . 137 Prism : altitude of 253 bases of 253 lateral area of 269 lateral edges 253 lateral faces of 253 oblique 254 right 254 truncated 264 volume of 294 Progression: arithmetical.. 211 geometrical 215 124 THIRD- YEAR MATHEMATICS [References are to sections, not to pages] Pyramid 247 altitude of 247 base of 247 frustum of 265 inscribed in a cone 295 lateral area of 270 lateral edges of 247 lateral faces of 247 regular 249 vertex 247 volume of 304 Quadrant 31 Quadratic equation 81 graph of quadratic in two unknowns 227 in two unknowns 225 nature of the roots of . . . . 92 Quadratic function 12, 13 simultaneous 232 solution by completing square 83 solution by formula 83 Radian measure 46 Radical 120 reduction of 121 Rationalizing denominator . 129 Regular polyedrons 325 Relation between the roots and coefficients 93 Remainder theorem 19, 20 Right section 256 Roots of a quadratic equa- 89 tion relation between roots and coefficients . . 93 square roots 84 square root of a radical expression 131 Section 255 of a cone 268 Segment, spherical 310 Signs of functions 33 Similar cones 275 Similar cylinders 274 Simultaneous quadratics . . . 232 Sines, law of 180 Slant height, of a frustum of a pyramid 266 Slide rule 163 Solution: of oblique tri- angles 187 of right triangle 174 Sphere : surface of 280 volume of 309 Spherical cone 313 Spherical excess 316 Spherical polygon 323 Spherical sector 315 Spherical segment 310 Spherical triangle 335 Square root : of polynomials 84 of a radical expression. . . 131 Subtraction theorem 195 Sums of sines and cosines. 197 Symmetrical polyedral angles 342 Symmetrical polygons 343 Synthetic division 16 Table of logarithms . . . 152, 170 Tangent : of angle 32 of half -angle 185 law of 183 Tetraedron 245 Triangle: birectangular. . . 335 polar 236 INDEX 125 [References are to sections, not to pages] spherical 335 solution of right 174 trirectangular 335 Triedral angle 321 Trigonometric equations 88, 134, 202 Trigonometric functions ... 32 Truncated prism 264 Variable 3 Variation: direct 11 direct and inverse 27 inverse 24 joint 2G Volume : of a cone 306 of a cylinder 298 of a prism 294 of a pyramid 304 of a sphere 309 of frustum of a cone .... 307 Zone. 281 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. rc-ctfiveD REC'D LD JUL 251963 j REC'D LD . I, 7 ' B4 -9 AM MAR 2 3 1968 $ REC'D LD MAR^ 'b6-ffl otm e i:CT5 7 *v UNIVERSITY OF CALIFORNIA LIBRARY