Essentials of plane trigonometry and analytic geometry

Antherton H. Sprague9781406703740, 1406703745

ESSENTIALS OF PLANE TRIGONOMETRY AND ANALYTIC GEOMETRY BY ATHERTON H. SPRAGUE PROFESSOR OP MATHEMATICS AMHEBST COLLEGE NEW YORK PRENTICE-HALL, INC. 1946 COPYRIGHT, 1934, BY PRENTICE-HALL, INC. 70 FIFTH AVENUE, NEW YORK ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAT BB REPRODUCED IN ANY FORM, BY MIMEOGRAPH OR ANT OTHER MEANS, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHERS. First Printing June, 1934 Second Printing April, 1936 Third Printing September, 1938 Fourth Printing October, 1939 Fifth Printing April, 1944 Sixth Printing August, 1946 PRINTED IN THE UNITED STATUS OF AMERICA PREFACE THE purpose of this book is to present, in a single volume, the essentials of Trigonometry and Analytic Geometry that a student might need in preparing for a study of Calculus, since such preparation is the main objective in many of our freshman mathematics courses. However, despite the connection between Trigonometry and Analytic Geometry, the author believes in maintaining a certain distinction between these subjects, and has brought out that distinction in the arrangement of his material. Hence, the second part of the book supple mented by the earlier sections on coordinate systems, found in the first part would be suitable for a separate course in Analytic Geometry, for which a previous knowl edge of Trigonometry is assumed. The oblique triangle is handled by means of the law of sines, the law of cosines, and the tables of squares and square roots. However, the usual law of tangents and the r formulas are included in an additional chapter, Supple mentary Topics. There is included in the text abundant problem material on trigonometric identities for the student to solve. The normal form of the equation of a straight line is derived in as simple a manner as possible, and the per pendicular distance formula is similarly derived from it. The conies are defined in terms of focus, directrix, and eccentricity and their equations are derived accordingly. In the chapter Transformation of Coordinates are discussed the general equation of the second degree and the types of conies arising therefrom. An attempt has been made to present rigorously, but without too many details, the material necessary for distinguishing between the types vi PREFACE of conies by means of certain invariants, which enter naturally into the discussion. Although this chapter may be omitted from the course, it is well included if time permits. ATHERTON H. SPBAGUE Amherst College CONTENTS PLANE TRIGONOMETRY CHAPTBB PAGV I. LOGARITHMS 3 1. Exponents 3 2. Definition of a logarithm 7 3. Laws of logarithms 8 4. Common logarithms 10 5. Use of the logarithmic tables 12 6. Interpolation 13 7. Applications of the laws of logarithms, and a few tricks 15 II. THE TRIGONOMETRIC FUNCTIONS 21 8. Angles 21 9. Trigonometric functions of an angle 21 10. Functions of 30, 45, 60 23 11. Functions of 90 – 0 25 12. Tables of trigonometric functions ……. 26 III. SOLUTION OF THE RIGHT TRIANGLE 29 13. Right triangle 29 14. Angles of elevation and depression 30 IV. TRIGONOMETRIC FUNCTIONS OF ALL ANGLES …. 35 15. Positive and negative angles 35 16. Directed distances 35 17. Coordinates 36 18. Quadrants 37 19. Trigonometric functions of all angles 37 20. Functions of 0, 90, 180, 270, 360 40 21. Functions of as varies from to 360. … 42 22. Functions of 180 6 and 360 0 45 23. Functions of -0 48 Vll viii CONTENTS CHAPTBK PAQB V. THE OBLIQUE TRIANGLE 51 24. Law of sines 51 25. Applications of the law of sines 52 26. Ambiguous case 53 27. Law of cosines, and applications 57 VI. TRIGONOMETRIC RELATIONS 66 28. Fundamental identities 66 29. Functions of 90 0 71 30. Principal angle between two lines 73 31. Projection 73 32. Sine and cosine of the sum of two angles …. 74 33. Tan a ft 76 34. Functions of the difference of two angles …. 78 35. Functions of a double-angle 79 36. Functions of a half-angle 81 37. Product formulas 86 VII. SUPPLEMENTARY TOPICS 92 38. Law of tangents 92 39…