By Antherton H. Sprague

Necessities OF aircraft TRIGONOMETRY AND ANALYTIC GEOMETRY by means of ATHERTON H. SPRAGUE PROFESSOR OP arithmetic AMHEBST university long island PRENTICE-HALL, INC. 1946 COPYRIGHT, 1934, via PRENTICE-HALL, INC. 70 5th road, long island ALL RIGHTS RESERVED. NO a part of THIS publication MAT BB REPRODUCED IN ANY shape, via MIMEOGRAPH OR ANT different ability, with out PERMISSION IN WRITING FROM THE PUBLISHERS. First Printing June, 1934 moment Printing April, 1936 3rd Printing September, 1938 Fourth Printing October, 1939 5th Printing April, 1944 6th Printing August, 1946 published within the UNITED prestige OF the USA PREFACE the aim of this booklet is to give, in one quantity, the necessities of Trigonometry and Analytic Geometry pupil may need in getting ready for a research of Calculus, on account that such education is the most goal in lots of of our freshman arithmetic classes. notwithstanding, regardless of the relationship among Trigonometry and Analytic Geometry, the writer believes in protecting a definite contrast among those topics, and has introduced out that contrast within the association of his fabric. consequently, the second one a part of the publication supple mented by means of the sooner sections on coordinate platforms, present in the 1st half will be compatible for a separate direction in Analytic Geometry, for which a prior knowl fringe of Trigonometry is believed. The indirect triangle is dealt with by way of the legislation of sines, the legislation of cosines, and the tables of squares and sq. roots. despite the fact that, the standard legislation of tangents and the r formulation are integrated in an extra bankruptcy, Supple mentary subject matters. there's integrated within the textual content plentiful challenge fabric on trigonometric identities for the coed to resolve. the conventional type of the equation of a directly line is derived in as basic a fashion as attainable, and the in step with pendicular distance formulation is in a similar way derived from it. The conies are outlined by way of concentration, directrix, and eccentricity and their equations are derived for that reason. within the bankruptcy Transformation of Coordinates are mentioned the overall equation of the second one measure and the categories of conies bobbing up therefrom. An try has been made to provide carefully, yet with no too many information, the fabric valuable for distinguishing among the kinds vi PREFACE of conies through convinced invariants, which input evidently into the dialogue. even if this bankruptcy can be passed over from the direction, it really is good integrated if time allows. ATHERTON H. SPBAGUE Amherst collage CONTENTS aircraft TRIGONOMETRY CHAPTBB PAGV I. LOGARITHMS three 1. Exponents three 2. Definition of a logarithm 7 three. legislation of logarithms eight four. universal logarithms 10 five. Use of the logarithmic tables 12 6. Interpolation thirteen 7. purposes of the legislation of logarithms, and some tips 15 II. THE TRIGONOMETRIC features 21 eight. Angles 21 nine. Trigonometric services of an perspective 21 10. capabilities of 30, forty five, 60 23 eleven. services of ninety - zero 25 12. Tables of trigonometric capabilities ....... 26 III. resolution OF definitely the right TRIANGLE 29 thirteen. correct triangle 29 14. Angles of elevation and melancholy 30 IV. TRIGONOMETRIC services OF ALL ANGLES .... 35 15. optimistic and unfavorable angles 35 sixteen. Directed distances 35 17. Coordinates 36 18. Quadrants 37 19. Trigonometric capabilities of all angles 37 20. services of zero, ninety, one hundred eighty, 270, 360 forty 21. capabilities of as varies from to 360. ... forty two 22. features of one hundred eighty 6 and 360 zero forty five 23. capabilities of -0 forty eight Vll viii CONTENTS CHAPTBK PAQB V. THE indirect TRIANGLE fifty one 24. legislations of sines fifty one 25. functions of the legislations of sines fifty two 26. Ambiguous case fifty three 27. legislations of cosines, and purposes fifty seven VI. TRIGONOMETRIC relatives sixty six 28. primary identities sixty six 29. capabilities of ninety zero seventy one 30. relevant perspective among strains seventy three 31. Projection seventy three 32. Sine and cosine of the sum of 2 angles .... seventy four 33. Tan a feet seventy six 34. capabilities of the variation of 2 angles .... seventy eight 35. features of a double-angle seventy nine 36. capabilities of a half-angle eighty one 37. Product formulation 86 VII. SUPPLEMENTARY subject matters ninety two 38. legislations of tangents ninety two 39...

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23. Functions of ( 0) . What relations hold between the functions of a negative angle and the functions of the corresponding positive angle? Consider Figure 28, FUNCTIONS OF ALL ANGLES r x y - r' = x = -y' *---V I/ sin (-0) 0) ( = -x = x r r cot $ = +cos = esc = sec ( 0) = -cot (-0) 0) ( sec - _ sin r t/ v = - = -^v = -- = -tan tan (~0) csc H r r' cos 49 Problems Example Find: sin - 1 300. sin - = -sin 300 - -sin (360 - 60) = -(-sin 60) 300 sin 60 V3 as "^' Example 2 Find all - 2 sin 2 and 360 angles between 1 = 0.

Sin csc In a comparison of the observed that 8). it is 30 cos 30 and so on; that = i 30 = = = cos 60 sin 60 sec 60 the functions of 30 are the corresponding co-functions of 60. This conclusion suggests a relation is, between the functions of sponding co-functions of any acute angle its We 6 and the complement. have, in Figure sin = - = 7, cos (90 - 0) c cos 6 = c 6 Figure 7. and so on. = sin (90 - 6) corre- PLANE TRIGONOMETRY 26 Hence we have: Theorem. Any function of an acute angle 6 equals the corresponding co-function of (90 0) equals the (90 12.

Consider Figure 28, FUNCTIONS OF ALL ANGLES r x y - r' = x = -y' *---V I/ sin (-0) 0) ( = -x = x r r cot $ = +cos = esc = sec ( 0) = -cot (-0) 0) ( sec - _ sin r t/ v = - = -^v = -- = -tan tan (~0) csc H r r' cos 49 Problems Example Find: sin - 1 300. sin - = -sin 300 - -sin (360 - 60) = -(-sin 60) 300 sin 60 V3 as "^' Example 2 Find all - 2 sin 2 and 360 angles between 1 = 0. 3 sin - Hence: sin = = or: sin 0=1. If sin 2 sin 2 3 sin + 1 then sin If then Hence: 1. In Problems (a) to Find: the tables.