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examples at the ends of the chapters have been carefully graded, beginning with those which are easy, and extending to those which are more and more difficult. These examples illustrate every part of the subject, and are intended to test, not only the student's knowledge of the usual methods of computation, but his ability to grasp them in the many forms they may assume in practical applications. Among these examples are some of the most elegant theorems in Plane and Spherical Trigonometry.

The Chapters on De Moivre's Theorem, and Astronomy, Geodesy, and Polyedrons, will serve to introduce the student to some of the higher applications of Trigonometry, rarely found in American text-books.

In writing this book, the best English and French authors have been consulted. I am indebted especially to the works of Todhunter, Casey, Lock, Hobson, Clarke, Eustis, Snowball, M'Clelland and Preston, Smith, and Serret.

It remains for me to express my thanks to my colleagues, Prof. R. W. Prentiss for reading the MS., and Mr. I. S. Upson for reading the proof-sheets.

Any corrections or suggestions, either in the text or the examples, will be thankfully received.

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E. A. B.

RUTGERS COLLEGE,
New Brunswick, N. J., April, 1892.

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CHAPTER III.

TRIGONOMETRIC FUNCTIONS OF Two ANGLES.

42. Fundamental Formulæ.

43. To find the Values of sin ( x + y) and cos (x + y)

44. To find the Values of sin (x y) and cos (x - y).

45. Formulæ for transforming Sums into Products..

46. Useful Formulæ ...

47. Tangent of Sum and Difference of Two Angles.

48. Formulæ for the Sum of Three or More Angles

49. Functions of Double Angles.....

50. Functions of 3 x in Terms of the Functions of x..

51. Functions of Half an Angle ......

52. Double Values of Sine and Cosine of Half an Angle..

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