42. To represent the angle geometrically when the ratios are given 43. Connection between angles and trigonometric ratios 59. The aid of logarithms in the solution of triangles 60. The use of logarithms in Cases I., II. 61. Relation between the sum and the difference of any two sides of a triangle. The law of tangents. Use of logarithms in Case III. . 108 85. General expression for all angles which have the same sine 86. General expression for all angles which have the same cosine 87. General expression for all angles which have the same tangent 88. Inverse trigonometric functions 89. Sum and difference of two anti-tangents. Exercises on inverse 92. Functions of twice an angle. Functions of half an angle 93. Functions of three times an angle. Functions of an angle in terms 157 158 159 APPENDIX. NOTE A. Historical sketch NOTE B. Projection definitions of trigonometric ratios. PLANE TRIGONOMETRY. CHAPTER I. LOGARITHMS: REVIEW OF TREATMENT IN ARITHMETIC AND ALGEBRA. 1. There is a large amount of computation necessary in the solution of some of the practical problems in trigonometry. The labour of making extensive and complicated calculations can be greatly lessened by the employment of a table of logarithms, an instrument which was invented for this very purpose by John Napier (1550-1617), Baron of Merchiston in Scotland, and described by him in 1614. From Henry Briggs (1556–1631), who was professor at Gresham College, London, and later at Oxford, this invention received modifications which made it more convenient for ordinary practical purposes.* Every good treatise on algebra contains a chapter on logarithms. This brief introductory review is given merely for the purpose of bringing to mind the special properties of logarithms which make them readily adaptable to the saving of arithmetical work. A little preliminary practice in the use of logarithms will be of advantage to any one who intends to study trigonometry. A review of logarithms as treated in some standard algebra is strongly recommended. * The logarithms in general use are known as Common logarithms or as Briggs's logarithms, in order to distinguish them from another system, which is also a modified form of Napier's system. The logarithms of this other modified system are frequently employed in mathematics, and are known as Natural logarithms, Hyperbolic logarithms, and also, but erroneously, as Napierian logarithms. See historical sketch in article Logarithms (Ency. Brit. 9th edition), by J. W. L. Glaisher. |