Plane Trigonometry and Numerical Computation

9. Introduction. At the beginning of the preceding chapter we described the fundamental problem of trigonometry to be the "solution of the triangle," i.e. the problem of computing the unknown elements of a triangle when three of the elements (not all angles) are given. This problem can be solved by finding relations between the sides and angles of a triangle by means of which it is possible to express the unknown elements in terms of the known elements. In order to establish such relations, it has been found desirable to define certain functions of an angle. One such function -- the tangent -- was introduced in § 3 by way of preliminary illustration.

In the present chapter, we shall give a new definition of the tangent of an angle and also define two other equally important functions -- the sine and the cosine. It should be noted that the definition given for the tangent in § 3 applies only to an acute angle of a right triangle. For the purposes of a systematic study of trigonometry we require a more general definition which will apply to any angle, positive or negative, and of any magnitude. Such definitions are given in the next article, in which the notion of a system of coördinates plays a fundamental role, the notion of a triangle not being introduced at all. After considering some of the consequences of our definitions in §§ 11-13, we consider the way in which these definitions enable us to express relations between the sides and angles of a right triangle. These results are then immediately applied to the solution of numerical problems by means of tables and to applications in surveying and navigation.