Plane Trigonometry and Numerical Computation

1. The Uses of Trigonometry. The word "trigonometry" is derived from two Greek words meaning" the measurement of triangles." A triangle has six so-called elements (or parts) viz., its three sides and its three angles. We know from our study of geometry that, in general, if three elements of a triangle (not all angles) are given, the triangle is completely determined.^[1] Hence, if three such determining elements of a triangle are given, it should be possible to compute the remaining elements. The methods by which this can be done, i.e. methods for "solving a triangle," constitute one of the principal objects of the study of trigonometry.

If two of the angles of a triangle are given, the third angle can be found from the relation A + B + C = 180° (A, B, and C representing the angles of the triangle); also, in a right triangle, if two of the sides are known, the third side can be found from the relation a² + b² = c² (a, b being the legs and c the hypotenuse). But this is nearly the limit to which the methods of elementary geometry will allow us to go in the solution of a triangle.

Trigonometry^[2] is the foundation of the art of surveying

[1] What exceptions are there to this statement?

[2] Throughout this book we shall confine ourselves to the subject of "plane trigonometry," which deals with rectilinear triangles in a plane. "Spherical trigonometry" deals with similar problems regarding triangles on a sphere whose sides are arcs of great circles.