Elements of Geometry and Trigonometry from the Works of A. M. Legendre

Let A, B, and C be the three fixed points on shore, and P the position of the boat from which the angles APC=33° 45', CPB=22° 30', and APB=56° 15', have been measured.

Subtract twice APC- 67° 30' from 180°, and lay off at A and O two angles, CAO, ACO, each equal to half the remainder = 56° 15'. With the point 0, thus determined,

as a centre, and OA or OC as a radius, describe the cir cumference of a circle: then, any angle inscribed in the segment APC, will be equal to 33° 45'.

Subtract, in like manner, twice OPB=45°, from 180°, and lay off half the remainder = 67° 30', at B and C, determining the centre of a second circle, upon the cir cumference of which the point P will be found. The required point P will be at the intersection of these two circumferences. If the point P fall on the circumference described through the three points A, B, and C, the two auxiliary circles will coincide, and the problem will be indeterminate.

ANALYTICAL

PLANE

TRIGONOMETRY.

40. WE have seen (Art. 2) that Plane Trigonometry explains the methods of computing the unknown parts of a plane triangle, when a sufficient number of the six parts is given.

To aid us in these computations, certain lines were employed, called sines, cosines, tangents, cotangents, &c., and a certain connection and dependence were found to exist between each of these lines and the arc to which it be longed.

All these lines exist and may be computed for every conceivable arc, and each will experience a change of value where the arc passes from one stage of magnitude to another. Hence, they are called functions of the arc; a term which implies such a connection between two varying quantities, that the value of the one shall always change with that of the other.

In computing the parts of triangles, the terms, sine, cosine, tangent, &c., are, for the sake of brevity, applied to angles, but have in fact, reference to the arcs which measure the angles. The terms when applied to angles, without reference to the measuring arcs, designate mere ratios, as is shown in Art. 88.

41. In Plane Trigonometry, the numerical values of these functions were alone considered (Art. 13), and the arcs from which they were deduced were all less than 180 degrees. Analytical Plane Trigonometry, explains all the processes for computing the unknown parts of rectilineal triangles, and also, the nature and properties of the circular functions, together with the methods of deducing all the formulas which express relations between them.

Then,

42. Let C be the centre of a circle, and DA, EB, two diameters at right gles to each other-dividing the circumference into four quadrants. AB is called the first quadrant; BD the second quadrant; DE the third quadrant; and EA the fourth quadrant. All angles hav ing their vertices at C, and to which we attribute the plus sign, are reckoned from the line CA, and in the direc tion from right to left. The arcs which measure these angles are estimated from A in the direction to B, to D, to E, and to A; and so on.

43. The value of any one of the circular functions will undergo a change with the angle to which it belongs, and also, with the radius of the measuring arc. When all the functions which enter into the same formula are derived from the same circle, the radius of that circle may be regarded as unity, and represented by 1. The circular functions will then be expressed in terms of 1: that is, in terms of the radius. Formulas will be given for finding their values when the radius is changed from unity to any number denoted by R (Art. 87).

44. We have occasion to refer to but one circular func tion not already defined. It is called the versed sine.

The versed sine of an arc, is that part of the diameter intercepted between the point where the measuring arcs begin and the foot of the sine. It is designated, ver-sin.

45. The names which have been given of the circular functions (Art. 11) have no reference to the quadrants in which the measuring arcs may terminate; and hence, are equally applicable to all angles.