Geometric The above indicated courses of variation of the trigonometric func tions may be, readily represented graphically, namely, by the so-called curves of sine, cosine, tangent, etc. These are found by the familiar method of analytical geometry (see GEosurray, CearrEstAN) for plotting the graph of a func tion of a real variable. A convenient unit of length is chosen to represent the radian. or unit angle. Angle values are then laid off on the X-axis and corresponding function values parallel to the axis of Y. A sufficient number of points being thus determined, a smooth curve, called the graph of the function, is drawn through them. The curve in Fig. 2 is part of the sine curve, or sinusoid. The undulations extend rightward and leftward (for negative angles) indefinitely. The curve of cosines is identical in form with the sinusoid and may be obtained in position from the latter by trans lating it as a rigid figure leftward through 2 units of length. For graphs of the remaining functions, the reader is referred to recent text books of trigonometry and analytical geometry. Ali the graphs in question are transcendental curves (see CURVES, Hicuait PLANE; CALCU LUS), being intersected by any straight line of the plane in an infinite set of points, real or imaginary.
Functions of Negative By ence to Fig. 1 and their definitions, it is im mediately seen that the functions of a and the corresponding functions of a are equal in value but excepting cos and sec, reversed in sign. Thus, symbolically, sin a = sin ( a), cos a = cos ( a). Accordingly, cos and sec are called even, while the others are called odd, functions of the angle, in obvious analogy with the behavior of signs in case of powers of positive and negative quantities.
Complements, Supplements, Explements. Two angles are called explements, supplements or complements, of each other, according as their sum is 27r, tr or 2 . Any pair of explements are representable by the symbols a a and a; any two supplements by + a and a; any two complements by 4 a and a, The query is natural: how are the values of a function of two explements, or two supplements, or two complements, related? By reference to Fig. 3, it is plain that sin ( -I- a) = sin (rr a) and that cos Or a) = cos or a ), since y = y', x x' and r = r'; hence: sines of explements are equal in value and opposite in sign; cosines of explements are equal. In like manner may be detected all the relationships in question, tabulated below: A Table of Critical Values. Certain critical values of the angle occur so fre, quently alike in theory and in practice that it seems desirable to tabulate the cor responding function values. The latter may be readily found from the function definitions by help of Fig. 1 and the three
accompanying tables. Other angle values of Note that in Fig. 4, r y= x= that in Fig. 5, OP = OP'= r = r', and that, from the laws of equality among triangles, x = y', x'= y. Note, too, that in any of the tables, the sine and cosine relations determine the others, as should be so, since tan and cot are the ratios, while cosec and sec are respec tively the reciprocals, of sin and cos. The mentioned determination is but a special mani festation of the Interdependence of the Functions. There is in trigonometry only a single fundamental or characteristic notion, as, say, that of sine a fact that accounts for the proverbial high de gree of plasticity or malleability of the subject matter. Any one of the six functions is de rivable from, and expressible in terms of, each of the others. Of such interdependence the definitions themselves of the functions afford illustration. At the same time they furnish the clue to its explicit determination. Thus, squaring and adding corresponding members of the defining equations of sin and cos, and noting that y'= r', one finds the fundamental rela tion: sin 'a cos'a = I, whence sin a = V 1 cost a, cos a =± V 1 from which, if either sin a or cos a be known, the other may be found. In any of the better recent textbooks of trigonometry may be found a readily constructible table expressing each func tion in terms of each of the remaining ones. The theory of the six trigonometric functions is ultimately the doctrine of a single one of them.
notably frequent occurrence are or and oe To find the value of sin observe, Fig. 1, that, if a= v is the half-chord of 60°, i.e., y = ; whence sin f. From this, by means of the relation, cos a = V 1 sinta, it is seen that cos 30° = VS; from this last, since cos + a ) sin = a) , sin 60° = V.T; and. 4 similarly, sin 30° =1. Thence, by definitions 1 and foregoing formulae, tan 30°.=. cot 60°= 3' tan 60° = cot 30° VT, sec 30° = coseco 60° = 2 3' 6 sec 60° = coseco 30° = 2. Analogously and by means of relationships to be subsequently given,function values corresponding to many other angles, as well as solutions of the inverse prob lem, admit of explicit determination. Such methods are however, in general, inconvenient, and, in problems of computation, the deter minations in question are, as a rule, effected by means of logarithmic tables. See ALEGERA, and below.
The Laws of Sine, Cosine and Tangent. Such may be named the three famous formula, now to be presented, that serve for the solution of any triangle, i.e., for the explicit deterrnina tiOn of the remaining parts of a triangle of which any three independent parts are known.