From the nature of the circle it is apparent that the tangent AT increases more slowly about the middle of the quadrant AH than in any other part of it, and that it increases fast as the point B approaches near H, having no limit to its increase, like the sine, but admitting all possible degrees of magnitude from 0 to infinity. 100. The secant becoming negative as often as the revolving radius CB passes the centre C, changes its sign like the cosine, and is affirmative in the first and fourth quadrants, and negative in the second and third quadrants. In the first quadrant AH the secant increases from radius to infinity. In the second quadrant HD it is negative; for the secant has its origin at the centre C, and its length is computed from the centre to its concourse with the tangent. In the first quadrant the length of the secant is computed from C toward B and T; but in the second quadrant it is reckoned, on the revolving radius, from C toward E and t, in a contrary direction, and therefore is negative. During the second quadrant this negative secant decreases from infinity till it becomes equal to radius. In the third quadrant DL it increases from radius to infinity, but continues negative; for the intersection which this revolving radius (produced) makes with the tangent continues on the same side (on that revolving line), with respect to the centre C, both for the second and third quadrants. In the fourth quadrant LA this intersection changes to the opposite part of the revolving radius (produced), namely, the same as at first; therefore in the fourth quadrant the secant is affirmative, and decreases from infinity till it becomes equal to radius, as it was at the beginning of the first quadrant. Thus the secant has the same algebraic signs as the cosine, in the same quadrants. 101. We may observe here, that the sine and cosine never exceed radius, that the secant and cosecant are never less than radius, and that the tangent admits all degrees of magnitude. Moreover, all these lines change their directions as often as they become either infinite or nothing. When they become infinite, their increase is at its utmost limit; after which they change their direction, and decrease. When they become O, their decrease is at its utmost limit, and then they increase again in a contrary direction. Thus, these quantities change their algebraic signs, whenever they pass through a state of infinity, or a state of nothingness. 30. As the cotangent HK is computed from the point H, in the same manner as the tangent AT is computed from A, the former will evidently vary in its direction and length like the latter, being affirmative in the first and third quadrants, AH, DL, and negative in the second and fourth quadrants, HD, LA. In the first quadrant AH the cotangent HK decreases from infinity to O, and in the second quadrant HD it increases negatively from 0 to infinity. It becomes affirmative in the third quadrant DL, and decreases from infinity to 0. In the fourth quadrant LA it increases negatively from 0 to infinity, as in the second quadrant. 102. The cosecant CK changes its sign in the same manner as the sine, being affirmative in the first and second quadrants AH, HD, and negative in the third and fourth quadrants DL, LA. In the first quadrant AH it decreases from infinity to radius, and in the second quadrant HD it increases from radius to infinity. In the third quadrant DL it decreases negatively from infinity to radius, and in the fourth quadrant LA it again increases negatively from radius to infinity. 103. The versed sine AF increases from O during the first and second quadrants AH, HD, till it becomes the diameter AD, which is its utmost limit. It then decreases during the third and fourth quadrants DL, LA, till it becomes O. Being always computed in the same direction, from A toward D, it is always affirmative. 104. The changes of the algebraic signs of the several trigonometrical lines are exhibited in the following table. 105. The values of these lines at the end of each quadrant of the circle may be exhibited in a table, as follows. 106. In many analytical investigations it is common to employ, indifferently, arcs of all magnitudes, whether positive or negative, greater or less than 360°; in which cases their sines, cosines, &c., may be derived from fig. 1 in the same manner nearly as those of the simple arcs. Thus, if to any arc AB there be added one or more circumferences of the circle, it is manifest that they will terminate again exactly in the point B, and that the arc AB, so augmented, will have the same positive or negative sine, cosine, &c. as the single arc AB. Whence, if C denote the whole circumference of a circle, or 360°, and x any arc AB; then sine x = sine (C+x) sine (2 C+x)=sine (3 C+x) = &c. The case is the same in respect of the cosine, tangent, &c. = 107. Also, if two arcs AB, AE, be taken in opposite directions on the circumference of the circle AHDL, one arc being considered positive, the other negative; their sines, in this case, will be equal, but will be affected with contrary algebraic signs; and the cosines of both arcs will be the same, namely, CF. Hence, if any negative arc AE be denoted by -a, then Sine a) == sine a Tan. If 360° a) a) === - tan. a = sec. a Cos. (a)=+cos. a == cot. a Cosec. (-a)= a be substituted for · cosec. a. - a, in any expression of this kind, it will be necessary only to consider those arcs which are positive. 108. By the inspection of fig. 1 it appears that the sine, tangent, and secant of any arc AB, are of the same magnitudes as the cosine, cotangent, and cosecant of its complement BH, and vice versa; therefore the values of the former of these lines may be expressed in terms of the latter, as follows. Sine a=cos. (90° — a), cos. a= =sine (90° — a), &c. Moreover, the sine, cosine, &c. of any arc are of the same magnitudes as the sine, cosine, &c. of its supplement (34, &c.); therefore these lines may be expressed, in a similar manner, with their proper signs, as follows. Sine a=sine (180° − a), cos. a=—cos. a), &c. (180° 109. By substituting 90°-a for a in these last forms, it will appear that the sine, cosine, &c. of any arc or angle below 90°, is equal to the sine, cosine, &c. of an arc or angle as much above 90° as the former is under 90°. Thus, sine (90° − a)= sine (90°+a), cos. (90°- a) = cos. (90°+a), &c. In each of these forms we must attend to the change of the signs, when the arc or angle a is greater than 90°. Since the cos. a=sine (90°—a), it is plain that, if we know how to value the sine in all possible cases, we shall be able to value the cosine, and thence all the rest of the trigonometrical lines. 110. The changes of the signs of the trigonometrical lines may be shown in a neat and concise manner, as follows. Vince's Trigonometry. Suppose the arc AB to begin at A, and in the quadrant AH assume the sine, cosine, tangent, secant, cotangent, and cosecant, all positive. If a line, supposed to be positive in one direction, vanish, and then be set off in a contrary direction, it becomes negative. Hence CF, the cosine of the arc AB, is positive; but Cf, the cosine of the arc Ab or As, is negative. In the arc AHD the sine is set off in the same direction, and therefore is positive; but at D it vanishes, and in the arc DLA is set off in an opposite direction; therefore it now becomes negative. The tangent radius X sine cosine (41). Now the radius is always positive, and from A to H the sine, cosine, tangent, &c. are positive by supposition. At H the cosine is O, and the tangent and secant vanish, for they become parallel, and therefore never meet. Hence a quadrant, or an arc of 90 degrees, has neither a tangent nor a secant (according to the definition). From H to D the sine is positive, and the cosine is negative; therefore the tangent is negative. From D to L the sine and cosine are both negative, therefore the tangent is positive. From L to A the sine is negative, and the cosine is positive; therefore the tangent is negative. The secant radius cosine; therefore the secant has always the same sign as the cosine. The cotangent: = radius tangent; therefore the cotangent has always the same sign as the tangent. = The cosecant radius2sine; therefore the cosecant has always the same sign as the sine. The versed sine is always set off from A in the same direction, and therefore continues positive through the whole circle. Hence the changes of the algebraic signs of the several trigonometrical lines may be conveniently exhibited in a table, as in art. 104. 111. If the arc AB be considered as positive, the arc AE, set off from A in a contrary direction, will be negative. The arcs AB, AE are equal, and have the same cosine CF, and the same versed sine AF; but the sine EF being set off in a direction contrary to that of the sine BF, becomes negative; that is, the sine of a negative arc AE has a sign contrary to that of the sine of a positive arc AB of the same value. The tangent radius X sine cosine, and the cotangent= radius x cosine sine, and the cosecant radius2 sine. Hence it appears from what has been said respecting the sine of a negative arc, that the tangent, cotangent, and cosecant will have the sign for a negative arc different from the sign which they have for a positive arc. The secant radius cosine; therefore the secant has the same sign both for a positive arc and a negative arc; for it has been shown that the sign of the cosine is the same for both arcs. Hence, if A, B denote two arcs, of which A is the less, the signs of the sine, tangent, cotangent, and cosecant of A-B will be contrary to those exhibited in the preceding table. 112. Any arc or angle and its supplement have the same sine (34); therefore, in trigonometrical computations, when the quantity required is found in terms of the sine, the case is ambiguous, unless the ambiguity be removed by some other consideration. But if the quantity sought be expressed by a cosine, tangent, or cotangent, there is no ambiguity, because a positive cosine, tangent, or cotangent denotes an arc or angle less than 90°; but a negative cosine, &c. denotes an arc or angle between 90° and 180° (for in trigonometry every arc or angle must be less than 180°). In all trigonometrical calculations it is necessary to attend to the signs of the quantities. II. Trigonometrical Theorems.* PROP. 113. If there be three such arcs (plate, fig. 3), AB, AC, AD, that BC, the difference between the first and second arcs, is equal to CD, the difference between the second and third; then the radius is to the cosine of the common difference, BC, as the sine of the middle arc, AC, is to half the sum of the sines of the extreme arcs, AB, AD. Draw CE to the centre E. Draw BF, CG, DH perp. to the radius AE, and they will be the sines of the arcs AB, AC, * This subject is treated with perspicuity and elegance by Legendre, Elements de Geometrie, p. 338. I |