the cosine of an arc and its supplement are equal with the contrary sign. 26. The cosine of 0° (being equal to the sine of the complement of 0° which is 90°) is R. The cosine of 90° is equal to the sine of 0° which is 0. The cosine of 180° being the distance from the foot of the sine to the centre, and being also on the left of the vertical diameter is R, as may be seen from the preceding diagram. The cosine of 270° being the distance from the foot of the sine to the centre, since the sine falls on the centre, is 0. The least value of the cosine is 0; the greatest value is R. When the sine has its least value, the cosine has its greatest; and vice versa. 27. Before noticing the cotangent and cosecant, let us consider the manner of treating negative arcs. mencing at the point Such arcs com positive arcs, i. e., downwards. Let us for simplicity M suppose the arc in question to be less than a quadrant; being laid off downward, such an arc will terminate in the fourth quadrant. Hence we see that the trigonometrical lines of a negative arc must be affected with the same signs as those of an arc in the fourth quadrant. Thus the sine of a negative arc will be the cosine +, the tangent ——, the secant +. Secondly, suppose the given negative arc to be greater than a quadrant; were it positive, some of its trigonometrical lines would be negative. The rule given above, which determines the sines of its trigonometrical lines by those of an arc in the 4th quadrant will apply with this modification, that when the trigonometrical line is + in the fourth quadrant, the corresponding trigonometrical line of the negative arc has the same sign as that of a positive arc of the same magnitude, and when the trigonometrical line is in the fourth quadrant, a contrary sign. The truth of this assertion may be seen, by trying negative arcs of various magnitudes upon the diagram, laying them off downwards from the right point of the circle, and observing in which quadrant their extremities fall. They will be found in every case to give results agreeable to the rule just stated. THE COTANGENT AND COSECANT. 28. The cotangent of 0° is (art. 23) and is therefore co. to the tangent of 0° and is 0. equal to the tangent of 90° equal to the tangent of 90° The cotangent of 90° is equal The cotangent of 180° is 180° the tangent of = 90° ∞ since - 90° is a negative arc, and terminates at the bottom of the circle, or the 270° point. The tangent of 270° = the tangent of 90° 270° the tangent of 180° = 0. = When the tangent has its least value which is 0, the cotangent has its greatest which is co, and vice versa. When the secant has its least value which is R, the cosecant has its greatest, which is oo, and vice versa. The cotangent and cosecant have their greatest values together and their least values together, viz., that of the one 0, of the other R, at the top and bottom of the circle, and both co at the right and left points. 30. With regard to the signs of the cotangent and cosecant in the different quadrants, they will be most conveniently discovered from the analytical expressions for these lines which we shall presently have. We add here, however, which so far as the cotangent and cosecant are concerned must be for a moment taken for granted, that the six trigonometrical lines may be arranged in three pairs, each pair having always the same algebraic sign. We have seen that the secant and cosine go together in this way; so do also the cosecant and sine; and so do the tangent and cotangent. The positive sines and cosecants are separated from the negative, by the horizontal diameter; the positive cosines and secants from the negative, by the vertical diameter; and the tangent and cotangent are together + and - alternately in the successive quadrants. 31. The following algebraic notation is employed for the six trigonometrical lines. Let a be the algebraic expression for the number of degrees in any arc, then the trigonometrical lines of the arc a will be expressed thus; sin a, tan a, sec a, cos a, cot a, cosec a. Cot a tan a = R is read, the cotangent of the arc a multiplied by the tangent of the same arc is equal to the square of the radius of the circle in which these trigonometrical lines are supposed to be drawn. Cot a and tan a are expressions for straight lines, and the equation above expresses that the rectangle formed by the tangent and cotangent of an arc is equivalent to the square formed upon the radius. The two members of the above equation contain the same number of dimensions, and are therefore homogeneous. This ought to be the case in all trigonometrical equations; because a line cannot be equal to a surface, nor either of these to a solid. Sometimes in analytical investigations R is supposed to be equal to 1; R2 and R3 would also be equal to 1. Whether this 1 is a unit of length, of surface, or of solidity, must be determined by what is required to preserve the homogeneity of the equation. 32. The tangent, secant, cotangent and cosecant may be expressed in terms of the sine and cosine. The values of the four former in terms of the two latter are derived geometrically as follows: whence multiplying the means and dividing by the first term, we obtain the last If R that is the tangent of any arc is equal to radius multiplied by the sine divided by the cosine of the same arc. be made equal to 1, then 34. In the triangles CMP and CED, which have their sides respectively parallel, and are therefore similar, we have the proportion* 36. In the expressions for the tangent and cotangent which we have here derived, it will be observed that we have the quotient of the sine and cosine, and that therefore when the sine and cosine have contrary signs, the tangent and cotangent will be negative. This occurs in the second and fourth quadrants. It appears hence, that the cotangent changes its sign always with the tangent. Also that both the tangent and cotangent of an arc are equal to those of its supplement with contrary signs. From the expressions for the secant and cosecant, it appears that the former must always have the same sign as the cosine, and the latter the same as the sine. The formulæ derived in the last four articles should be committed to memory. *The homologous sides are those which are parallel. (Geom. B. 4, Prop. 21, Schol.) |