D E 42. Let C be the centre of a circle, and DA, EB, two diameters at right angles to each other-dividing the circumference into four quadrants. Then, AB is called the first quadrant; BD the second quadrant; DE the third quadrant; and EA the fourth quadrant. All angles having 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. First quadrant. If CA=1 PM: = sin a, CM = cos a, AT tan a, CT sec a, P M T that is, sec a, ver-sin a. PM = sin a, CM cos a, AT = tan a, = sec a, AM = ver-sin a. PM2 + CM2 Regarding the radius CP of the cir cle as unity, and denoting it by 1 (Art. 43); we have in the right-angled triangle CPM, 2 = R2 = 1, 2 sina + cos a = 1, * (1) AT CA tan a = = T 46. We will now proceed to established some of the important general relations between the circular functions. T PM P M (2) that is, 47. Since the triangles CPM and CTA are similar, we have, P P M A T A A T * The gymbols sin2 a, cos2 a, tan2 a, &c., signify the square of the sine, the square of the cosine, &c. 48. Substituting in equation (2), 90 sin (90 tan (90 that is, that is (Art. 12), cot a = a) sec a = CT CA 51. Substituting for a, cos a sin a (3) 49. Multiplying equations (2) and (3), member by member, we have, cosec a = = tan a X cot a = 1. : (4) 50. From the two similar triangles CPM and CTA, we have, sec (90 - a) 1 COS a 90 CP CM cosec2 a = 1 + cot2 a. - = 2 2 sec a = 1 + tan a. ver-sin a 1-cos a. 1 that is, sin a 52. In the right-angle CTA, we have, CT2 = CA2 +AT2; a for a, we have, a) a) a, we have, 1 cos (90-a) (7) that is, 53. Substituting (90-a) for a, in equation (7) and recol lecting that scc (90-a)=cosec a, and tang (90-a) = cot a, we have T (8) D (9) 54. We have, AM equal to the versed sine of the are AP; hence, M A 55. These nine formulas being often referred to, we shall place them in a table. They are used so frequently, that they should be committed to memory. -. sin a R2 = = 1 1 COS a 1 sin a 2 =1+tan a. = 1. 2 : 1+ cot a. cos a. 56. We will now explain the principles which determine the algebraic signs of the trigonometrical functions. There are but two. 1st. All lines estimated from DA, upwards, are consid ered positive, or have the sign + and all lines estimated from DA, in the opposite direction, that is, downwards, are considered negative, or have the sign 2d. All lines estimated from EB along CA, that is, to the right, are considered positive, or have the sign + and all lines estimated from EB along CD, that is, in the opposite direction, are considered negative, or have the sign 57. Let us determine, from the above principles, the algebraic signs of the sines and cosines in the different quadrants. 58. In the first quadrant. PM sin a, Pm CM= cos a, = and are both positive, the former being above the line DA, and the latter being estimated from C to the right (Art. 56). First quadrant. 59. In the second quadrant, PM = sin a, Second quadrant. Pm = CM = COS α: and PM = CM Pm 60. In the third quadrant, sin a, Third quadrant. = and M the sine is negative, falling below the DM cos a: Fourth quadrant. 61. In the fourth quadrant, sin a, PM = D and the sine is negative, falling below the line DA, and the cosine is positive, falling on the right of EB. Hence, we conclude, that P |