43. The Tangent of an Arc. The tangent of an arc is the perpendicular to the primary diameter, produced from the primary origin, till it meets the prolongation of the diameter through the terminus of the arc. OR is the tangent of the arcs OT and OT". OR' is the tangent of the arcs OT' P and OT". T = C T O' S T The arcs OT and OT" are in the first and third quadrants, respectively, and their tangent, OR, is estimated upward, and is therefore positive; hence, The tangent of an arc in the first or third quadrant is positive. The arcs OT' and OT"" are in the second and fourth quadrants, respectively, and their tangent, OR', is estimated downward, and is therefore negative; hence, The tangent of an arc in the second or fourth quadrant is negative. Let the arc OT be equal to the arc T'P. Then, since T'P is the supplement of OT', OT will be the supplement of OT'; but the arc T"O is the supplement of OT'; hence, OTT""O, and the angle OCT is equal to the angle OCT". The angle COR is equal to the angle COR', since each is a right angle. Hence, the two triangles COR and COR' have two angles, and the included side of the one equal to two angles and the included side of the other, each to each, and are therefore equal in all their parts. Hence, OR, opposite the angle OCR, is equal to OR', opposite the equal angle OCR'. Since OR is estimated upward, and OR' downward, they have contrary signs; hence, OR -OR'. But OR is the tangent P' R of the arc OT, and OR' is the tangent of the arc OT', the supplement of OT; hence, The tangent of an arc is equal to minus the tangent of its supplement. The tangent of 0° is 0. As the arc increases from 0° to 90°, the tangent increases from 0 to +∞. As the arc increases from 90° to 180°, the tangent passes through ∞, changes its sign from to, and decreases numerically, but increases algebraically from -0. ∞ to As the arc increases from 180° to 270°, the tangent passes through 0, changes its sign from to, and increases from 0 to +∞. As the arc increases from 270° to 360°, the tangent passes through ∞, changes its sign from to, and decreases nu+ merically, but increases algebraically from - ∞ to -0. Hence, for the limiting values of the tangent we have tan 0° = 0, tan 90° tan 180° – 0, tan 270° +∞09 tan 360° - 0. 44. The Co-tangent of an Arc. The co-tangent of an arc is the perpendicular to the secondary diameter, produced from the secondary origin, till it meets the prolongation of the diameter through the terminus of the arc. O'S is the co-tangent of OT and OT". The arcs OT and OT" are in the first and third quadrants, respectively, and their co-tangent, O'S, is estimated to the right, and is therefore positive; hence, The co-tangent of an arc in the first or third quadrant is positive. The arcs OT' and OT"" are in the second and fourth quadrants, respectively, and their co-tangent, O'S', is estimated to the left, and is therefore negative; hence, The co-tangent of an arc in the second or fourth quadrant is negative. The word co-tangent is an abbreviation of complementi tangens, the tangent of the complement. In fact, O'S, the co-tangent of OT, is the tangent of O'T, the complement of OT; hence, The co-tangent of an arc is the tangent of its complement. OR, the tangent of OT, is the co-tangent of O'T, the complement of OT; hence, The tangent of an arc is the co-tangent of its complement. Let the arcs OT and T'P be equal. Then, since T'P is the supplement of OT', OT will be the supplement of OT'. The arcs O'T and O'T' are equal, since they are complements of the equal arcs OT and T'P; hence, the angles O'CT and O'CT', measured by these equal arcs, are equal. The angles CO'S and CO'S' are equal, since each is a right angle. Hence, the two triangles CO'S and CO'S' have the common side CO', and the two adjacent angles equal, and are therefore equal in all their parts; and O'S, opposite the angle O'CS, is equal to O'S', opposite the equal angle O'CS'. Since O'S is estimated to the right, and O'S' to the left, they have contrary signs; hence, O'S O'S'. But O'S is the co-tangent of OT, and O'S' is the cotangent of OT'', the supplement of OT; hence, The co-tangent of an arc is equal to minus the co-tangent of its supplement. The co-tangent of 0° is +∞. As the arc increases from 0° to 90°, the co-tangent decreases from to +0. As the arc increases from 90° to 180', the cotangent passes through 0, changes its sign from + to -, and increases numerically, but decreases algebraically from -0 to ∞. As the arc increases from 180° to 270°, the co-tangent passes through ∞, changes its sign from to +, and decreases from +∞ to +0. As the arc increases from 270° to 360°, the co-tangent passes through 0, changes its sign from to —, and increases numerically, but decreases algebraically from -0 to = ∞. Hence, the limiting values of the co-tangent are cot 0° cot 90° = +0, cot 180° = -cot 270° 0, :+∞, 00, cot 360° 45. The Secant of an Arc. The secant of an arc is the line drawn from the center of the circle to the terminus of the S' T T R tangent. CR is the secant of OT and OT". CR' is the secant of OT' and OT". P The arcs OT and OT"" are in the first and fourth quadrants, respectively, and their secants, CR and CR' are estimated from the center toward the termini of the arcs, and are therefore positive; hence, The secant of an arc in the first or fourth quadrant is positive. The arcs OT' and OT" are in the second and third quadrants, respectively, and their secants, CR' and CR, are estimated from the center, from the termini of the arcs, and are therefore negative; hence, The secant of an arc in the second or third quadrant is negative. Let the arcs OT and T'P be equal. Then, since T'P is the supplement of OT', OT is the supplement of OT"; but T"O is the supplément of OT'; therefore, TO is equal to OT, and the angle T"CO, measured by T""O, is equal to the angle OCT, measured by the equal arc OT. The right angles COR and COR' are equal. Hence, in the triangles having the common side CO, and the two adjacent angles equal, CR is equal to CR'; but CR, the secant of OT, is positive; and CR, the secant of OT", the supplement of OT, is negative; hence, CRCR; hence, The secant of an arc is equal to minus the secant of its supplement. The secant of 0° is +- 1. As the arc increases from 0° to 90°, the secant increases from +1 to +co. As the arc increases from 90° to 180°, the secant passes through, changes its sign from to, and decreases numerically, but increases algebraically from to 1. As the arc increases from 180° to 270°, the secant increases numerically, but decreases algebraically from 1 to As the arc increases from 270° to 360°, the secant passes through, changes its sign from to +, and decreases from +∞ to +1. Hence, for the limiting values of the secant we have sec 0° +1, sec 90° = ∞o, sec 180° —— sec 270' —— ∞, sec 360° -1, + 1. 46. The Co-secant of an Arc. The co-secant of an arc is the line drawn from the center of the circle to the terminus R of the co-tangent. CS is the co-secant of OT and OT". CS' is the co-secant of OT' and OT"". P The arcs OT and OT' are in the first and second quadrants, respectively, and their co-secants CS and CS' are estimated from the center toward the termini of T T O' S C the arcs, and are therefore positive; hence, The co-secant of an arc in the first or second quadrant is positive. |