39. The Sine of an Arc. The sine of an arc is the perpendicular distance of its terminus from the primary diameter. MT is the sine of the arc OT. M'T' is the sine of the arc OT'. M'T" is the sine of the arc OT". MT" is the sine of the arc OT". By the arcs OT"" and OT", we are to understand the positive arcs, and not the negative arcs designated by the same letters. The sine of an arc is the sine of the angle measured by that arc. Thus, MT, the sine of the arc OT, is the sine of the angle OCT, which is measured by the arc OT; and similarly for the other arcs and angles. The arcs OT and OT' are in the first and second quadrants, respectively, and their sines MT and M'T' are estimated upward, and are therefore positive; hence, P T Τ M' N N P M T T The sine of an arc in the first or second quadrant is positive. The arcs 0T" and OT"" are in the third and fourth quadrants, respectively, and their sines, M'T" and MT"", are estimated downward, and are therefore negative; hence, The sine of an arc in the third or fourth quadrant is negative. Let the chord TT' be parallel to the primary diameter OP, then will M' T' be equal to MT, and the arc OT will be equal to the arc TP; but the arc T'P is the supplement of the arc OT'; therefore, the arc OT is the supplement of the arc OT'; but M' T', the sine of the arc OT', is equal to MT, the sine of the arc OT, the supplement of OT'; hence, The sine of an arc is equal to the sine of its supplement. The sine of 0° is 0. As the arc increases from 0° to 90°, the sine. increases from 0 to +1. As the arc increases from 90° to 180°, the sine decreases from +1 to 0. As the arc increases from 180° to 270°, the sine passes through 0, changes its sign from to, and increases numerically, but decreases algebraically from 0 to - 1. As the arc increases from 270° to 360°, the sine decreases numerically, but increases algebraically from 1 to 0. Hence, for the limiting values of the sine, we have sin 0°- = 0, sin 90° 1, sin 180° +0, sin 270° -1, sin 360°--0. P T T 40. The Co-sine of an Arc. The co-sine of an arc is the perpendicular distance of its terminus from the secondary diameter. NT is the co-sine of the arc OT M' N N P M T = 11/ T first and fourth quadrants, respective ly, and their co-sines NT and N'T"" are estimated toward the right, and are therefore positive; hence, The co-sine 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 co-sines, NT' and N'T", are estimated toward the left, and are therefore negative; hence, The co-sine of an arc in the second or third quadrant is negative. The word co-sine is an abbreviation of complementi sinus, the sine of the complement. In fact, NT, the co-sine of OT, is the sine of O'T, the complement of OT; hence, The co-sine of an arc is the sine of its complement. MT, the sine of OT, is the co-sine of O'T, the complement of OT; hence, The sine of an arc is the co-sine of its complement. Since the radius CO' is perpendicular to the chord TT', NT and NT' are numerically equal; but since NT is estimated toward the right, and NT' toward the left, they have contrary signs; hence, NT—— NT'; but NT is the co-sine of OT, and NT" is the co-sine of OT', the supplement of QT; hence, The co-sine of an arc is equal to minus the co-sine of its supplement. It is evident that CN is equal to the sine of OT, or of OT', and that CN' is equal to the sine of OT", or of OT""; hence, The sine of an arc is equal to that part of the secondary diameter from the center to the foot of the co-sine. It is evident that CM is equal to the co-sine of OT, or of OT", and that CM' is equal to the co-sine of OT' or of OT"; hence, The co-sine of an arc is equal to that part of the primary diameter from the center to the foot of the sine. The co-sine of 0° is + 1. As the arc increases from 0° to 90°, the co-sine decreases from +1 to + 0. As the arc increases from 90° to 180°, the co-sine passes through 0, changes its sign from to, and increases ✈ numerically, but decreases algebraically from 0 to 1. As the arc increases from 180° to 270°, the cosine decreases numerically, but increases algebraically from 1 to -0. As the arc increases from 270° to 360°, the co-sine passes through 0, changes its sign from to, and increases from 0 to + 1. Hence, for the limiting values of the co-sine, we have cos 0° + 1, cos 90° = +0, cos 180° = - 1, cos 270° cos 360° +1. = – 0, 41. The Versed-Siné of an Arc. The versed-sine of an arc is the perpendicular distance of the primary origin from the sine. MO is the versed-sine of the arc OT, and of the arc OT". M'O is the versed-sine of the arc OT', and of the arc OT". The versed-sine of an arc, in any quadrant, is estimated to the right, and is therefore positive; hence, The versed-sine is always positive. The versed-sine of 0° is 0. As the arc increases from 0° to 90°, the versed-sine increases from 0 to 1. As the arc increases from 90° to 180°, the versed-sine increases from +1 to +2. As the arc increases from 180° to 270°, the versed-sine decreases from 2 to +1. As the arc increases from 270° to 360°, the versed-sine decreases from +1 to +0. Hence, the limiting values of the versed-sine are vers 0° = 0, vers 90° + 1, vers 180° + 2, vers 270° +1, vers 360° + 0. What are the least and greatest values of the sine, and what are the corresponding arcs? P T T M' N P M T T What are the least and greatest values of the co-sine, and what are the corresponding ares? What are the least and greatest values of the versedsine, and what are the corresponding arcs? 42. The Co-versed-sine of an Arc. The co-versed-sine of an arc is the perpendicular distance of the secondary origin from the co-sine. Thus, see diagram of the last article, NO' is the coversed-sine of the arc OT, and of the arc OT'; N'O is the co-versed-sine of the arc OT", and of the arc ОТ"". The co-versed-sine of an arc in any quadrant is estimated upward, and is therefore positive; hence, The co-versed-sine is always positive. The word co-versed-sine is an abbreviation of complementi versatus sinus, the versed or turned sine of the complement. In fact, NO, the co-versed-sine of OT, is the versed-sine of O'T, the complement of OT; hence, The co-versed-sine of an arc is the versed-sine of its complement. MO, the versed-sine of OT, is the co-versed-sine of OT, the complement of OT; hence, The versed-sine of an arc is the co-versed-sine of its complement. The co-versed-sine of 0° is 1. As the arc increases from 0° to 90°, the co-versed-sine decreases from 1 to 0. As the arc increases from 90° to 180°, the coversed-sine increases from 0 to +1. As the arc increases from 180° to 270°, the co-versed-sine increases from 1 to +2. As the arc increases from 270° to 360°, the co-versed-sine decreases from 2 to +1. Hence, the limiting values of the co-versed-sine are, covers 0° +1, covers 90° +0, covers 180°: + 1, covers 270°: = +2, covers 360° : + 1. What are the least and greatest values of the coversed-sine, and what are the corresponding arcs? = Trace the arcs from 0° to 360°, and the changing functions. |