FUNCTIONS OF NEGATIVE ARCS. 62. Let AM", estimated from A toward D, be numerically equal to AM; then, if we denote the arc AM by a, the arc AM" will be denoted by a (Art. 48). B A being the middle point of the arc M"AM, the radius OA bisects the chord MM at right angles (B. III., P. VI.); therefore, PM" is numerically equal to PM, but PM"" being measured downward from the initial diameter is negative, while PM being measured upward is positive, and, therefore, PM"" PM; OP is equal to the cosine of both AM" and AM (Art. 61); hence, we have, = sin a, cos (-a) = cos a. . (2.) Dividing (1) by (2), member by member, and then dividing (2) by (1), member by member, we have (formulas 6 and 7, Art. 61), tan (-a) = tan (a); cot (-a) = cot a. Taking the reciprocals of the members of (2), and then the reciprocals of the members of (1), we have (formulas 11 and 12, Art. 61), sec (-a) = sec a; cosec (-a) = cosec a. FUNCTIONS OF ARCS FORMED BY ADDING AN ARC TO, OR SUBTRACTING IT FROM, ANY NUMBER OF QUADRANTS. 63. Let a denote any arc less than 90°. By definition, Draw lines, as in the figure. Then PM sin a; OP = cos a ; ON = P'M' = sin (90° + a); NM' = cos (90° + a). The right-angled triangles ONM' and OPM have the angles NOM' and POM equal (B. III., P. XV.), the angles ONM' and OPM equal, both being right angles, and therefore (B. I., P. XXV., C. 2), the angles OM'N and OMP equal; they have, also, the sides OM' and OM equal, and are, consequently (B. I., P. VI.), equal in all respects hence, ON OP, and NM' PM. = These are nu merical relations; by the rules for signs, Art. 58, ON and OP are both positive, NM' is negative, and PM positive; and hence, algebraically, ON = OP, and NM' = therefore, we have, sin (90° + a) = cos a; PM; (1.) (2.) Dividing (1) by (2), member by member, we have, In like manner, dividing (2) by (1), member by member, we have, Taking the reciprocals of both members of (2), we have (formulas 11 and 12, Art. 61), In like manner, taking the reciprocals of both members of (1), we have, From these equations (1) and (2), and formulas (6), In like manner, the values of the several functions of the remaining arcs in question may be obtained in terms of functions of the arc a. Tabulating the results, we have the following 65 It will be observed that, when the arc is added to, or subtracted from, an even number of quadrants, the name of the function is the same in both columns; and when the arc is added to, or subtracted from, an odd number of quadrants, the names of the functions in the two columns are contrary: in all cases, the algebraic sign is determined by the rules already given (Art. 58). By means of this table, we may find the functions of any arc in terms of the functions of an Thus, arc less than 90°. sin 115° = sin (90° 25°) = cos 25°, sin 284° sin (270° + 14°) &c. tan 210° = tan (180° + 30°) PARTICULAR VALUES OF CERTAIN FUNCTIONS. 64. Let MAM' be any by arc, denoted 2a, MM its chord, and OA a radius drawn perpendicular to M'M: then will PM = M'M, and AMMAM (B. III., P. VI.). But PM is the sine of AM, or, PM that is, the sine of an sin a: hence, M arc is equal to one half the chord of twice the arc. Let M'AM = 60°; then will AM = 30°, and M'M will equal the radius, or 1 (B. V., P. IV.): hence, we have sin 30° that is, the sine of 30° is equal to half the radius. |
