PLANE TRIGONOMETRY. 





111. NUHBER OF THE VERB AFTER A COLLECTIVE 

 NOUN. 



(1.) Every verb having, as its subject, a general collective 

 noun [ 3, (0. )], preceded by the definite article, such aa U 

 totalite, 1'infinite, etc., takes the number of that noun : 



L'armee de iufidi-lo fat en- 



,.t il.'-truite. 



L'ACADiMIB. 



lA multitude de bouuea 

 chose* quo Ton trouve dam un 

 onvrage, flit prdro de vue la 

 des mauvuises. 



CAMINADE. 



The army of the infidel t vat en 

 tirtly destroyed. 



The multitude of the good thing* 

 which we find in a work, make* us 

 lost tight of the multiplicity of the 

 lad one*. 



When a partitive collective noun [ 3, (6.)] occurs at 



il>joot of a clause, the verb agrees with that noun, ii 



it occupies the first rank in the thought of the speaker or 



writer. 



The verb agrees, on the contrary, with the plural noun fol- 

 lowing the collective, if the latter acts only a secondary part, 

 or if it is employed only to add an accessory idea of number : 



./inf irit/i the Collective. 



Une troupe d'assassius entra 

 i fliumbre de Coligny. 



VOLTAIRE. 

 A gang of assassins entered Co- 



li'jny't chamber. 



Une nuee de traits obscurcit 

 1'air. 



A cloud of arrows darkened the 

 air. 



Cette espece de paons parait 

 avoir eprouvt! les m.'-ines effets par 

 la nit'-me cause. BTJFFON. 



That species of peacocks seems to 

 have experienced the same effects 

 through the same cause. 



Agreement uith tlic foUo\ring Noun. 



Une troupe de nymphes cou- 

 ronudes de fleurs, nageaient 

 antour de son char. FKNKLON. 



A troupe of young nymphs, 

 crowned with flowers, were swim mi 

 around her chariot. 



Une nuee do barbares .v 



lerent le pays. L'ACADiMlE. 



A cloud of barbarians desolated 

 the country. 



Cette espece de chiens qu'ou 



appelle cliieus de Laconic, ne 

 Vivent que dir ans. BOILEAU. 



That species of dogs which they 

 call Laconmn dogs, live only ten 

 years. 



112. DUMBER OF THE VERB &TRE AFTER THE 

 PRONOUN CE. 



(1.) The verb dtre preceded or followed by ce, as the gram- 

 matical subject, takes the number of the noun placed after the 

 verb ( 105, (3.)] : 



Ce SOnt les mceurs qui font la 

 bonue compagtiie. 



LA CHAUSSKB. 



Sont-ce des religieux et des 

 pn'tresqui parlent ainsi ? sont-ce 

 des chrdtiens ? PASCAL. 



If is morals which form good 

 company. 



Are they monks and priests who 

 speak so f are they Christians ? 



(2.) The verb dtre, with ce as subject, is also put in the 

 plural when it precedes the pronouns eux and elles : 



Ce SOnt eux qui vionnent. j It is they who come. 



Before nous and vous similarly placed, the verb is always in 

 the singular : c'est nous ; c'est vons. 



(3.) When the verb 6tre having ce for subject is need inter- 

 rogatively and followed by a personal pronoun, it remains in 

 the singular even before pronouns of the 3rd person plural : 

 Est-ce lui ? Is it he ? Est-ce eux ? Est-ce elles ? Is it they ? 

 Est-ce nous ? Is it we ? Est-ce vous ? Is it you ? 



113. THE VERB BELATING TO SEVERAL SUBJECTS OF 

 DIFFERENT PERSONS. 



A verb having several subjects in different persons, is 

 put in the plural, and assumes the termination of the first 

 person in preference to that of the second, and that of the 

 second in preference to that of the third. It may then be 

 preceded by the plural pronoun of the person preferred, which 

 sums up in one word all the other subjects and governs the 

 verb : 



Vote* jW-re et mol, nooi a von* 

 4t4 looftemp* otmemiH 1'nn do 

 1'autre. FM*U>M. 



Allez ; VOUS et roc Mmblablea 

 n'etes point fait* pour etre trans- 

 plant^. MOKTEftO.fi KU. 



Your ftthtr and 1 UM Iff* far 

 iumU* to each otker. 



Go ; you an-l tueh at you aw not 

 fit to be 



PLANE TRIGONOMETRY. II. 



COMPLBMKNTAL ANGLES SUPPLEMENTAL ANGLES -TRIOOXOMK- 

 TKICAL CONCEPTION OK AN ANGLE NEGATIVE ANOLEK. 



IV. Complemental Angle*. It was explained in Section IL 

 that the complement of an angle (i.e., of an acute angle) U the 

 difference between it and a right angle, or, in other words, it* 

 defect from a right angle ; and it was elated that the function 

 of an angle is the co-function of its complement that u, 

 sin. A = cos. (90 A), 

 cos. A = sin. (90 - A) ; and BO on. 

 Or, in circular measure, 



sin. A = cos. ( "~ ^ ) 



..(26) 



This is perhaps apparent enough by inspection of Fig. 2, but 

 Fig. 3 shows it more clearly. The complement of A in that is 



B, and it is plain that, just as cos. A is -, so cos. B is , since 



c c 



a is the adjacent side to B. But = sin. A. 



. . sin. A = cos. B 

 cos. A = sin. B 



And so on for other functions. 

 The above may, however, be thus proved geometrically : 



(27) 



In Fig. 4, let c A B = A ; then B A F - A. 



a 



Make G A F 



CAB (whence c A o = B A F). Note that A B = A o = radius. 

 A G H and BAD are easily shown to be similar triangles, whence, 



BD : AB: : AH : AG 



and 

 AD:AB: I G H : A <; 



sin. c A B = cos. c A G } 



and 



cos. c A B = sin. c A o } 

 But c A B = A, 



andc A G = -- A ; 

 2 



whence 







t 



Fig. 4. 



2 



V. Numerical Values of cer- 

 tain Trigonometrical Ratios. 

 It was stated in the last lessor. 

 (Section II.) that the ratios of 

 certain angles could be worked 



out geometrically. These angles are 45, 60 (and therefore 30, 

 its complement), 18 (and therefore 72). We select 45, 60, 

 and 30 as specimens, and work to five places of decimals : 



By (7), sin* 45 + cos. ! 45 = 1. 



But since complement of 45 = 45 (for 90 - 45 = 45), 

 sin. 45 = cos. 45, and sin. 2 45 = cos. 2 45, 



.2 sin. 1 45" = 1, and 2 cos. 1 45 = 1- 



.-. sin. 5 45 =, and sin. 45 = 



0-70710. 



Similarly, cos. 45<- = 0'70710. 



By (14), see. 45 = 



And by (10), cotan. 45 = 1. 

 1 1 



cos. 45 0-70710 

 Whence, also, cosec. 45 = 1-41421. 



= 1-41421. 



