PLANE TRIGONOMETRY. 



77 



. 

 A = , sec. will have always the same sign aa cos., and 



sin. A 

 cosec. the same as sin. 



It ia clear from this section that if we know the signs of both 

 sine and cosine of an angle, wo know the quadrant to which it 

 belongs. 



VIII. Value of Functions of 0, 90, 180, and 270. Let 



angle A = c A B in Fig. 6. Then sin. A = B -?. Now if A = 



AB 



(i.e., represents no opening at all), A B must coincide with AC, 

 uid B c disappear altogether ; 







sin. = 



= 0. 



Since is the utter negation of all quantity, it is impossible 

 to attach a sign to it. This accounts for the absence of the 

 minus sign evidently required by the symmetry of the above 

 table against sin. and tan. 180, and COB. and cot. 270. From 

 this cause erroneous values (as regards sign) would be obtained 

 for cosec. 180 and sec. 270 if we trusted in their case to 

 formulae (14) and (15), lately adverted to. To find cosec. 180: 



By (20), cosec. 180-= 



__ 



_ v'sec.'ISO -! 



To find sec. 270. By (23), (10), and (24), 



Fig. 7. TABLE SHOWING THE VARIATION IN RATIO OF SINE, COSINE, TANGENT, ETC. 



Again, cos. A=- 



A B 



.'. cos. 0= 1. 

 Whence, by (11), ton. = sm ' 



But if A = 0, AC = AB. 



cos. 1 



And by (12), cot. = 9??!!. * = co (infinity). 



sin. 



Similarly, by (14) and (15), sec. 0= 1 ; cosec. = co. 

 Now let A = 90 ; then (referring to same figure), B c will 

 plainly coincide with and be equal to A B, and A c disappear. 



Then, sin. 90 --- = 1 ; 



A B 



cos.90 = A^ == _ = 



Whence, by the formulae above quoted 



AB 



tan. 90 = eo, 

 cot. 90 = 0, 



sec. 90 = 

 cosec. 90 = 1. 



When, at 180, A B (or A D) again coincides with A G, D G dia- 

 DG 



appears, and 



also COB. 180 = 



sin. 180= = 0; 



A O 

 A O 





But A G is negative ; 



A D 



.-. cos. 180= - 1. 



If A B (represented now by A K) revolve further to 270, E o 

 coincides with A E, and A G disappears. 



Then sin. 270 = - = - 1 (for E G is negative), 

 ooa. 270 = A a = JL = 0. 



AS AE 



This proves indirectly that sin. and tan. 180, and cos. and 

 cot. 270, have merely lost their minus sign through the acci- 

 dent of being represented, as to value, by 0. 



It will be observed in the above table that no ratio changes 

 its sign ezcept in passing through the values or e*. 



The above curious diagram (Fig. 7) shows at a glance the 

 fluctuations in the value of the several ratios in passing through 

 the four quadrants, and will be more easily borne in mind by 

 many than any written account. Its evident symmetry and 

 completeness also indicate the justice of employing the signs + 

 and in the arbitrary manner before explained. The propriety 

 of so using those signs in dealing with lines can, however, b 

 proved mathematically. Trigonometry, in its higher form, has 

 been defined as "the consideration of alternating or periodic 

 magnitude," and these words will be more easily grasped by the 

 pupil with this diagram before him. 



IX. Negative An.jks. An angle starting from below the base- 

 line A c in Fig. 6, by the movement of its free side in a direction 

 contrary to the arrows, is called a negative angle, and takes the 

 minus sign. Its four quadrants are, of course, reckoned the 

 reverse way ; whence it follows, since the first quadrant of a 

 negative is tho fourth of a positive angle, and the second of a 

 negative is the third of a positive angle, that for any given 

 quadrant of a negative angle the sine differs in sign from the 

 corresponding quadrant of a positive angle, but the cosine 

 is always the same. This is plain from inspection of Fig. 6. 

 Thus we say, generally 



i.e. sin. (A) = sin. A;") 

 but cos. (A) = cos. A ) 



(32) 



