6 



THE POPULAE EDUCATOE. 



it all other functions can be readily obtained by means of the 

 equations in the next section. Tables have also been computed 

 of the logarithms of these numerical values, including every 

 function of all the angles just mentioned. By substituting the 

 logarithmic values for the natural or actual values of the ratios, 

 the processes of calculation are immensely facilitated, just as 

 lengthy calculations of natural numbers are often solved with 

 little trouble by the aid of their logarithms. In the next lesson 

 we will find the natural sines, etc., of two or three angles which 

 can be solved geometrically ; but, as stated above, the solution 

 in most cases rests upon other and more abstruse grounds. 



III. Relations of Trigonometrical Ratios to one another. 

 Since the square of the hypothenuse of a right-angled triangle 

 = the squares of the other two sides (Euclid I. 47), we have, by 

 Fig. 3 



TV -A- i. * a 2 /*' 2 c z /a\ 2 ,/ 

 Dividing by c; . + _=_; i.e., ( ) + ( 



c 2 c 2 c z \ c / \ 



or, sin. 2 A + cos. 2 A = 1 



Dividing the first equation by 6 2 , we get 



or, reversing the order, sec. 2 A = 1 -f tan. 2 A. ... (8) 

 Dividing the same by a 2 , we get 1 + ( ) = ( ) ' 

 or, reversing as before, cosec. 2 A = 1 + cot. 2 A. ... (9) 



Since-- . - = 1, tan. A . cot. A= 1. ... (10) 

 o a 



Again, tan. A = *- = .' tan. A = 



/i r * 



sin. A 

 cos. A 



... (11) 



Again, cot. A = - =_, ... co t. A = l .. (12) 



a a tan. A 



Again, cot. A = - = -L, .-. cot. A = - A . 

 a a sin. A 



Again, sec. A =_ = _, .-. sec. A = 



cos. A' 



Again, cosec. A = _=._, .-. cosec. A = 



sin. A 



(13) 

 (14) 

 (15) 



From these equations, (7) to (15), we can find the value of 

 any function in terms of any other function, as in the following 

 examples : 



It has already been shown, in (5) and (6), that 



vers. A = 1 cos. A. 

 covers. A = 1 sin. A. 



To shoiv sin. A in, terms of cos. A, and vice versa: 

 From (7) we get sin. 2 A= 1 cos. 2 A. 



. . sin. A = V 1 cos. 2 A. 



And similarly, cos. A = VI sin. 3 A. 

 Cot. in terms of sin. By (13) and (17), 



cos. A V 1 - sin. 2 A 



cot. A = 



sin. A 



sin. A 



(16) 

 (17) 



(18) 



Cos. in terms of tan. By (14), cos. A =- 

 whence, by (8), cos. A 



sec. A ' 

 1 



. ... (19) 



<v/l+:tan. 2 A 

 Cosec. in terms of sec. Using consecutively (15), (16), and (14), 



1 "* 



cosec. A = = . 



sin. A 



'^/sec.-A- 1 

 sec. A 



.'. cosec. A = 



sec. A 



. (20) 



Sin. in terms of tan. By (11) and then (19), 



i 

 sin. A = tan. A cos. A = tan. A ' 7^ 



</ 1 + tan. 2 A' 



'. sin. A = 



tan. A 



V 1+ tan. 2 A 

 Other important results are 



From (8), tan. A -/sec.* A - 1. 

 sec. A = -/I -f- tan.* A. 



From (9) cot. A = v'cosec.* A 1. , 

 cosec. A = 



(21) 



(22) 

 (23) 

 (24) 

 (25) 



-f- cot. 2 A. 



The learner should take the trouble to express every function 

 in terms of every other function, writing down both reasoning 

 and results in each case, and will thus acquire a great and most 

 useful familiarity with the ratios existing between the various 

 functions. Only the plain rules for solving simple equations 

 are required for this. 



EXERCISE 1. 



1. If tan. A = 0'8, calculate sin. A (say to four places of decimals). 



tan. A '8 '8 '8 



By (21) sin. A= -^ = ~== = ^^ = ffflOB = AM - 



2. If coe. A = 0'45, calculate sin. A. 



3. If tan. A = 0'22, calculate cos. A. 



4. What is the value of sin. A when cosec. A = 1'25 ? 



5. Calculate cot. A on the assumption that tan. A = . 



6. If versin. A = , calculate all the other functions of A. 



7. Show that cosec. A sin. A = cos. A cotan. A. 



8. Show that l_+~t* = L__ 



sin.- A 1 cos. A. 



LESSONS IN GREEK. XLII. 



STRENGTHENED STEMS (continued). 



III. Verbs whose Pure, Stem is in the Present and Imperfect 

 strengthened by the insertion of av (less often aiv) before 

 the terminations. 



(a) av or a.iv is introduced without any other change. 

 All verbs of this kind form their tenses from a triple stem- 

 namely, the present and imperfect from the strengthened stem, 

 the second aorist from the pure stem, the future and perfect 

 from a third stem which arises from the pure stem and an added 

 e, which in the inflection passes into 77. The o in the termination 

 avca is short. 

 1. aiffda.vofj.ai, I feel, aor. yad-o/j-riv, atcrBecrdai ; perf. r>ffOrifj.ai, fut. 



2. a/uLapravu, I miss the mark, fail, sin, aor. 2 fipapTov, fut. 

 a/j.apTr]cro/j.a,i, perf. r]fj.apTr]Ka, perf. pass. y/j.apTrjfj.ai, aor. pass. 



3. airexOavo/jiai, I am hateful, aor. airnx^o/jL-nv, inf. airtxdfO'Bai, 



fut. <r7rex0?7<ro 1 uat, perf. aiT7JX^W at (I am hated). 



4. avavca (and au|), I increase, fut. avrj(ra>, aor. 1 rju|rj<ra (perf. 



Tjufjjica), perf. pass. 7)u|7jjua, fut. pass. av^-r}crofj.ai, aor. pass. 

 7ju|rj07jv. 



5. ^\affravw, I sprout, aor. 2 ej8\a<TTOf, fut. 0\affTyffw, perf. 



eA.a<m7/ca and /8e/8A.aerT7jKa. 



6. SapOavca, commonly as a compound KaraSapQavoi, I sleep, 



aor. 2 KareSapBov, fut. Kara5apOii(ro/j.ai, perf. /caTa8e5ap07j/ca. 



7. o\tffdavw, I slip, I slide, aor. 2 wKurOov, fut. o\iff6r]ff<a, perf. 



8. off(jjpaivo/j.ai, I smell, aor. 2 oxr^po/uTji/, fut. oc 



9. ofyMffKavw, I am liable, I owe, aor. 2 <a<pKov, fut. o<>/\.rj<reo, 



perf. iacp\riKa, perf. mid. or pass. oc(p\ri/j.ai. Mark the 

 double strengthening in ICTK and av. 



(b) a.v is added, together with the insertion of the nasal v, before 



the characteristic consonant of the pure stein. 

 Thus in \ai/6ayca, pure stem \a.6-, between a and 6, v is intro- 

 duced, forming \av6-, to which av is added, forming \av0av-. 

 The short vowel in the pure stem passes in the tenses (except 

 the second aorist) into the corresponding long one : pav8avw is 

 an exception. The v before a p sound and a fc sound undergoes 

 the usual changes. 



10. Qiyyavu (pure stem 617), I touch, aor. 2 eOtyov, fut. 0i|o/xcu. 



11. Kayx -" 03 ) I obtain by lot, aor. 2 \axov, fut. \ijo[j.ai, perf. 



perf. mid. or pass. eiATjy/xai, aor. pass. 



