220 



THE POPULAR EDUCATOR 



KEY TO EXERCISES IN LESSONS IN GERMAN. 



EXERCISE 181 (Vol. III., page 379). 



1. 3d) nranfcije 3f)nen eincn gnten 2lbenb, 2. 3$ 6,abe baS SSergnugen, 

 Sfyncn einen guten 3)lorgcn ;u tounfcfyen. 3. 3n ber grembe gebenfen totr 

 eft nut Stebe unfcrcr grcunbe in ber etmatf). 4. 3cf> gcbcnfe nacbflen 

 Q)imtat nacb, bent gefllar.be ju geljen. 5. cbenfen <2ie lange bort ju Met- 

 ben ? 6. Sleis, tcb. gcbcn!e nich. t tange bcvt gu bhiben, icty tocrbe bait 

 juructfefyrcn. 7. @r ticrfuctjte feinm greunb bet ber rlcrnimg ber beut. 

 f.ten tirade cinju^olcn, abet er fornte eS nictyt, benn fein greunb h>ar ju 

 ro'cif sorgeruch. 8. cbenfen ic, 3I;ren Sruber auf feiner Sleife etnju. 

 Ijokn? 9. 3cf> fyolte meinen S3rubcr nad) ciner breitagigen JJfeife tin. 10. 

 23or fect> aiionaten tear i$ tm Scgtiffe, nod; Slmcrifa 311 ge|en; nun aber 

 bin i$ fe|r frofc,, bap icfy in ber eimat&, geblicben bin. 



EXERCISE 182 (Vol. III., page 379). 



1. When Rudolph of Hapsburg had become emperor of Germany, 

 the internal dissensions and the so-called club-law ceased in this 

 empire. 2. After they had killed a few stags, they desisted from 

 hunting. 3. It ceases raining, and we now can continue our journey. 

 4. My brother is at home ; he has already been a week in bed. 5. In 

 Germany there are other manners and customs than in America. 6. 

 The imperial diets were held at Ratisbon in later years. 7. The high- 

 school at Breslau is among the best in Germany. 8. They were just 

 dining as we arrived there. 9. They were not accustomed to take 

 their supper until they had done all their day's work. 10. They took 

 their dinner in summer during fine weather under a linden-tree, which 

 stood in the yard. 11. When the cholera raged in Paris, thousands 

 upon thousands died of it. 12. The soldiers take the field. 13. In 

 the last storm several ships sank. 14. The beggar goes from door to 

 door, and from village to village. 15. This redounds to iny honour, 

 to his disgrace. 16. You might do it for my gratification. 17. The 

 enemy steers with all sails towards the east. 18. That is too good for 

 him. 19. I am only too certain that it will happen so. 20. That 

 may be done too when we have first regulated our own affairs. 21. 

 Friend, life is an earnest business ; suffer its hardships : thus only 

 will the voyage be easy to you. 22. Finally, thou landest, after all, 

 safely on shore in thy harbour ; it is called the grave. 23. He has 

 ruined his own and his friends' fortune. 24. He has ruined his health 

 by these labours. 25. Nelson destroyed the French fleet. 26. If he 

 is not careful, his whole business may be ruined in a short time. 



PLANE TRIGONOMETRY. V. 



SOLUTION OF OBLIQUE-ANGLED PLANE TRIANGLES. 



XXI. Solution of Oblique-angled Plane Triangles. It lias 

 been already explained (Section X.) that any plane triangle can 

 be computed when three out of its six " elements " are given, 

 provided that at least one side be given. By aid of the formulae 

 developed in the last section, we proceed to show this in the 

 three following cases, which include all that can be presented ; 

 viz. : 



1. Where three sides are given. 



2. Where two sides and one angle are given. 



3. Where one side and two angles are given. 



1. Given the three sides a, b, c. Find A, B, and C. 

 simplest way to effect this is by (76), 



The 



s (s a) 



whence, by the table of logarithmic sines, tangents, etc., in 

 Galbraith and Haughton's mathematical tables, 4 A, and there- 

 fore A, can be found. Similarly, by (76), 



A and B being now known, C of course is known also. 



Familiarity with the use of logarithms is necessarily assumed 

 in the student, who will remember that, as 10 is added to all 

 logarithms of trigonometrical ratios (to avoid the necessity of 

 entering negative indices in the tables, which would otherwise 

 arise from the fact that many of the ratios are less than unity), 

 it is also necessary to deduct 1 from them before using them 

 in calculations, or (what is the same thing) to add 10 to the 

 other side of any equation in which they may appear. This 

 has been done above. The use of logarithms is fully explained 

 in our " Lessons in Logarithms " in the POPULAR EDUCATOR. 



EXERCISK 4. 



1. Given a = 26, b = 31, c = 43. Find the angles. 



2. Given a = 16'22, b = 15'33, c = 21'56. Find the angles. 



3. Given a = 1110, b = 13i2, c = 1500. Find the angles. 



4. Given a = T32, b = 1, c = 075. Find the angles. 



2. This case appears in two forms 



First, given two sides, a and b, and tlie included angle C. 

 Find A, B, and c. 



i (A + B) = 

 Again, from (67), tan. % (A 



(180- C) = 90- i-C. 

 _- B) _tan 1 i (A + B) X (a - 



a + b 

 .-. log. tan. (A - B) = log. tan. (90 - i C) -f log. (a - I) 



- log. (a + b). 



There being a logarithmic ratio on each side of this equation , 

 of like sign, there is no occasion to allow for the added tens, 

 which balance each other. 



We have now obtained (A -f- B) and J (A B), the sum 

 and difference of which, by the well-known rule, give the values 

 of A and B respectively. 



T> /r*r\ i tt . sin. C 



By (G5), we have c= ; 



sin. A 

 .'. log. c = log. a + log. sin. C - log. sin. A. 



Here again, owing to difference of sign, the " added tens " 

 balance each other. 



Secondly, given two sides, a and b, and an angle, A, not 

 included between them: (This is called " the ambiguous case.") 



Find B, C, and c. 



We find B readily from (65) ; viz., S !~ B = -; 



sin. A a 



whence log. sin. B = log. sin. A -f- log. b log. a ; 



and C = 180 - (A + B), 

 c is found from (65), as in the last example. 



Now since sin. B is also sin. (180 B), the above equation 

 for log. sin. B always admits of two values of B (except when 

 B = 90), one greater and the other less than a right angle ; 

 and other data have to be considered in determining which is 

 the correct one ; thus 



(a) If the given angle A is a right angle, or greater than a 

 right angle, B must be less than 90, and no doubt arises. 



(/3) Again, if A, though less than a right angle, together with 

 the greater value of B, be not less than 180, it is clear the Zess 

 value of B must be adopted. For instance, if A = 80 and B 

 = 70 or 110 (i.e., 180- 70), it is plain that 110 is an 

 inadmissible value for B ; consequently, B = 70 and C = 30. 



(7) But if A, together with the greater of the two values of 

 B, be less than 180, it is plain that the data given apply to 

 two triangles. Thus, if A = 80 and B = 85 or 95, we may 

 have either 



A = 80; B = 85; C = 15 ; 

 or A = 80 ; B = 95 ; C = 5 ; 



these alternative values being quite consistent with the fact 

 that a and b are fixed values, as appears by Fig. 14, where both 

 the triangles ABC and A B' c corre- 

 spond with the data given. 



It appears, however, by inspection 

 of Fig. 14, that the ambiguity can 

 never arise when a is greater than b, 

 since then one of the two equal lines 

 which may still be drawn from c to 

 A B (or A B produced) will fall to the 

 left of A, an impossible position for 

 a side of a triangle in which A is an angle. This is a re-state- 

 ment of (/3) in a more convenient form. 



We may thus sum up : The ambiguity can only occur when 

 the given angle is acute, and when the side opposite to it is less 

 than tlie other given side. 



When these conditions are fulfilled, both values of B must be 

 worked out, causing two values of C and two of c. 



EXERCISE 5. 



1. Given a = 218, b = 156, and C = 38 21' 20". Find A, B, and c. 



2. Given a = 53 '24, b = 31'27, and C = 126 36' 6". Find A, B, and c. 



3. Given b = 173, c'= 123, and A = 22 13' 30". Find B, C, and a. 



4. Given b = 156, a = 130, A = 42 25'. Find B, C, and a (give both 

 solutions). 



'B' 



Fig. 14. 



