SOUTTIOX or TRIANGLES.] MA THEM ATICS. TRIGONOMETRY. (HI 



CHAPTER XIII. 



THE NUMERICAL SOLUTION OF RIGHT-AXGLED TRIANGLES. 



THE parts of a right-angled triangle are the three sides, 

 two angles and the right angle ; if any two of the former 

 five, one of the two being a side, are given, we can cal- 

 culate the remaining three. The following are the 

 methods employed in the various cases, which are these. 



(1). Given the base and perpendicular. 



(2). Given the hypothenuse and another side. 



(3). Given the base or perpendicular and an angle. 



(4). Given the hypothe- Kg. 23. 



nuse and an angle. 



(1). Given the Base and 

 Perpendicular; i.e., given 

 o. 6. find A. B. c. 



tan. A = -=- gives A. 



B = 90 A Gives B. 



Cos. A=-. 

 c 



Gives c. 



These can be put into forms adapted for logarithmic 

 calculation, as follows : 



log. tan. A = log. a log. 6. 

 . 10 + log. tan. A = log. a + 10 log. 6. 



or, L. tan. A = log. a + ar. : comp. : log. b. (o). 

 Similarly, 



Log. e = log. 6. log. cos. A. 



= log. 6. + 10 (10 + log. cos. A) 

 = log. 6. + 10 L. cos. A. 



.". log. c = log. 6. + ar. : comp. : L. cos. A. (6). 



(a) and (b) are the formulas actually used in calcula- 

 tion. Thus, 



Given a = 7564 5 yds. 6 = 3987 4 yds. 

 Find A. B. and e. 

 To find A. from (o) we have, 



Log. tan. A = log. a + ar. : comp. : log. 6. 

 = log. 7564-5 = 3-8787802 



+ ar . : comp. : log. 3987 '4 6 -3993102 



L. tan.6212' 



diff. for 1 



10-2780904 

 10-2779915 



/. A 



62 

 90 



19* 



51-05)989-00(19 

 5105 



47850 

 45945 



and B = 27 47' 41* 



To find c we have, 

 Log. c = log. b + ar. : comp. : L. cos. A 



log. 6 



ar. : comp. : L cos. 62 12 7 19* = ar. : comp. : 



0-6686623 = 



8551-2 



3 '6006898 

 3313377 



3-9320275 

 9320271 



08 diff. - 



/.c- 8551-208 



(2). Given the Hypothenuse and another aide, 

 e. g. , given a c to find A B and 6. 



We have. sin. A = 

 c 



.'. log. sin. A = log. o log. e. 

 .". L. sin. A = log. a + ar. : comp. . log. c (a) 

 which gives A and .'. B which equals 90" A. 



Tot. I. 



Again cos. A = . 



C 



6 = c cos. A. 



log. 6 = log. c + L. cos. A 10. (6) 

 Ex. Given a = 724 -5. c = 1005 -4. 

 We shall have A = 46 6' 17'. B=43 53' 43*. 

 6 =697086. 



(3). Given the Base or Perpendicular and an A ngle. 



e. g., given a. A., find B. 6, c. 

 We have B = 90 - A. 



Again c = -r- 



sin. A 



.'. log. e = log. o log. sin. A 

 .'. log. c = log. a + ar. : comp. : L. sin. A. (a) 

 6 



Again c = 



cos. A 



.'. log. 6 = log. c + L. cos. A - 10. (6) 



In using formula (6), it must be remembered that in 

 the calculation of formula (a) we have already found 

 log. c. 



Ezample. Given a = 7643-5 A =37. 18'. 



We shall find B = 52. 4^ 



c = 12613-4 

 6 = 10033-53 



(4). Given the Hypothenuse and an Angle. 



e. g., given c. A, find B. 6. a. 

 We have B = 90 - A. 



6 = . cos. A. 



.'. log. 6 = log. c+L. cos. A 10. (a) 



Again a c sin. A. 



.'. log. a = log. c + L. sin. A 10. (6) 



Ezample. Given c = 7234 5 A = 33.19'. 



We have B = 66.4l'. 



6 = 6045 493. 

 o = 3973 665. 



The cases of oblique triangles are in like manner 

 reducible to four, viz. 



(1). Given three sides to find the angles. 



(2). Given two sides and the included angle. 



(3). Given two sides and the angle opposite to them. 



(4). Given one side and 

 two angles. 



(1). Given three sides to 

 find the angles, i. e., given 

 a 6. c. to find A B. C. 

 We have 2s = a + 6+c. 



Fig. 24. 



B 



2 ,. (.-) 



.'. 2 log. tan. - = log. (s 6) + log. (3 c) log. i 

 log. (s a) 



.'. 21og. tan. + 20= log. (- 



10 



log. $ + 10 log. (s a) 



.'. 2 L. tan. A = log . ( s _ 6) + log. (s - c) + ar. : 

 comp. : log. t + ar. : comp. log. (s a) 



This formula gives us - We shall have to find B 

 from the formula 



T> 



2 L. tan. -^ = log. (s-c) + log. (s-a) + ar. : comp. : 



log. + ar. : comp. : log. (s b) 



It will be observed that all the logarithms needed for 

 finding B and C have been already used in finding A, 

 whence, when A is known, B and C are found with very 

 little trouble. 



4 N 



