MATHEMATICS. TRIGONOMETRY. 



[INVERSE RATIOS. 



(Remember that tan. A - " ^ ). 



(Remember that tan. 'A-tan. >B - (tan. A + 

 tan. B)(tau. A-tan. B.) 



nain. a cos. (a z) 



* eotan. ftm ootan . a 



Tan. z 



n-f-m. 



If + 0+ y- - 90 show that 

 (a) Tan. tan. V^ + tan. y* tan. + tai 



Tan. 9 + tan. + tan. y- = tan. 6 + tan. 



tan. y- + sec. sec. sec. y 

 (Remember that sin. (0 -f + 

 and cos. (0 + + 



(8) . Km.*"; ("-*) ntan '* 



I 



= 1. 

 = 0. 



oos.'x 



_ 

 ooa7(a-x) 



Then, 



sin. 2x m* 



and tan. (a 2x) 



tan. n. 



n -)- m 



(9). If tan. A + 3 cos. A = 4, show that A has two 

 series of values, one of which is 45, 225, 405, 585, 

 J... (i.e., tan. A=l). 



(10). If cos. nA + cos. (n 2) A = cos. A, show that 

 A must have the values 



30 330 390 690" 750 

 n 1' n l"n 1' n 1'n 1" 



(11). If sin. (x + a) + cos. (x + a) = sin. (z a) + 

 cos. (x n), then x must be 45, 225, 405 



(12). If sin. A + sin. (0 A) + sin. (20 + A) = sin. 

 (0 + A) 4- sin. (20 A.) 



(13). If 2 sin. (0 0)=1, and sin. (0 0)= cos. 

 (0 + 0), then we shall have = 45, = 15. 



_ #2 1 



(14). If Tan. = tan. 3 and cos. *0 = - ^. 



M O 



m 



,i _ 1 /cos. S sin. S 0\* 



3 \s.*0+sin.$0' 



(15). If sin. s 2 sin. ! = J. 

 Show that sin. = '^ , and show that the values 



of are given by the series, 



18 162 378 528 



54 126 416 486 



198 342 558 702 



234 306 594 666 



(16). Show that the series of angles in No. 9 can be 

 expressed by the formula TO. 180 + 45. 



m. 360 + 30 

 In No. 10, by the formula, ~ . 



In No. 11, by the formula, m. 180 + 45. 

 In No. 15, by the formulas, m. 180 18. 

 m. 180 64. 



31. On Inverse Trigonometrical Ratios Explanatory. 



The following notation, which is part of a general 

 system of notation originally proposed by Sir J. Her- 

 schel, is very generally adopted, and is very convenient : 



If tan. 0=p. 



Then0 -tan. ~ l p. 



i.e., tan. l p. means the angle whose tangent is p. In 

 like manner sin. *p means the angle whose sine is p, 

 and coa. -->p the angle whose cosine is p. 



The system of notation originally proposed was the 

 following : If sin. 0p and an angle, the arc subtend- 

 ing which, divided by radius, is equal to p; then, sin. 

 ii the sin. of p, and therefore sin. 0,= sin. (sin. 0), and 

 Sir J. Honchel proposed to write 



Sin. (sin. 0)- (sin.*0),. 



reserving the notation (sin. A) 3 for the squares of the 

 ain. of 0. Upon this principle, 



Sin. (sin. (sin. . (. . . sin. ) ) ) sin" 0, 

 and evidently 



sin." (sin." 0) sin." + " 



or the notation follows the law of indices,* and the 

 interpretation that sin. l p must obtain, is that it is the 

 angle whose sine is p. 

 in like manner 



log. (log. (log. a) is written log. 'a 



and log. * a, signifies the number whose logarithm is a. 

 Of this system the only part that has obtained any 

 extensive currency is that given above in the case of the 

 inverse trigonometric ratios. 



32. Formulas connecting inverse Trigonometric Ratios. 



There are some formulas in which these inverse ratios 

 occur that are worthy of notice. 



(a). Tan.-'m + tan.-i = tan.-i m +*. 



1 mn 

 For if tan. = m and tan. = n 



tan. (0 + 0) 



.. 

 1 -tan. 0tan. 1 -mn 



Now = tan. 1 m and = tan. 1 i. 

 .". -(- = tan. 'm -f- tan. 'n 



But + - tan.-i ?L+?. 

 1 mn 



..tan.-im+tan.-in=tan.- 1 ^"-". 



I -mil 



..(28). 



Hence, 



2 tan.-*m = tan.-' 



2m 



and 



Tan. 'm tan. J n 



tan.-i -J5Z? 



1 mn. 

 Again, 



sin. 'm +sin. ] n = sin. '{m */l n s 4" n ijT~- 

 For let sin. im = 0, sin. 'n = 0. 



.*. m = sin. 71 = sin. 0. ^/ 1 m 1 



= cos. ^/ 1 n j = cos. 0. 

 Now sin. (0 -f 0) = sin. 0. cos. + cos. 0. sin. 



= m, 



0+0 = sin,- 1 {ro VI n 2 + nv/l m 2 } 



i.-'n = sin.- 1 {TO^/I n ! +n /l m 2 } 



Or, 



Sin. -'m4 

 (29). 



Similarly, 



Cos. -'TO cos.- 1 n = cos. -' {mn ^/l m'y/1 n*}. 

 Cos. -' m + sin. -' n = sin. -' {mn + V'l n j </l m s j. 



33. Examples. 

 Show that 



Tan.-' J -(- tan. -^=,45 



For tan.-i J + tan. -1 A = tan.-'- 

 2 3 



1-LL 

 2 3 



-tan.-i 



tan.-T 1. = 45 



It is to be observed that when we say tan. 1 = 45 we 

 mean that this is one value. All the values of tan. 1 L 

 are, of course, given by the formula ml80 +45. 



Show that 



175 "3 " 



(2). Sin. -i m - tan.-* _ 



(3). Cos.-'m + tan. -im - tan.-* 



Sec antr, p. 46S. 



m m ,J 1 m 1 



