MATHEMATICS. TRIGONOMETRY. 



[AMBIGUOUS RESULTS. 



(6). Again, we have 

 Co*. 0-l- 



_ 



n . 



" 



Hence it appears that for a given value of cos. there 

 will be two value* of sin, -^ , likewise two values of cos. 



equal in magnitude, but with different signs. This is, 



as before, necessarily the case, if we only know the value 

 of cos. 0. For if we have a given value p of cos. 6, so 



th.it 



Cos. - p, 



and if 1 be a value of which satisfies this equation, 

 then all the values of 0, which satisfy this equation, 

 are expressed by, 



..m(3GO 01). 



.'. sia.-= sin. (m 180 ~) 

 2 2 



which 



sin. if m be even, 

 2 



or, + sin. if m be odd, 

 2 



under any circumstances ; therefore there must be two 

 f\ 



values of sin. -s equal in magnitude, but with different 



sums the same result as that we obtained from the 







equations for sin. -g **" cos. -jj 



6 

 (c). Again, we have given sin. to find sin. y and 



0_ 

 2 

 We have 



cos. y 



sin. J y + cos. * y = 1 



2 sin. -g cos. y sin. 



Adding these equations we obtain, 



ft Q O ft 



sin. * y + 2 sin. -jj- cos. y + cos. y = 1 + sin. 



And, subtracting the second from the first, 



G e . 



sin.* y 2 sin. y cos. y + cos. 2 y = 1 sin. 



Eitracting the square root of each of these equations, 

 and we have 







. 

 in. y + < 2"= 



1 + si n. 



sin. cos. - l_ 



/. Adding 





 2 sin. y- 



and subtracting 



(> 

 2cos. y - 



_ 

 in. 0+ ^1 sin. 



_ 

 0_ ^l sin.0 



_ , _ 



.'. sin. y - J ( ,/ i + Bin . + ./l sin.O 



and cot. 



_ ) 

 ^1 sin. 9 } 



And since each square root has two sines, it follows 

 that if we have given merely the value of sin. 0, t. e., 



n 



tin. - p. we have four different values of sin. 



p. 



^vpO, 1( v/ 1 +P- -v' 1 -P-) 



We may prove, as before, that this amount of in- 

 determinateness is involved in the data : for if 



sin. = p, 



and if ff be a value of 0. which satisfies this equation, 

 then all the values of which satisfy the equation, are 

 given by the formulas. 



" 2 m. 180 + 

 and _ (2 m. + 1) 180 ff 



where m is any whole number whatever. 





 .'. sin. y may be either one of the two forms, as 



follows : 



0\ 0'\ 



Sin. (ro.l80+ y/ or sin. (m. 180 + 90 y/ or ac- 

 cording as m is odd or even, i.e., according as m is 2 n 

 or 2 n -f 1. 

 One of the four 



Sin. (2 n. 180 + y) sin. (2 + L 180 + y) 

 Sin. (2n. 180 + 90 y) or sin. (2n + l. 180 



And these are respectively equal to 



. ff . ff ff ff 



Sin. , sin. -g, cos. y, and cos. y. 



i.e., it may have one of the four different values 



ff & 0' 



Sm. y) Sin. yi COS. y COS. y' 



If, however, we know the limits within which the 

 value of lies, this indeterminateness vanishes. Thus, 

 f\ 



if 7 90V 180, then y 7 45 90, and therefore sin. 



1 



y must be 7 -T<J- 



this case, 



1. and be positive. Hence, in 







. __ 



y {+ J 1 + sin. + ^/ 1-sin. 0}- 



Sin. 



1 



For when = 90 this gives sin. y = -/g 





 = 180 sin. y = 1. 



and between these limits the value of the formula con- 

 tinually increases. 

 If, however, 7 270 / 360, we have 



Sin. y= y {--v/l + sin.0 + ^l-sin.0}. 



The student will do well to verify this for himself. 

 The same kind of reasoning applies to the formula for 

 



cos. y 



(</). Again the formula 



Tan. 



2 tan. y 







enables us to determine tan. y when tan. is given. For 



we can easily put the equation into the form 

 02 



r+cr*"-** 1 



o 



2 



1.0 



== sir*--*"-* 



1 



tan. 



