SINKS, BTC., OF MOT.TIPLB ANGLES.] MATHEMATICS. TRIGONOMETRY. 



619 



In the game manner, 



Cos. (A + B + C) = cos. A cos. B cos. C cos. A sin. 

 B. sin. C cos. B sin. C sin. A cos. C sin. A sin. B. 



And hence, 



Tan. (A + B + C) 



^tan A + tan. B + tan C tan. A tan. B tan. C 

 1 tan. B tan. C tan. C tan. A tan. A tan. B. 



26. Certain other Formulas. 

 Again, since 







2 



cos. - 



. 00 



~ 

 0-0 



f*f\a 



+ 0' . 



cos. E. sin. 



m 



< 





 ~ 



(20). 



Sin. + sin. 0=2 sin. 



Sin. sin. = 2 . 



Sin. + sin- 

 Sin. + sin. 



= tan. 5-^- cotan. 

 In like manner, 



Cos. cos. + 1.. x 01N 



COS.0+COS.0- an " -2- " a" (21) " 



There are many similar combinations of the trigono- 

 metric ratios besides those above given. These are of 

 very frequent occurrence, and the student who has 

 thoroughly mastered the above will be at no loss in 

 investigating other combinations that may occur in his 

 subsequent reading. 



27. The Sines, Cosines, <tc., of Multiples of given Angles. 

 We have already seen that 

 Sin. (A + B) sin. A, COB. B + sin. B, cos. A. 



ThU being true of all values of A and B is true when 

 A - B, and .'. when A + B = 2A. 



.'. Sin. 2A = 2 sin. A, cos. A. ... (22). 

 Similarly, since 



Coe. (A + B) cos. A, COB. B sin. A, sin. B. 

 .'. COB. 2A= cos.A-sin.A (23). 



Now, 1 = cos.*A + sin. 'A. 

 Add this equation to (23), and we obtain 

 Cos. 2A = 2cos. ! A L 



And Cos. 2A = 1 - 2 sin.*A. 

 Similarly, 



Tan. 2A = , 2 tan - * (24)' 



1 tan. 'A 



Again, 



Sin. 3A - sin. (2A + A). 



= sin. 2 A, COB. A + cos. 2A, sin. A 



= 2 sin A, cos. A, cos. A+(Cos. *A sin. *A)sin. A. 



+ 3 sin. A, cos'A sin. 3 A. 



3 sin. A -4 sin. 'A (25). 



Similarly, 



Cos. 3 A =3 Cos. A + 4 co8.A (26), 



and 



3 tan. A-tan.'A 



Tan. 3 A - 



1 3 tan. 'A 





28. Determination of Sine, <te., of an Angle in terms of 

 the Sine, <fec ., of the Sub-multiples of that A ngle. 



From these expressions we may derive others expres- 

 sing the sines, <tc., of an angle in terms of the sines, .to. 



9 



of the submultiples of that angle. Thus, writing -= for 



m 

 A, we have 



From (22) Sin. 



2 sin. cos. -. 

 2 2 



From (23) Cos. = cos. -_ - sin. 2 - 

 2 2 



= 1-2 sin. 2 

 2 



= 2cos. 2 -_L 



| 



From (24) Tan. = 



And writing for we have 

 3 



From (25) Sin. = 3 sin. ^_4 sin. 3 

 o 3 



From (26) Cos. = -3 cos. + 4cos. 3 ^. 

 o 3 



From (27) Tan. 



29. On the "Ambiguities" resulting from the use of the 

 above Formulas. 



These formulas enable us to solve the following ques- 

 tion : Having given the sine, <bc., of an angle, we can 

 find from it the sine, &c. , of double that angle ; and 

 conversely having given the sine, <tc , of an angle, we 

 can find the sine, etc. , of half that angle. 



(a). Thus, having given sin. A = p, to find cos. 2 A, 

 we have 



Cos. 2A = 1-2 sin. s A = l-2p 



and so, having given sin. A = p to find sin. 2 A. 



Since sin, A = p cos. A ^/l p* 

 .'. sin. 2 A = 2/> VT P- 



It will be seen, from the above formulas, that for one 

 given value (p) of sin. A, there is one value of cos. 2 A, 

 while there are two of sine 2 A equal in magnitude, but 

 of different signs. This is sometimes spoken of as an 

 ambiguity. It will be observed, however, that the am- 

 biguity in the determination of sin. 2 A arises necessarily 

 from the data, since it appears by considering the values 

 of A which satisfy the equation. 

 Sin. A = p, 



that there will be one value of cos. 2 A and two values 

 of sin. 2 A resulting from the data. Thus, if A 1 be one 

 angle which satisfies the equation, then all the values of 

 A are included in the two formulas. 



A = 2m 180 + A' 

 and A = (2m + 1)180"+ A' 

 m being any integer whatever ; 



' cos. 2 A = cos. (2m360' +2AO = cos. 2 A*, 



or = cos. (2m + 1) 360-2 A 1 ) = cos. 2 A 1 

 Under all circumstances, therefore, 



Cos. 2A = cos. 2Ai, 

 and therefore has but one value. Whereas 



Bin. 2 A = sin. (2m 360^ + 2 A 1 ) = sin. 2 A 1 . 



or = sin. {(2^+1) 360 2 A 1 )) sin. 2 A'. 



.". sin. 2 A has the two values + sin. 2 A 1 and 

 Isin. 2 A 1 . 



If, however, we know A, or even the limits between 

 which A lies, as well as that sin. A = p, then all inde- 

 terminateness vanishes from the expression for sin. 2 A. 

 Thus, if A be less than 90, then 2 A is 180 and sin 

 2 A must be positive. And, again, if A ~? 90 ^ 180, 

 then 2 A 7 180 360, and the sin. 2 A is negative in 



the former case ; therefore, 



Sin. 2 A = 2p J 1 -p*. 



In the latter, 



Sin. 2 A = -2p V-V- 



