SIXES OF AXGLES 15, 30, ETC.] MATHEMATICS. TR IGOKTOMETRY. 



621 



It will be observed that for one value of tan. we have 





 two values of tan. - The ambiguity is involved in the 



data in this case as in others. For if 

 tan. p. 



then ff being one value of 0, all the values of are in- 

 cluded in the formula 



= TO. 180 + 1 . 



.'. according as m is odd or even, t. ., according as m is 

 equal to 2n or 2n + 1, 



That is, 



or- =n. 180 + 90+ 



Tan. |-=tan. n!80+^- 



or, tan. (n 180 + 90 + -) 



2 

 according as m is odd or even ; and therefore 



tan. | _ tan. | or tan. (90+ ^) 



O f & 



= tan. or cotan. _ 



2 2 



Two different values, as also appeared from the 

 formula. 



30. On the Numerical Value of the Sines, Cosines, <kc. t 

 of the Angles 15, 30* 45, 60, 75. 



We have already investigated the value of the trigo- 

 nometrical ratios of 45, 30, and CO . By the aid of 

 the above formulas we can investigate the values of 

 many others. For example, of 15 and of 18. 



Sin. 15 - sin. (45 -30) - sin. 45 cos. 30- sin. 30, 

 ooe.45. 



which also equals cot. 75, 



since 75 = 90 -15. 

 Cos. 15 - cos. (45-30) - cos. 45 cos. 30 + sin. 



Cosec. 15 = 



The student will observe that this investigation of the 

 ratios of an angle of 15, together with those previously 

 investigated, gives the ratios of the series of angles 15 , 

 30, 45 S , 60, 75, 90. 



31. To investigate the Trigonometrical Hallos of an Angle 



of 18, 36, 64, 72. 

 Since 54 = 90 -36, 



if we write for 18 3 , we have 



30 = 90-20, 

 and therefore 



Cos. 30 = sin. 20. 



.'. 4 Cos.0-3cos. - 2 sin. 0, cos. ft 



.'. 4 Cos. 2 0-3cos. = 2sin. 0. 

 .'. 1-4 siu. 3 = 2sin. 0. 

 4 Sin. 2 + 2 sin. 6-= 1. 



/. 4 Sin. 2 6 + 2 sin. + - = - 



Sin. _ ~- 



4 



This is a case of ambiguity similar to those above 

 explained, and if we only had the equation 



4 Sin. S + 2 sin. = 1, 

 we should have the two values of sin. just given, viz., 



4 4 



But as we not only have the equation, but also know 

 that = 18, this enables us to choose the only admis- 

 sible value. 



Sin. 18 = nl_^J. 



4 



For the other value of sin. being negative cannot be 

 the sine of 18. Hence, 



Cos. 2 18 = 1 sin. "18 =. 

 , 3 1 V6 5+ 

 ~ - -- 



8 



Hence, sin. 36 = sin. 2x18=^ ^^ 

 and cos. 36 = ^ 6 + 1 . 



and sin. 36 = cos. 54, and sin. 18 = cos. 72. 

 Hence we evidently can obtain the trig, ratios of the 

 angles 18, 36, 64, 72. 

 Again 



Sin. 3 = (18 15) 

 - sin. 18 cos. 15 - cos. 18 sin. 15. 



, 



" 



4 2^2 4 



_ ( VIS - 1) (.y/3 + 1) - 0^/3 - 1) 



10 + 2 V 6 



S.v/2. 



Hence we may clearly obtain numerical values for the 

 trigonometrical ratios of the series of degrees, 3, 6, 9, 

 12, 15, &c. 



N.B. In reading the preceding pages the student 

 will have observed, that in many instances, when a 

 method of reasoning has been applied to one case, it has 

 been merely indicated that the same method is appli- 

 cable to a similar case. In all these instances he will do 

 well to write out at full length the reasoning in these 

 similar cases. By this means he will ensure a thorough 

 comprehension of this part of the subject, and become 

 familiar with the various combinations that trigonometric 

 ratios can form. In regard to this very subject, Dr. 

 Peacock observes : "It should be the first lesson of a 

 student, in every branch of science, not to form his own 

 estimate of the importance of elementary views and 

 propositions, which are very frequently repulsive or 

 uninteresting, and such as cannot be thoroughly mas- 

 tered and remembered without a great sacrifice of time 

 and labour." To assist in obtaining this familiarity he 

 may perform the following exercises : 



(1). i + cos. 2 A=2 sin. (60 A) sin. (60 + A.) 

 (Remember that cos. 120= |.) 



?). 4 cos. m cos. n cos. r = cos. (m + n + r) 

 cos. (m + n r) + cos. (m n + r) 6 + cos. 

 (m n + r) 9 



(Remember that cos. (A + B) + cos. (A B) 



= 2 cos. A cos. B 

 (3). 4 sin. cos. = cos. cos. 3 0. 



(4). Tan. A + tan. B 



cos. A cos. B 



