SERIES.] 



MATHEMATICS. TRIGONOMETRY. 



627 



CHAPTER XII. 

 TRIGONOMETRICAL SERIES AND TABLES. 



(46) GENERAL EXPLANATIONS, 



THE preceding pages give the theory of plane trigono- 

 metry. To complete this branch of mathematical science 

 it remains to explain the mode of deducing numerical 

 results from the above theory, and to perform the actual 

 calculations of the parts of triangles from which the 

 science derives its name. We have already seen that 

 the trigonometrical ratios of certain angles are known ; 



for example, we know that sin. 45 = ~~ ""** 



= ^ and some others, and knowing these values 



9 



we can determine log. sin. 45, log. sin. 60, and so on . 

 Now instead of knowing only the trigonometrical func- 

 tions of certain angles, we want to know the trigono- 

 metrical functions of every angle from up to 90. 

 And as the calculations are for the most part carried on 

 by means of logarithms, we require to know also the 

 logarithms of these functions. These values have been 

 calculated and arranged in tables in a similar manner to 

 the tables of logarithms as before explained ; we pur- 

 pose in the following pages to explain the principles on 

 which this calculation has been effected.* 



In our article on series and logarithms, we found it 

 necessary to investigate certain algebraical series before 

 proceeding to discuss the nature of logarithms ; in like 

 manner we shall find it now necessary to investigate 

 certain trigonometrical series before explaining the con- 

 struction of trigonometrical tables. In the following 

 article on trigonometrical series we shall always keep 

 this object strictly in view, a circumstance that will 

 account for the absence of certain series that are often 

 given in treatises of trigonometry. There is a very 

 large number of such series ; we treat the series here 

 simply as the means of arriving at the numerical results, 

 and on this principle make the following selection from 

 that larg;e number. The student will do well to observe, 

 in studying any branch of mathematics, in which alge- 

 braical formulas occur, that unless he understands the 

 method of obtaining arithmetical results from his for- 

 mulas, he has not as yet mastered the subject. 



The first article is an example of a limiting value. 

 We would recommend the student to consider it very 

 carefully. The proposition to be proved is that the 



limiting value of 5 "" 1, where, of course, is in cir- 

 cular measure. The following will sufficiently explain 

 the meaning of the statement. When becomes very 



small, : -becomes very nearly equal to 1, and the 



smaller 6 becomes the more nearly j- becomes equal 



to 1 ; but so long as 8 has any value, however small, 

 sin. 6 



IT 



can never actually equal 1. Then the value which 



limits the value of 5 is 1, while the value which 

 I limits the values of is 0, and we assume that if 9 

 actually equals 0, *-^r- actually equals 1. The assump- 

 tion is in point of fact an axiom. We cannot, however, 

 discuss the questions here that this statement gives rise 

 to. There are several methods by which the proposition 

 can be proved. The following is a modification, of New- 

 ton's sixth Lemma. 



(47). To show that when = we must have ^ = 1 



Let A P. P be the arc of a circle, the centre of which 

 U O. A T a tangent, A P a chord. Produce A P to p, 



TaMn of natural linei and corinet. and logarithmic lines, cosinei, 

 tangent*, tea., will be found at the end of this section. 



and AT lo t, draw PT iaid.pt parallel to AO, drawjpo 

 parallel to P O. Then angle Apo = APO = PAO. 

 Since A O = P O, and therefore Ao = op, with centre o 

 and radius op describe a circular arc, Ap. Then the 

 angle A P being equal to Aop, we have 

 arc AP _ arc Ap 

 ~AO~~ Ao"' 



since each measures the equal angles. 

 Again, draw P N and pn. parallel to At. 

 AT PN 



Then sin. AOP. 



(a) 



At _ }>n 



AT_Aj 

 " A0~ Ao 

 ' arc AP _ arc Ap 

 AT At 



Fig. 22. 

 Ti T 



Now, suppose P to move along the arc to P., and sup- 

 pose pt to remain fixed, produce the chord A P, , to meet 

 f/t in p, make the chord A P 1 to meet pt in />, , make the 

 angle APO -> A^O.. Then as before 



arc A P t _ arc Aj>, 



~~AT~ At 



so that in all cases the equation (a) holds good. Now 

 when P moves up to A, p moves up to t, and when P 

 coincides with A, p coincides with t, and then p and t 

 coinciding, the changing arc Ap coincides with At, and 

 therefore in the extreme or limiting case. 

 arc Ap 



~~ST 



and hence on the limiting case, when A P vanishes, 

 arc AP 



1 



AT 



1. 



NowletAOP* 



'0, where 9 is in circular measure, then- 

 arc A P 



and 



Now 



AO 



AT = PN 

 AO PO 

 arc AP 



arcAP 



0. 



AT 



PO 



AT 



PO 



sin. 6. 



sin. 

 ' 



