A 02 A3 à fortiori, sin 6 > 2 (1-4), i.e. >0 0 A2 .(3). 4 Hence sin 0 lies between 0 - 483 and 0. 486. We have cos 0 = 1 – 2 sino 10, .. substituting from (1), cos 6>1 – 2 (10)”, i.e.>1 - .......(4); , >1 A3 12 but substituting from (3), cos <1 - 2 CO 2 32 A2 A4 A6 512' A2 64 ..(5). 16 2 + + 487. We have, by division, 2 tan 10 2 tan 10 (1 + tan’ 10+ tan* 10 + ...), a convergent series, since 10< 45°, and .. tan 10 <1. A5 A2+1 :: substituting from (1), tan 0 >0++ e 4" < + + + +... 16 Limits. and of sin tan 488. The limit of . when 6 is indefinitely A 0 diminished, is unity. For sin 0, 0, tan 0 are in ascending order of magnitude. 0 1, tan ? are in ascending order of magnitude, But, as decreases, sec 0 and cos o may each be made as near to unity as we please. Hence 0/sin & which lies between 1 and sec 0; and 0/tan 0 which lies between cos 0 and 1; may each be made as near to unity as we please. A Thus the limits, when 0 is indefinitely diminished, of sin ? sin 0 0 tan A are each unity. A tan 0' 489. The limit of the ratio of the sines or tangents of two vanishing angles is equal to the limit of the ratio of the two angles. sin 0 0 (1 -- 2) Then sin $$ (1 - y) Now, as 0 and $ are indefinitely diminished, and y are also indefinitely diminished by last article. sin A A sin • 490. The limit of the ratio of sin ko to k, when k is indefinitely diminished and 0 is finite, is 0. sin ko sin ko For -0. k ko Now, since 0 is finite, .. k0 is indefinitely diminished when k is indefinitely diminished. .. limit of sin ko - kO is unity. 491. The results of the last three articles should be carefully examined. In Art. 488, observe first, that 0, not sin 0, is to be made indefinitely small. If 0 =for instance, sin 0 vanishes, but the limit of sin 0 = 0 would be, not unity, but zero. Observe secondly, that 0 is the circular measure of the angle not any other measure. For instance, if the angle 6 radians = A degrees, then 180 п sin Ao A a sin Ac T 180' In Art. 489, however, 0 O radians A degrees since (say), $ $ radians B degrees sin Ao А .. limit of - limit of sin Bo B As before in Art. 488, A and B here must diminish indefinitely, not only sin Ao and sin Bo. In Art. 490, on the other hand, o may have finite value. Thus 0 might be the circular measure of an angle greater than two right-angles. For here the angle which is to be made small is kó, not 0; and provided 0 is finite, ko is made small by diminish ing k. 492. The results of the three articles, 488, 489, 490 are usefully summarised in the statement When 0 = 0, sin 0 = 0. But in this statement we must distinguish two distinct propositions; viz., (1) When 0 is small, sin is approximately equal to 0. sin 0 (2) When is indefinitely diminished, the limit of is exactly unity. : COS COS COS COS 28 . The first proposition may be used in arithmetical and approximate calculations : the second in algebraical and exact theorems. The propositions in the following articles will exemplify these distinctions. 493. To show that, when n is indefinitely increased and O is finite, the limit of A o A ө sin A A ө 0 COS 22 sin COS COS 2 2 2 22 and so on. 0 COS sin 0/2 2n sin Ꮎ . 2 0/21 And, as in Art. 490, when n is indefinitely increased and : 0/2" indefinitely decreased, the limit of the above is 0. 0 0 sin e Hence, in the limit, cos 2 22 23 0 Example. Put 0 = . Then sin 0 =1; 2 cos to=12; 2 cos 10 = (2+12); 2 cos 38={2+1(2+12)}; and so on. Thus, the product of the endless series of factors 12 1(2+2) [ {2+/(2+/2) 2 2 . COS COS . . (1 - 0) (1 m), 494. To show that sin 0 >0 - 303, if o denotes an acute angle. 02 23 25 A2 A2 23 25 Ꮎ 1 1 A2 1-1/4" + > COS COS COS 82 A2 >1 6 6.4n? 02 6 0 02" 2 4 2n sin Thus, is less than the product of these cosines, but may be made as nearly equal to it as we please by increasing n. sin 0 A2 Hence >1 0 6 i.e. sin 6 > 0-10%. = > ) 2 tan 10 tan A 1 - tan’ 10' 2 tan: 10 2 (10) .. tan 0 – 2 tan 10 by Art. 483. 1 - tan” 10 1-(10) 0.402 1-442 A2, 1 A2 4n tan 10 tan 10 1 82 1 162 2 1 42 n 10 42'4 447 421 and so on. + + + tan 10 + + 24 +.. . 10 > 10 |