.. substituting from (1), cos 0 > 1 – 2 (} 0)2, i.e. > 1 - — ......(4); 02 2 a convergent series, since 10° < 45°, and .. tan 10 < 1. .. à fortiori, tan 0> sum of any number of these terins. For sin 0, 0, tan 6 are in ascending order of magnitude. Hence, dividing each by sin and by tan 0, both 0 1, sin ' are in ascending order of magnitude, But, as decreases, sec 0 and cos may each be made as near to unity as we please. Hence and 0/tan which lies between cos 0 and 1; may each be made as near to unity as we please. Thus the limits, when is indefinitely diminished, of Ө sin ' 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. Now, as and are indefinitely diminished, x and y are also indefinitely diminished by last article. 490. The limit of the ratio of sin ko to k, when k is indefinitely diminished and 0 is finite, is 0. Now, since is finite, .. ko is indefinitely diminished when k is indefinitely diminished. .. limit of sin k✪ ÷ k✪ is unity. .. limit of sin ko÷k is 0. 491. The results of the last three articles should be carefully examined. Ꮎ In Art. 488, observe first, that 0, not sin 0, is to be made indefinitely small. If 0 for instance, sin vanishes, but the limit of sin÷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 ✪ radians = A degrees, then As before in Art. 488, A and B here must diminish indefinitely, not only sin A° and sin Bo. In Art. 490, on the other hand, ✪ may have any finite value. Thus might be the circular measure of an angle greater than two right-angles. For here the angle which is to be made small is ko, not ; and provided is finite, ko is made small by diminishing 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 is small, sin is approximately equal to 0. sin 0 (2) When is indefinitely diminished, the limit of is exactly unity. 0 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 0 is finite, the limit of And, as in Art. 490, when n is indefinitely increased and .. 0/2" indefinitely decreased, the limit of the above is 0. Example. Put . Then sin =√(2+√2); 2 cos 30=√{2+√(2+√2)}; and so on. Thus, the product of the endless series of factors √2 √(2+√2) √{2+√√(2+√2)} 2 π 494. To show that sin 0>0 – 103, if 0 denotes an acute angle. .. cos + + 24 42 02 1-1/4" "81-1/4 4n |