Increasing indefinitely the number of these inequalities, adding them, and observing that the term on the right involving

and à fortiori tan 0> sum of any number of these terms.

497. The results in these last three articles are nearer approximations to the values of the ratios than the corresponding results in Arts. 485, 486, 487.

1 when n is indefinitely large, we have here

n

to evaluate a limit of the form 1° —the index being a function of the base. See Art. 468.

Now, if x is any quantity greater than 1, the following three quantities are in ascending order of magnitude, viz.,

For if the last quantity is expanded by the binomial theorem,

Ө

its first two terms are 1 + tan2 and the remaining terms are

n

all positive.

Raising each to the power n, the following three are in ascending order, viz.,

n

.. dividing by (sin )"; 1, (0/0)", (sec)"

n

(sin [n

0/n

Hence, by the last article, the limits of in

0/п

(tan O/n)

n

and of their reciprocals are each unity.

500. The above limits—and others of the form 1"-may

n

be

found from the known theorem that the limit of (1 +

(

Thus (sec )+0 = [(1 + tan3 0) cot3]xTM tan2 0÷0TM.

Hence, when = 0, the expression in the square brackets becomes equal to e, and the limit of (sec )+ is as follows:

501.

0

n

=

Ꮎ .

sin 0/n 0/n

Ꮎ ; .. n sin2. = 0 in the limit.

n

To expand cos 0 and sin 0 in powers of 0. We have

Now let n increase indefinitely while remains constant and finite so that a decreases indefinitely.

reaches its limit 1 independently of the number of them, therefore the limit of their product is 1.

Similarly since the diminution of a in

tan

pendent of the index r, therefore the limit of

α

is inde

cos

Exactly in the same manner, we may prove that

502. The student should observe that the expansions of and sin in powers of are obtained from those of cos no and sin no in powers of cos 0 and sin 0 in the same way that the expansion of e" is obtained from the binomial theorem. And hence that

The terms of cos 0 and of sin 0 are taken alternately but with alternate change of sign from the expansion of eo.

503. The above expansions of sin and cos hold for all finite values of whatever. For the ratio of any term to the

preceding is

02

n (n − 1) :

: which, when n is large enough, is clearly less than a quantity less than 1.

Now if we increase ◊ by 2π, the values of cos ◊ and of sin are unaltered; hence the values of these series are unaltered by adding 2 to 0.

504. All the theorems with respect to the sine and cosine that have been investigated from the definition by means of an angle, may be proved from the above series. A few cases may

be worked out.

505. Since every index in sin 0 is odd, .. sin (— 0) = − sin 0. Since every index in cos 0 is even, .. cos (- 0) = cos 0.

506. If measures an acute angle, 02 <1π2; :, à fortiori

Hence each of the above brackets is positive;

.. sin 00; but > 0-103; but < 0 - 103 +12005; and so on.

Hence, since the series beginning at any term has the same sign as that term, the difference between sin 0 and any number of its first terms is less than the first term omitted.

Similar propositions hold with respect to the cosine.