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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
i.e. <1
2 16

512'

A2 64
à fortiori < 1
2

..(5). 16

2

+

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487. We have, by division,

2 tan 10
tan A =
1 - 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

4"
.. à fortiori, tan 6 > sum of any number of these terins.

<

+

+

+

+...

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.
Hence, dividing each by sin 6 and by tan 6, both

0
1, sec 0; and cos ,
sin 0

1,

tan ? are in ascending order of magnitude,

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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.

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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
Hence, limit of ratio = limit of ratio-

sin •

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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.
.. limit of sin ko - k is 0.

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
A x 0.

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sin Ao

A

a sin Ac

T
whose limit is not unity.
0 x 180°

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
any

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

.

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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
2 4 8 2n

A ө 0
For sin A = 2 sin

COS

22 sin COS COS 2 2

2 22 and so on.

0
Thus sin A = 2^ sin COS COS

COS
an 2 22 2n
A

sin 0/2
But

2n sin .

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
2

2

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.

COS

COS

.

.

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(1 - 0) (1 m),

494. To show that sin 0 >0 - 303, if o denotes an acute angle.
By Art. 486, cos 6 > 1-402.
0 0

02
.. cos
2
4

23 25

A2 A2
à fortiori > 1

23 25
Proceeding in this way,
0

1

1
>
2 4 2n 2 14 42 4n

A2 1-1/4"
8 1-1/4

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+

>

COS

COS

COS

82 A2 >1

6 6.4n?

02
à fortiori >1-

6
Now as in the last article,
sin A sin 0/2" A 0

0
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%.

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=

>

)

2 tan 10
496.
We have

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
tan A
tan 10 A2 4212

A2,
o 10 4 4
tando 1 02 1 022

1 A2
44 42 4

4n tan 10 tan 10 1 82

1 162 2

1 42 n 10 42'4 447

421 and so on.

+

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tan 10

+

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24

+..

.

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10

>

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10

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