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10. The series x sin A + 2a sin 20 + ... + och sin n A + ...
A. If x>1, is divergent.
B. If x < 1, it is convergent and =

a sin 0

1- 2 cos 8+ C. If x=1, and 0/7 is incommensurable, it is finite but wholly indeterminate.

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D. If x=1, and 0/1 =p/q where p and q are prime to one another, it = cot 10 - 4 cosec 10 cos (n + }) , an expression

,
which has (q + 1) values if р

is
even,

but

9
values if

P

is odd. 11. The series 1 + x cos 0 + xc* cos 20 + ... + 20n cos no +... A. If x> 1, is divergent.

1- x cos 0 B. If x < 1, it is convergent and =

1 – 2x cos 0 + x2

+ C. If x=l, and 0/1 is incommensurable, it is finite but wholly indeterminate.

D. If x=1, and 0/1 =plq where p and q are prime to one another, it = 1 cosec 10 sin (n + 3) 0 + ], an expression which has 9+1 values if

9 + P

is
even,

but

9

values if

2

9 +p is odd.

12.

The series

nxn

is convergent or divergent according as

n

2

+

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+

(sima)*+

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n

x is or is not less than e-1.

sin? 0 n (n + 2) (sino 13. sec=l+n

2 12

2
n (12 + 2) (n + 4) /sin6) 3
3

2
14. secA=1 + tan? A +

n (n-- 2)

tano 0
2

2.4
n (n - 2) (n - 4)

6

n (n - 2)
= tan0 +
tan” A+

tan”–4 + ...

2.4 according as is less or greater than 17.

+

tan6 A + ...

2.4.

n

n

[blocks in formation]

or

2

+

15. Prove directly that the series

2c2 a3 1 + x +

12 3 is convergent for all finite values of x. 16. Prove directly that the series

X – La2 + }ac – 124 + . is convergent if x has any value from - 1 to +1, excluding the first but including the last.

1 1 1 Hence show that loge 2

2 3.4 5.6

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+

+

+...

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1

1

24. log. (x + 1) - log. (x - 1) 26+ + ...)

) – w – =

1 3x3

+

+

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25. 2 loge X – loge (x + 1) – loge (oc – 1)
1
1

1
2x2 - 1

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-2 (*

3 (2a* — 1)3 + 5 (222 - 1)*+...}.

26. loge n=m{(m/n-1) - (m/n - 1)2 + } (m/n - 1)3 – ...}.
27. If {loge (1 + x)} = a- jaga? + * - az.2 + ...

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28. If a, ß be the roots of pacé + qx + r = 0, loge (2-qx + raa) = loge p +(a+B) - (a? +B) + } (a®+) 2.-.. P

} B2x2 a3B3208 – 29. From the known expansion

1 - C2
Ž
1 – 2x cos A + 22

1 + cos 0 + x* cos 20 + ac cos 30 + ... show that - } loge (1 – 2. cos 0 + xc) = x cos 0 + 3x2 cos 20 + f2c3 cos 30 + ...

30. If n is odd, and p=7|n, then loge (1 – 2x cos 20 + aco) + loge (1 – 2.c cos 4$ + aca)

+ ... + loge {1 - 2x cos (n-1) + + = 2 + 12 + face... - (och + f2c2n + f2c3n + ...).

1
sec 20 - tan 20 cos O

= 1 + 2 tan o cos 0 + 2 tano $ cos 2A + 2 tancos 30 + ... if $ lies between 0 and 1.

1 32. Expand

in a series of cosines of multiples 1 + y cos 0

1 of A: and show that the constant term is

and the

(1-y)

2X" co-efficient of cos no is

where I is the larger root

(1 – y)' of 2* + 2xc/y + 1 = 0.

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31.

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J. T.

23

CHAPTER XIX.

RELATIONS BETWEEN THE CIRCULAR MEASURE AND THE TRIGONOMETRICAL RATIOS OF ANGLES.

§ 1. RELATIONS OF INEQUALITY.

481. The symbol 6c represents an angle containing & radians; i.e. an angle whose circular measure is 0, the unit of circular measurement being the radian. When we are referring to the angle Ac it is often convenient to drop the symbol for the unit and to write o simply to denote the angle. The context will always show when 0 means “the angle whose circular measure is 0," i.e. “the angle equal to 0 radians.” Without such context, O denotes simply a number. See Chapter II.

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482. If 0 represents the circular measure of an angle, we know, by Art. 51, that 0 = the ratio of the arc-subtended by the angle at the centre of a circle—to the radius.

We shall now apply the proposition of Art. 59 to a comparison between the circular measure and the trigonometrical ratios of an angle.

483. The circular measure of an acute angle lies in magnitude between the sine and the tangent of the angle.

At the centre o of a circle, let the acute angle AOH be equal to A radians.

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Draw HB, AT at right-angles to 01.
Then, by Art. 59,

perp. BH < arc AH < tangent AT.
Divide each by the radius of the circle. Thus

BH AH AT

OH ОА OA
i.e. sin 0 <A <tan 0..........

.(1).
Also we have
cos 0 < 1....

(2). These are the first approximations to the values of the ratios of an acute angle in terms of its circular measure.

The student must particularly observe that in the above inequalities A, standing by itself, means a number ; but 0, standing after sin, cos, or tan, means an angle. The inequalities are, of course, relations between mere numbers or ratios.

484. By expressing the ratios of 8 in terms of those of 10, we obtain closer relations between the ratios and the circular

See next three articles. We confine ourselves in this section (except in Arts. 490 and 493) to acute angles.

measure.

485. We have sin 0 = 2 sin 10 cos 10 = 2 tan 10 cosa 10 = 2 tan 10 (1 - sin? 10). Now, by (1) Art. 483,

tan 10 > 18 and sin 10 < 10, .. substituting in the above,

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