10. The series x sin + x2 sin 20+

+xn sin n + ...

A. If x 1, is divergent.

B. If x1, it is convergent and =

x sin 0 1-2x cos 0 + x2°

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

D.

If x=1, and 0/π = p/q where p and q are prime to one another, it = 1 cot 10 − 1 cosec 1⁄2 0 cos (n + 1) 0, an expression Ө which has (9+ 1) values if 20 Ρ is even, but q values if Ρ is odd.

=

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

D. If x= 1, and 0/T p/q where Ρ and q are prime to one another, it cosec 10 sin (n + 1) 0+1, an expression which has q + 1 values if q+p is even, but զ values if

=

nnxn

9+p is odd.

12. The series is convergent or divergent according as

n

is convergent if x has any value from 1 to + 1, excluding the first but including the last.

+ ...}.

25. 2 log, x-log, (x + 1) − loge (x − 1)

1

= 2

+

2x2-1

1

1

3 (2x2 - 1)31 5 (2x2 – 1)5

26. log, n=m {("/n − 1) − } (TM/n −

27.

then

+

28. If a, ß be the roots of px2 + qx + · loge (p-qx+rx2) = log. p + (a+ẞ) x − (a2 +

1

r

= 0,

+x cos 0+x2 cos 20+ x3 cos 30 + ...

- log. (1 - 2x cos 0 + x2) = x cos 0 + x2 cos 20+ 1x3 cos 30 + ...

+ + loge {1 - 2x cos (n − 1) + x2} = x + x2 + x3... - (x2 + x2 + x3 + ...).

1

sec 24-tan 24 cos 0

=

1+2 tan cos 0 + 2 tan2 + cos 20+ 2 tan3 & cos 30+

if & lies between 0 and 1π.

CHAPTER XIX.

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

§ 1. RELATIONS OF INEQUALITY.

481. The symbol 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 it is often convenient to drop the symbol for the unit and to write simply to denote the angle. The context will always show when ✪ means "the angle whose circular measure is 0," i.e. "the angle equal to 6 radians." Without such context, denotes simply a number. See Chapter II.

482. If 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 radians.

perp. BH <arc AH < tangent AT.

Divide each by the radius of the circle. Thus

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 in terms of those of 10, we obtain closer relations between the ratios and the circular measure. See next three articles.

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

485. We have

sin 02 sin cos 0=2 tan 10 cos2 10 = 2 tan 10 (1 − sin2 10). Now, by (1) Art. 483,

tan 100 and sin 100,

.. substituting in the above,