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

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

=

which has (9+ 1) values if p is even, but q values if

11. The series 1 + x cos 0 + x2 cos 20+

A. If x 1, is divergent.

>

B. If <1, it is convergent and =

p is odd.

+an cos no + ...

1 - x cos 0

1- 2x cos 0 + x2·

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

D.

If x=

1, and 0/= p/q where p and q are prime to one another, it cosec 10 sin (n + 1) +1, an expression which has = 1⁄2 0 is even, but 9 values if q+p is odd.

q + 1 values if q +p

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

=

22. log, /10-x+(--]

2n

gn

2n+1

26.

27.

log, n = m {("/n − 1 ) − 1 (~/n − 1 )2 + } (~/n − 1 )3 — ... }.

If {loge (1 + x)}2 = x2 − ¦ ¤ ̧ñ3 + ‡ α¿x1 — 2 α ̧Ñ3 + .......

1

r

then

28.

If a, ẞ be the roots of px2 + qx + r = 0,

loge (p − qx + rx2) = loge p + (a+ẞ) x − 1 (a2 + ẞ2) x2 + 1⁄2 (a3 +ß3) x3 –

– 1⁄2 log. (1 − 2x cos 0 + x2) = x cos 0 + x2 cos 20+ x3 cos 30+

30.

= 1 + 2 tan cos 0 + 2 tan2 cos 29 + 2 tan3 & cos 30+...

CHAPTER XIX.

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

§ 1. RELATIONS OF INEQUALITY.

0

481. The symbol c 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 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 of a circle, let the acute angle AOH be equal to radians.

H

B

A

Draw HB, AT at right-angles to OA.

Then, by Art. 59,

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

tan 100 and sin 100,

.. substituting in the above,