Again, the series for cos. x must = 1 when x = O, and therefore the series must contain a term independent of x, and it must be 1; also the series can contain no odd powers of x, for if we suppose

-

cos. x=1+a,x+a,x2+a ̧x2+a ̧x1+ then cos. (-)=1—a ̧x+a,x2————α ̧x3+α ̧x*———— but cos. (x)=cos. x

=1+a ̧x+a ̧x2+a ̧x3+a ̧x*+

...

(2) - (3)

(4)

2

3

Hence cos. +sin. x=1+a,x+a2x2+a ̧xa+a ̧x*+açx® cos. sin. x=1—a‚x+a ̧x2— a ̧x3+a ̧x'—a ̧x3+ Now in equation (3) write x + h for x, and we have

x

4

cos.(x+h)+sin(x+h)=1+a,(x+h)+a ̧(x+h)2+a ̧(x+h)3+(5) but cos. (x+h)+sin. (x+h)=cos. x cos. hsin. x sin. h

+sin. x cos. h+cos. x sin. h`

=cos. h (cos. x+sin. a)+sin. h (cos. x—sin. x)
=(1+a„h2+a ̧h*+.....). (1+a ̧x+a2x2+a ̧x3+ .....)
(a2h+a ̧h3+a ̧h3+...) (1—a ̧x+a2x3—a ̧x3+..)
=1+a,x+a,x2+α ̧x3 +

+

3

2

5

Comparing equations (5) and (6) we have

(6)

+ ash3 +
+

+

and equating the coefficients of the terms involving the same powers of x and h, we have

and we have only to determine the value of a,. To effect this, we have

Now the value of x may be assumed so small that the series in the parenthesis, and sin. x, shall differ from 1 and x respectively, by less than any assignable quantities; hence ultimately

x=a,x, and therefore a,=1; whence

To develop tan. x and cot. x in a series ascending by the

powers of x.

The development may be obtained from those of sin. x and

cos. x, already found.

a 3

Hence, equating the coefficients of the like terms, we have

We shall here repeat the enunciations of the two propositions established in Chapter I.

PROPOSITION I.

In any right-angled plane triangle,

Io. The ratio which the side opposite to one of the acute angles has to the hypothenuse, is the sine of that angle.

2o. The ratio which the side adjacent to one of the acute angles has to the hypothenuse, is the cosine of that angle.

3°. The ratio which the side opposite to one of the acute angles has to the side adjacent to that angle, is the tangent of that angle.

Thus, in any right-angled triangle ABC,

In any plane triangle, the sides are to each other as the sines of the angles opposite to them.

We shall, frequently in treating of triangles, make use of the following notation; denoting the angles of the triangle by the large letters at the angular points, and the sides of the triangle opposite to these angles, by the corrcsponding small letters.

Thus, in the triangle ABC, we shall denote the angles, BAC, CBA, BCA, by the letters, A, B, C, respectively, and the sides BC, AC, AB, by the letters a, b, c, respectively.

According to this, we shall have, by the proposition,

PROPOSITION III.

In any plane triangle, the sum of any two sides, is to their difference, as the tangent of half the sum of the angles opposite to them, is to the tangent of half their difference.

Let ABC be any plane triangle, then, by Proposition II