TABLE OF THE MOST USEFUL ANALYTICAL VALUES OF
SIN. 8, COS. 8, TAN. 8.

15. sin. (60°+4)-sin. (60°-4) 30. cos. (60°+4)+cos. (60°-4)

2

From certain properties of the circles to be discussed in another volume, other important trigonometrical formulæ, may be deduced, furnishing us with more expeditious means of determining, numerically, the values of some of the trigonometrical lines, and ratios, all of which will occur in their order.

To develop sin. x and cos. x in a series ascending by the powers of x.

The series for sin. x must vanish when x=0, and therefore no term in the series can be independent of x, nor can the even powers of x occur in the series; for if we suppose

then sin. (x)=—a ̧x+a ̧x2— a ̧x2+a ̧x*—a ̧x3+..
but sin. (x)=-sin. x
x)=—

Again, the series for cos. x must = 1 when x = 0, 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 ̧x*+

then cos. (x)=1—a ̧x+a ̧x2— ɑ ̧x3+a ̧xa—

2

4

3

4

5

Hence cos. +sin. x=1+a, x+a2x2+a ̧xa+a ̧x* +ɑ ̧x° cos. sin. x=1—a ̧x+a,x2 — α ̧Ã3 +α ̧x*—a ̧x°+ Now in equation (3) write x + h for x, and we have

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. h—sin. x sin. h +sin. x cos. h+cos. x sin. h

cos. h (cos. x+sin. x)+sin. h (cos. x-sin. x) =(1+a,h2+a ̧h'+...).(1+a ̧x+a2x2+a ̧x3+ .....)

2

3

+ (a1h+a ̧h3+a ̧h3+.....) (1—a ̧x+a2x2—α ̧x3+..) =1+a1x+a ̧x2+ɑ ̧x3 +

2

2

3

Comparing equations (5) and (6) we have

(6)

and equating the coefficients of the terms involving the same

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

23+

1.2.3

1.2.3.4.5

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.

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,

1o. 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,