by verifying the following values of Sin. 4, Cos. 4, Tan. 6, which are extracted from the large work of Cagnoli.

TABLE OF THE MOST USEFUL ANALYTICAL VALUES
OF SIN. 6, COS. e, TAN. 0.

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

43. tan.(45°+5)-tan. (45°—

2

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

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

....

0, a3 = 0

Hence cos. x + sin. x = 1 + α1x + ɑ2x2 + α3Ñ3 + ɑ4x1 + ɑ5x5 +
COS. x- sin. x = 1-α1x + α2x2 — ɑzx3 + α4Xa . -α5x5 +
Now in equation (3) write x + h for x, and we have

but cos.

....(5)

cos. (x+h)+sin. (x+h)=1+a,(x+h)+a2(x+h)2+a3(x+h)3+ ·(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+a2h2+ah1+........). (1+a ̧x+а„x2+α3ñ3+.....) +(ah+ah3+a ̧h3+...) (1—α ̧x+α2x2—ɑ3x3+..) =1+a1x+ a2x2 + азx3 +.

+a,h-a,2xh+a1ax2h +
+ a2h2+a,a,xh2 + .
+ aзh3 +
+

Comparing equations (5) and (6) we have

1+a1x+ α2x2 + α323 +

+ah+2axh+3a3x2h+

+ a2h2 +3a,xh2 + .

+ agh3 +
+....

.....

...

(6)

and equating the coefficients of the terms involving the same powers of x and

h, 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 a respectively, by less than any assignable quantities; hence ultimately

xax, 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.

and the series will therefore be of the form x+а ̧Ã3+а5x2+α-x2+

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

CHAPTER III.

FORMULE FOR THE SOLUTION OF TRIANGLES.

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

PROP. I.

In any right-angled plane triangle,

I. 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, henceforth, in treating of triangles, make use of the following notation. We shall denote 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 corresponding 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,

a

..(B)

B

PROP. 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

To express the cosine of an angle of a plane triangle in terms of the sides of

the triangle.

Let ABC be a triangle; A, B, C, the three angles;

a, b, c, the corresponding sides.

1. Let the proposed angle (A) be acute.

From C draw CD perpendicular to AB, the base of the triangle.