Dividing Equation (A'), first by Equation (4), and then by Equation (3), member by member, we have,

Taking the reciprocals of both members of the last twe

67. If Formulas (A) and (B) be first added, member to member, and then subtracted, and the same operations be performed upon (C) and (D), we shall obtain,

and then substitute in the above formulas, we obtain,

From Formulas (L) and (K), by division, we obtain,

That is, the sum of the sines of two arcs is to their difference, as the tangent of one half the sum of the arcs is to the tangent of one half their difference.

all of which give proportions analogous to that deduced from Formula (1).

Since the second members of (6) and (4) are the same, we have,

That is, the sine of the difference of two arcs is to the difference of the sines as the sum of the sines to the sine of the sum.

All of the preceding formulas may be made homogeneous in terms of R, R being any radius, as explained in Art. 30; or, we may simply introduce R, as a factor, into each term as many times as may be necessary to render all of its terms of the same degree.

METHOD OF COMPUTING A TABLE OF NATURAL SINES.

68. Since the length of the semi-circumference of a circle whose radius is 1, is equal to the number 3.14159265 f we divide this number by 10800, the number of minutes n 180°, the quotient, .0002908882..., will be the length of the arc of one minute; and since this arc is so small that it does not differ materially from its sine or tangent, this may be placed in the table as the sine of one minute. Formula (3) of Table II., gives,

Having thus determined, to a tion, the sine and cosine of one formula of Art. 67, and put it under the form,

near degree of approxima minute, we take the first

thus

sin 4' 2 sin 3' cos 1'- sin 2' = .0011635526 ...

obtaining the sine of every number of degrees and minutes from 1' to 45°.

The cosines of the corresponding arcs may be computed by means of Equation (1).

Having found the sines and cosines of arcs less than 45o, those of the arcs between 45° and 90°, may be deduced, by considering that the sine of an arc is equal to the cosine of its complement, and the cosine equal to the sine of the complement. Thus,

sin 50° = sin (90° — 40°) = cos 40°, cos 50° sin 40°,

in which the second members are known from the previous computations.

To find the tangent of any arc, divide its sine by its cosine. To find the cotangent, take the reciprocal of the corresponding tangent.

As the accuracy of the calculation of the sine of any arc, by the above method, depends upon the accuracy of each previous calculation, it would be well to verify the work, by calculating the sines of the degrees separately (after having found the sines of one and two degrees), by the last proportion of Art. 67. Thus,