Elements of Geometry and Trigonometry

These functions of an arc have been defined on the supposition that the radius of the arc is equal to 1; in this case, they may also be considered as functions of the angle which the arc measures.

Thus, PM, NM, AT, and BT', are respectively the sine, cosine, tangent, and cotangent of the angle AOM, as well as of the arc AM.

30. It is often convenient to use some other radius than 1; in such case, the functions of the arc, to the radius 1, may be reduced to corresponding functions, to the radius R. Let AOM represent any angle, AM an arc described from O as a centre with the radius 1, PM its sine; A'M' an arc described from 0 as a centre, with any raradius R, and and P'M' its sine. Then, because OPM and OP'M' are similar triangles, we shall have,

PA P'A'

OM: PM :: OM' : P'M', or, 1: PM::R: P'M';

whence,

PM =

P'M'
R

and, P'M' = PM × R;

and similarly for each of the other functions.

That is, any function of an arc whose radius is 1, is equal to the corresponding function of an arc whose radius is R, divided by that radius. Also, any function of an arc whose radius is R, is equal to the corresponding func tion of an arc whose radius is 1, multiplied by the ra

dius R.

By making these changes in any formula, the formula will be rendered homogeneous.

TABLE OF NATURAL SINES.

31. A NATURAL SINE, COSINE, TANGENT, OR COTANGENT, is the sine, cosine, tangent, or cotangent of an arc whose radius is 1.

A TABLE OF NATURAL SINES is a table by means of which the natural sine, cosine, tangent, or sotangent of any arc, may be found.

Such a table might be used for all the purposes of trigonometrical computation, but it is found more convenient to employ a table of logarithmic sines, as explained in the next article.

32. A LOGARITHMIC SINE, COSINE, TANGENT, or COTANGENT is the logarithm of the sine, cosine, tangent, or cotangent of an arc whose radius is 10,000,000,000.

A TABLE OF LOGARITHMIC SINES is a table from which the logarithmic sine, cosine, tangent, or cotangent of any arc may be found.

The logarithm of the tabular radius is 10.

Any logarithmic function of an arc may be found by maltiplying the corresponding natural function by 10,000,000,000 (Art. 30), and then taking the logarithm of the result; or more simply, by taking the logarithm of the corresponding natural function, and then adding 10 to the result (Art. 5).

33. In the table appended, the logarithmic functions are given for every minute from 0° up to 90°. In addition, their rates of change for each second, are given in the column headed "D."

The method of computing the numbers in the column headed "D," will be understood from a single example. The

logarithmic sines of 27° 34', and of 27° 35', are, respectively, 9.665375 and 9.665617. The difference between their mantıssas is 242; this, divided by 60, the number of seconds in one minute, gives 4.03, which is the change in the mantissa for 1", between the limits 27° 34' and 27° 35'.

For the sine and cosine, there are separate columns of differences, which are written to the right of the respective columns; but for the tangent and cotangent, there is but a single column of differences, which is written between them. The logarithm of the tangent increases, just as fast as that of the cotangent decreases, and the reverse, their sum being always equal to 20. The reason of this is, that the product of the tangent and cotangent is always equal to the square of the radius; hence, the sum of their logarithms must always be equal to twice the logarithm of the radius, or 20.

The angle obtained by taking the degrees from the top of the page, and the minutes from any line on the left hand of the page, is the complement of that obtained by taking the degrees from the bottom of the page, and the minutes from the same line on the right hand of the page. But, by definition, the cosine and the cotangent of an arc are, respectively, the sine and the tangent of the complement of that arc (Arts. 26 and 28): hence, the columns designated sine and tang, at the top of the page, are designated cosine and cotang at the bottom.

USE OF THE TABLE.

To find the logarithmic functions of an arc which is expressed in degrees and minutes.

34. If the arc is less than 45°, 100k for the degrees at the top of the page, and for the minutes in the left hand column; then follow the corresponding horizontal line till you

come to the column designated at the top by sine, cosine,

tang, or cotang, as the case may be; the number there found is the logarithm required.

log sin 19° 55'

log tan 19° 55'

Thus,

45°, look for the degrees at for the minutes in the right

If the angle is greater than the bottom of the page, and hand column; then follow the corresponding horizontal line backwards till you come to the column designated at the bottom by sine, cosine, tang, or cotang, as the case may be; the number there found is the logarithm required.

Thus,

To find the logarithmic functions of an arc which is expressed in degrees, minutes, and seconds.

35. Find the logarithm corresponding to the degrees and minutes as hefore; then multiply the corresponding number taken from the column headed "D," by the number of seconds, and add the product to the preceding result, for the sine or tangent, and subtract it therefrom for the cosine or cotangent.

EXAMPLES.

1. Find the logarithmic sine of 40° 26′ 28′′.