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, COSINES, &c., is a table by means of which the natural sine, cosine, tangent, or cotangent of any arc, or angle, may be found.

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

TABLE OF LOGARITHMIC SINES.

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. This value of the radius is taken simply for convenience in making the table, its logarithm being 10.

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

Any logarithmic function of an arc, or angle, may be found by multiplying 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 mantissas is 242 millionths; this, divided by 60, the number of seconds in one minute, gives 4.03 millionths, 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.

But,

The arc, or angle, obtained by taking the degrees from the top of the page and the minutes from the left-hand column, is the complement of that obtained by taking the degrees from the bottom of the page, and the minutes from the right-hand column on the same horizontal line. by definition, the cosine and the cotangent of an arc, or angle, are, respectively, the sine and the tangent of the complement of that arc, or angle (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.

1

USE OF THE TABLE.

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

34. If the arc, or angle, is less than 45°, look 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. Thus,

If the arc, or angle, is 45° or more, look for the degrees at the bottom of the page, and for the minutes in the right-hand column; then follow the corresponding horizontal line backward 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 or angle which is expressed in degrees, minutes, and seconds.

35. Find the logarithm corresponding to the degrees and minutes as before; then multiply the corresponding number taken from the column headed "D," which is millionths, 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".

The same rule is followed for decimal parts, as in Art. 12.

2. Find the logarithmic cosine of 53° 40′ 40′′.

If the arc or angle is greater than 90°, find the required function of its supplement (Arts. 26 and 28).

3. Find the logarithmic tangent of 118° 18' 25".

4. Find the logarithmic sine of 32° 18' 35.

Ans. 9.727945.

5. Find the logarithmic cosine of 95° 18' 24".

Ans. 8.966080.

6. Find the logarithmic cotangent of 125° 23′ 50′′.

Ans. 9.851619.

To find the arc or angle corresponding to any logarithmic function.

36. This is done by reversing the preceding rule:

Look in the proper column of the table for the given logarithm; if it is found there, the degrees are to be taken from the top or bottom, and the minutes from the left or right hand column, as the case may be. If the given logarithm is not found in the table, then find the next less logarithm, and take from the table the corresponding degrees and minutes, and set them aside. Subtract the logarithm found in the table from the given logarithm, and divide the remainder by the corresponding tabular difference. The quotient will be seconds, which must be added to the degrees and minutes set aside in the case of a sine or tangent, and subtracted in the case of a cosine or a cotangent.

Examples.

1. Find the arc or angle corresponding to the logarithmic sine 9.422248.