EXERCISES.

Find the pairs of values of the angle a from the following values

18. Cases when the Function is very Small or Great.

When the angle of which we are to find the functions approaches to zero, the logarithms of the sine, tangent, and cotangent vary so rapidly that their values to five figures cannot be readily interpolated. The same remark applies to the cosine, cotangent, and tangent of angles near 90° or 270°. The mode of proceeding in these cases will depend upon circumstances.

In the use of five-place logarithms, there is little advantage in carrying the computations beyond tenths of minutes, though the hundredths may be found when the tangent or cotangent is given. Where greater accuracy than this is required, six- or seven-place tables must be used.

If the angles are only carried to tenths of minutes, there is no necessity for taking out the sine, tangent, or cotangent to more than four decimals when the angle is less than 3°, and three decimals suffice for angles less than 30'. The reason is that this number of decimals then suffice to distinguish each tenth of minute.

When the decimals are thus curtailed, an expert computer will be able to perform the multiplication and division for the tenths of minutes mentally. If, however, this is inconvenient, the following rule may be applied.

To find the log sine or log tangent of an angle less than 2° to four places of decimals:

RULE. Enter the table of logarithms of numbers with the value

of the angle expressed in minutes and tenths, and take out the logarithm.

To this logarithm add the quantity 6.4637.

The sum will be the log sine, and the log tangent may be assumed to have the same value.

Example 1. To find log sin 1° 22′.6.

This rule is founded on the theorem that the sines and tangents of very small arcs may be regarded as equal to the arcs themselves. Since, in using the trigonometric functions, the radius of the circle is taken as unity, an arc must be expressed in terms of the unit radius when it is to be used in place of its sine or tangent. Now, it is shown in trigonometry that the unit radius is equal to 57°.2958 or 3437.747 or 206 264".8. Hence we must divide the number of angular units in the angle by the corresponding one of these coefficients to obtain the length of the corresponding arcs in unit radius. Now,

log 3437.747 = 3.5363
co-log....... 6.4637

which may be added instead of subtracting the logarithm.

To find the cosine of an angle very near 90°, we find the sine of its complement, which will then be a very small angle, positive or negative.

If an angle corresponding to a given sine or tangent is required, the rule is:

From the given log sine or tangent subtract 6.4637 or add 3.5363. The result is the logarithm of the number of minutes.

Of course this rule applies only to angles less than 2°, in the value of which only tenths of minutes are required.

Find a from:

1. log sin a = 7.2243;

EXERCISES.

2.8816;

3. log tan a =
4. log cos a = 6.9218.

2. log cot a = 2.8816; When the small angle is given in seconds. Although the computer may take out his angles to tenths of minutes, cases often arise in which a small angle is given in seconds, or degrees, minutes, and seconds, and in which the trigonometric function is required to five decimals. In this case the preceding method may not always give accurate results, because the arc and its sine or tangent may differ by a greater amount than the error we can admit in the computation. Table II. is framed to meet this case. The following are the quantities given:

In the second column: The argument, in degrees and minutes, as already explained for Table III.

In the first column: This argument reduced to seconds. From this column the number of seconds in an arc of less than 2°, given in degrees, minutes, and seconds, may be found at sight.

Example. How many seconds in 1° 28′ 39′′? In the table, before 1° 28', we find 5280", which being increased by 39" gives 5319", the number required.

Col. 3. The logarithm of the sine of the angle. This is the same as in Table III.

Col. 4. The value of log sine minus log arc; that is, the difference between the logarithm of the sine and the logarithm of the number of seconds in the angle.

Col. 5. The same quantity for the tangent.

Cols. 6 and 7. The complements of the preceding logarithms, distinguished by accents.

The use of the tables is as follows.

To find the sine or tangent of an angle less than 2°:

Express the angle in seconds by the first two columns of the table. Write down the logarithm in column S or column T, according as the sine or a tangent is required.

Find from Table I. the logarithm of the number of seconds. Adding this logarithm to S or T, the sum will be the log sine or log tangent.

Example. Find log sin 1° 2′ 47′′.9.

1° 2' 47".9

S, 4.685 55

3767".9; log, 3.576 10

log sin 1° 2′ 47′′.9, 8.261 65

To find the arc corresponding to a given sine or tangent:

Find in the column L. sin. the quantity next greater or next smaller than the given logarithm.

Take the corresponding value of S' or T'' according as the given function is a sine or tangent, and add it to the given function.

The sum is the logarithm of the number of seconds in the required angle.

Example. Given log tan x = 8.401 25, to find x.

log tan x, 8.401 25

T',

5.314 33

When the cosine or cotangent of an angle near 90° or 270° is required, we take its difference from 90° or 270°, and find the complementary function by the above rules.

Remark. The use of the logarithms of the trigonometric functions is so much more extensive than that of the functions themselves that the prefix "log" is generally omitted before the designation of the logarithmic function, where no ambiguity will result from the omission.

TABLE V.

NATURAL SINES AND COSINES.

19. This table gives the actual numerical values of the sine and cosine for each minute of the quadrant.

To find the sine or cosine corresponding to a given angle less than 45°, we find the degrees at the top of a pair of columns and the minutes on the left.

In the two columns under the degrees and in the line of minutes we find first the sine and then the cosine, as shown at the head of the column.

A decimal point precedes the first printed figure in all cases, except where the printed value of the function is unity.

If the given angle is between 45° and 90°, find the degrees at the bottom and the minutes at the right.

Of the two numbers above the degrees, the right-hand one is the sine and the left-hand one the cosine.

For angles greater than 90° the functions are to be found according to the precepts given in the case of the logarithms of the sines and tangents.