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76. Use of Tables of Logarithmic Trigonometric Functions. Since the sines, cosines, tangents, etc., of angles are numbers, we may use the logarithms of these numbers in numerical calculations in which trigonometric functions are involved; and these logarithms are in practice much more useful than the numbers themselves, as with their assistance we are able to abbreviate greatly our calculations; this is especially the case, as we shall see hereafter, in the solution of triangles. In order to avoid the trouble of referring twice to tables - first to the table of natural functions for the value of the function, and then to a table of logarithms for the logarithm of that function- the logarithms of the trigonometric functions have been calculated and arranged in tables, forming tables of the logarithms of the sines, logarithms of the cosines, etc.; these tables are called tables of logarithmic sines, logarithmic cosines, etc.

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Since the sines and cosines of all angles and the tangents of angles less than 45° are less than unity, the logarithms of these functions are negative. To avoid the inconvenience of using negative characteristics, 10 is added to the logarithms of all the functions before they are entered in the table. The logarithms so increased are called the tabular logarithms of the sine, cosine, etc. Thus, the tabular logarithmic sine of 30° is

10+ log sin 30° = 10 + log=10- log 2 = 9.6989700.

In calculations we have to remember and allow for this increase of the true logarithms. When the value of any one of the tabular logarithms is given, we must take away 10 from it to obtain the true value of the logarithm. Thus in the tables we find

log sin 31° 15' = : 9.7149776.

Therefore the true value of the logarithm of the sine of 31° 15' is 9.7149776 - 10

=

1.7149776.

Similarly with the logarithms of other functions.

NOTE. English authors usually denote these tabular logarithms by the letter L Thus, L sin A denotes the tabular logarithm of the sine of A.

French authors use the logarithms of the tables diminished by 10. Thus,

log sin A = 1.8598213, instead of 9.8598213.

The Tables contain the tabular logs of the functions of all angles in the first quadrant at intervals of 1'; and from these the logarithmic functions of all other angles can be found.*

Since every angle between 45° and 90° is the complement of another angle between 45° and 0°, every sine, tangent, etc., of an angle less than 45° is the cosine, cotangent, etc., of another angle greater than 45° (Art. 16). Hence the degrees at the top of the tables are generally marked from 0° to 45°, and those at the bottom from 45° to 90°, while the minutes are marked both in the first column at the left, and in the last column at the right. Every number therefore in each column, except those marked diff., stands for two functions—the one named at the top of the column, and the complemental function named at the bottom of the column. In looking for a function of an angle, if it be less than 45°, the degrees are found at the top, and the minutes at the left-hand side. If greater than 45°, the degrees are found at the foot, and the minutes at the right-hand side.

On page 113 is a specimen page of Mathematical Tables. It gives the tabular logarithmic functions of all angles between 38° and 39°, and also of those between 51° and 52°, both inclusive, at intervals of 1'. The names of the functions for 38° are printed at the top of the page, and those for 51° at the foot. The column of minutes for 38° is on the left, that for 51° is on the right.

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* Many tables are calculated for angles at intervals of 10".

77. To find the Logarithmic Sine of a Given Angle.

Find log sin 38° 52′ 46′′.

We have from page 113

Let d =

log sin 38° 53' = 9.7977775

log sin 38° 52' = 9.7976208

diff. for 1'= .0001567

diff. for 46", and assuming that the change in the log sine is proportional to the change in the angle, we have

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... log sin 38° 52′ 46′′= 9.7976208+.0001201

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78. To find the Logarithmic Cosine of a Given Angle. Find log cos 83° 27' 23", having given from the table

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.. log cos 83° 27′ 23′′= 9.0571723 — .0004223

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79. To find the Angle whose Logarithmic Sine is Given. Find the angle whose log sine is 8.8785940, having given from the table

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diff. between 4° 20' and required angle; then

.0016639.0003086 :: 60: d.

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