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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 log

arithms 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.

Since the sines and cosines of all angles and the tangents of angles less than 45o 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 tabulur logarithms of the sine, cosine, etc. Thus, the tabular logarithmic sine of 30° is 10 + log sin 30°

10 – log 2 9.6989700.

= 10

+ lo

log2=

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 logarithin 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 45o and 0°, every sine, tangent, etc., of an angle less than 45o 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.

Thus we find

log sin 38° 29' = 9.7939907.
log cos 38° 45' = 9.8920303.
log tan 51° 18' : 10.0962856.

* 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

log sin 38° 53' = 9.7977775
log sin 38° 52' = 9.7976208

diff. for 1'= .0001567

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

60 : 46 ::.0001567 : d. 46 x .0001567

1.0001201.

60
.. log sin 38° 52' 46"= 9.7976208 +.0001201

= 9.7977409.

.. d.

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

log cos 83° 27'= 9.0571723
log cos 83° 28'= 9.0560706

diff. for 1'= .0011017

Let d= decrease of log cosine for 23"; then

60:23::.0011017 : d.

23 x .0011017 .:: d

= .0004223, nearly.

60
.. log cos 83° 27' 23'= 9.0571723 – .0004223

= 9.0567500.

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log sin 6° 33'= 9.0571723,
log sin 6° 32=9.0560706;
log sin 6° 32' 37".

find

Ans. 9.05675.

987 988 988 989 990

991

1610 9.8946317

992

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2598

1600 9.8972312

996 998 9.8948457 43 998 998

Sine. Diff. Tang. Diff. Cotang Diff. o 9.7893420

1616
9.8928098

10.1071902 1 9.7895036

2604 9.8930702

10.1069298 1616

2604 2 9.7896652

1614
9.8933306

2603

10.1066694 3 9.7898266

1614
9.8935909

2602

10.1064091 4 9.78998801

10.1061489

9.8938511 5 9.7901493 1611 1613

2603
9.8941114

2601
10.1058886

990
6
9.7903104

1611
9.8943715

2602

10.1056285 7 9.7904715

10.1053683

2бо 8 9.7906325 9.8948918

10.1051082

992 1608

2601 9 9.7907933

1608
9.8951519

2600
10.1048481

992
9.7909541

9.8954119
1607

2600
10.1045881

993
II
9.7911148

1606
9.8956719

2600

10.1043281 I2

994 9.7912754

9.8959319 1605

10.1040681

2599 13 9.7914359

995
1604
9.8961918

10.1038082 14 | 9.7915963

2599
9.8964517
1603

995
10.1035483

2599 15 | 9.7917566 9.8967116

995 1602

10.1032384 2598

997 16 9.7919168

1601
9.8969714

10.1030286 17 9.7920769

10.1027688 18 | 9.7922369 9.8974910

2598

10.1025090 1599 19 9.7923968

9.8977507

2597

10.1022493 20 9.7925566

1598
9.8980104

2597

10.1019896

2596 1597

1000 21 | 9.7927163 9.8982700

10.1017300
22 9.7928760
1597

999
2596
9.8985296

10.1014704
1595
2596

1001 23 9.7930355

9.8987892

10.1012108
1594
2595

1001 24 9.7931949 9.8990487

10.1009513 1594

ΙΟΟΙ 259.7933543

2595
9.8993082

10.1006918
1592
2595

1003 26 9.7935135

9.8995677

10.1004323 1592

1002 27 | 9.7936727

2594 9.8998271

10.1001729 28 9.7938317

1590
9.9000865
2594

1004

10.0999135
1590
2594

1004 29 9.7939907

1589
9.9003459

10.0996541
2593

1004 30 9.7941496

1587
9.9006052

10.0993948
2593

1005 31 9.7943083 9.9008645

10.0991355 32 9.7944670

1006
2592
9.9011237

10.0988763
2593

1007 33 9.7946256

10.0986170
9.9013830
1585

2592 34 | 9.7947841

1007 9.9016422

10.0983578 2591

1007 35 9.7949425

9.9019013 36 9.7951008

10.0978396

1009 37 9.7952590

9.9024195

10.0975805

2591 38 9.7954171

IOIO
10.0973214
2590

IOIO 39 9.7955751

9.9029376

10.0970624 40 9.7957330 1579 2590

IOIO

10.0968034
9.9031966
1579
2589

IOII 41 9.7958909

9.9034555

10.0965445
1577
2589

IOI2 42 9.7960486

9.9037144

10.0962856
1576
2589

1013 43 9.7962062

9.9039733

10.0960267

1013 44 9.7963638

9.9042321

10.0957679 1574

1013 45 | 9.7965212

9.9044910 1574

10.0955090

2587 46 9.7966786 9.9047497 47 | 9.7968359 1573

1015

10.0949915
| 9.9050085
1571
2587

Io16
48 9.7969930 9.9052672
1571

10.0947328 2587

1016 49 9.7971501

1570
9.9055259

10.0944741

2586 50 9.7973071 9.9057845

10.0942155

1018 51 9.7974640

1568
9.9060431

10.0939569

I017 52 9.7976208

1567
9.9063017

10.0936983

1019 53 9.7977775

10.0934397 54

I019 9.7979341

9.9068188

2585

10.0931812 55 9.7980906

1020
2585
1564
9.9070773

2584
10.0929227

1020 56 9.7982470

9.9073357

10.0926643 57 9.7984034

2584 9.9075941

10.0924059

2584 58 9.7985596

1021 9.9078525

10.0921475 59 9.7987158

I022
2584
9.9081109
1560

10.0918891 60 9.7988718 9.9083692

2583
10.0916308

1023
Cosine. Diff. Cotang: Diff. Tang

Diff.

Cosine. 9.8965321 60 9.8964334 59 9.8963346 58 9.8962358 57 9.8961369 56 9.8960379 55 9.8959389 54 9.8958398 53 9.8957406 52 9.8956414 51 9.8955422 50 9.8954429 49 9.8953435 48 9.895244047 9.8951445 | 46 9.8950450 45 9.8949453 44 9.8947459 42 9.8946461 41 9.8945463 40 9.8944463 39 9.8943464 38 9.8942463 37 9.8941462 36 9.8940461 35 9.8939458 34 9.8938456 33 9.8937452 32 9.8936448 31 9.8935444 30 9.8934439 29 9.8933433 28 9.8932426 27 9.8931419 26 9.8930412 25 9.8929404 24 9.8928395 23 9.8927385 9.8926375 9.8925365 20 9.8924354

19 9.8923342 18 9.8922329 17 9.8921316 16 9.8920303 15 9.8919289 14 9.8918274 13 9.8917258

1587 1586

10.0980987 1008

1584
1583
1582 9.9021604
1581
1580 9.9026786

2591 2591

22 21

1576

2588 2589

1014

2588 10.0952503

12 II

[ocr errors]

1016 9.8916242

[blocks in formation]
[blocks in formation]

9.8915226 10 9.8914208 9.8913191 9.8912172 9.8911153 6 9.8910133 5 9.8909113

4 9.8908092 3 9.8907071 9.8906049 9.8905026

Sine.
51 Deg.

2

I o

[blocks in formation]

4. Given log cos 44° 35' 20"= 9.8525789,

log cos 44° 35' 30"= 9.8525582; find

log cos 44° 35' 25".7. See foot-note of Art. 76.

9.8525671.

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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

log sin 4° 21'= 8.8799493
log sin 4° 20'= 8.8782854

diff. for 1'= .0016639
given log sine = 8.8785940
log sin 4° 20'= 8.8782854

diff. .0003086

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

.0016639 :.0003086 :: 60: d.

3086 x 60 d:

= 24, nearly.

16639
.. required angle 4° 20' 24".

..

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