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. * Many tables are calculated for angles at intervals of 10". 77. To find the Logarithmic Sine of a Given Angle. log sin 38° 53' = 9.7977775 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 = 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 diff. for 1'= .0011017 Let d= decrease of log cosine for 23"; then 60:23::.0011017 : d. 23 x .0011017 .:: d = .0004223, nearly. 60 = 9.0567500. log sin 6° 33'= 9.0571723, find Ans. 9.05675. 987 988 988 989 990 991 1610 9.8946317 992 2598 1600 9.8972312 996 998 9.8948457 43 998 998 Sine. Diff. Tang. Diff. Cotang Diff. o 9.7893420 1616 10.1071902 1 9.7895036 2604 9.8930702 10.1069298 1616 2604 2 9.7896652 1614 2603 10.1066694 3 9.7898266 1614 2602 10.1064091 4 9.78998801 10.1061489 9.8938511 5 9.7901493 1611 1613 2603 2601 990 1611 2602 10.1056285 7 9.7904715 10.1053683 2бо 8 9.7906325 9.8948918 10.1051082 992 1608 2601 9 9.7907933 1608 2600 992 9.8954119 2600 993 1606 2600 10.1043281 I2 994 9.7912754 9.8959319 1605 10.1040681 2599 13 9.7914359 995 10.1038082 14 | 9.7915963 2599 995 2599 15 | 9.7917566 9.8967116 995 1602 10.1032384 2598 997 16 9.7919168 1601 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 2597 10.1019896 2596 1597 1000 21 | 9.7927163 9.8982700 10.1017300 999 10.1014704 1001 23 9.7930355 9.8987892 10.1012108 1001 24 9.7931949 9.8990487 10.1009513 1594 ΙΟΟΙ 259.7933543 2595 10.1006918 1003 26 9.7935135 9.8995677 10.1004323 1592 1002 27 | 9.7936727 2594 9.8998271 10.1001729 28 9.7938317 1590 1004 10.0999135 1004 29 9.7939907 1589 10.0996541 1004 30 9.7941496 1587 10.0993948 1005 31 9.7943083 9.9008645 10.0991355 32 9.7944670 1006 10.0988763 1007 33 9.7946256 10.0986170 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 IOIO 39 9.7955751 9.9029376 10.0970624 40 9.7957330 1579 2590 IOIO 10.0968034 IOII 41 9.7958909 9.9034555 10.0965445 IOI2 42 9.7960486 9.9037144 10.0962856 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 Io16 10.0947328 2587 1016 49 9.7971501 1570 10.0944741 2586 50 9.7973071 9.9057845 10.0942155 1018 51 9.7974640 1568 10.0939569 I017 52 9.7976208 1567 10.0936983 1019 53 9.7977775 10.0934397 54 I019 9.7979341 9.9068188 2585 10.0931812 55 9.7980906 1020 2584 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 10.0918891 60 9.7988718 9.9083692 2583 1023 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 2591 2591 22 21 1576 2588 2589 1014 2588 10.0952503 12 II 1016 9.8916242 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. 2 I o 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. 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 diff. for 1'= .0016639 diff. .0003086 Let d= diff. between 4° 20' and required angle; then .0016639 :.0003086 :: 60: d. 3086 x 60 d: = 24, nearly. 16639 .. |
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