But on pp. 30, 49, the characteristic changes one unit in value at the places marked with bars. Above these bars the proper characteristic is printed at the top, and below them at the bottom, of the column.

25. On pages 28-49 the log sin, log tan, log cot, and log cos, of 1° to 89°, are given to every minute. Conversely, this part of the table gives the value of the angle to the nearest minute when log sin, log tan, log cot, or log cos is known, provided log sin or log cos lies between 8.24186 and 9.99993, and log tan or log cot lies between 8.24192 and 11.75808.

If the exact value of the given logarithm of a function is not found in the table, the value nearest to it is to be taken, unless interpolation is employed as explained in § 26.

If the angle is less than 45° the number of degrees is printed at the top of the page, and the number of minutes in the column to the left of the columns containing the logarithm. If the angle is greater than 45°, the number of degrees is printed at the bottom of the page, and the number of minutes in the column to the right of the columns containing the logarithms.

If the angle is less than 45°, the names of its functions are printed at the top of the page; if greater than 45°, at the bottom of the page. Thus,

If log sin = 9.47760

9.4776010, the nearest log sin in the table is 9.47774 — 10 (page 36), and the angle corresponding to this value is 17° 29′.

If log tan = 0.76520 = 10.76520 = 10.76520 — 10, the nearest log tan in the table is 10.76490 10 (page 32), and the angle corresponding to this value is 80° 15'.

26. If it is desired to obtain the logarithms of the functions of angles that contain seconds, or to obtain the value of the angle in degrees, minutes, and seconds, from the logarithms of its functions, interpolation must be employed. Here it must be remembered that,

The difference between two consecutive angles in the table is 60".

Log sin and log tan increase as the angle increases; log cos and log cot diminish as the angle increases.

Find log tan 70° 46′ 8′′.

Page 37. log tan 70° 46′ = 0.45731.

The difference between the mantissas of log tan 70° 46′ and log tan 70° 47′ is 41, and of 41 = 5.

60

As the function is increasing, the 5 must be added to the figure in the fifth place of the mantissa 45731; and

Therefore log tan 70° 46′ 8′′ = 0.45736.

Find log cos 47° 35′ 4′′.

Page 48. log cos 47° 35′ = 9.82899 10.

The difference between this mantissa and the mantissas of the next log cos is 14, and of 14 = 1.

60

As the function is decreasing, the 1 must be subtracted from the figure in the fifth place of the mantissa 82899; and

Therefore log cos 47° 35′ 4′′ = 9.82898 — 10.

Find the angle for which log sin=9.45359-10.

Page 35.

The mantissa of the nearest smaller log sin in the table is 45334. The angle corresponding to this value is 16° 30'.

The difference between 45334 and the given mantissa, 55359, is 25.

The difference between 45334 and the next following mantissa, 45377, is 43, and 25 of 60" = 35′′.

43

As the function is increasing, the 35′′ must be added to 16° 30′; and the required angle is 16° 30′ 35′′.

Find the angle for which log cot=0.73478.

Page 32. The mantissa of the nearest smaller log cot in the table is 73415. The angle corresponding to this value is 10° 27'.

The difference between 73415 and the given mantissa is 63.

71

The difference between 73415 and the next following mantissa is 71, and i of 60" 53".

As the function is decreasing, the 53′′ must be subtracted from 10° 27′; the required angle is 10° 26′ 7′′.

and

27. If log sec or log csc of an angle is desired, it may be found from the table by the formulas,

Page 42. log csc 59° 36′ 44′′ = colog sin 59° 36′ 44′′ = 0.06418.

28. If a given angle is between 0° and 1o, or between 89° and 90°; or, conversely, if a given log sin or log cos does not lie between the limits 8.24186 and 9.99993 in the table; or, if a given log tan or log cot does not lie between the limits 8.24192 and 11.75808 in the table; then pages 21-24 of Table III. must be used.

On page 21, log sin of angles between 0° and 0° 3', or log cos of the complementary angles between 89° 57' and 90°, are given to every second; for the angles between 0° and 0° 3', log tan=log sin, and log cos 0.00000; for the angles between 89° 57′ and 90°, log cot=log cos, and log sin=0.00000.

=

On pages 22-24, log sin, log tan, and log cos of angles between 0° and 1°, or log cos, log cot, and log sin of the complementary angles between 89° and 90°, are given to every 10".

Whenever log tan or log cot is not given, they may be found by the formulas,

log tan colog cot.

log cot colog tan.

Conversely, if a given log tan or log cot is not contained in the table, then the colog must be found; this will be the log cot or log tan, as the case may be, and will be contained in the table.

On pages 25-27 the logarithms of the functions of angles between 1o and 2°, or between 88° and 90°, are given in the manner employed on pages 22–24. These pages should be used if the angle lies between these limits, and if not only degrees and minutes, but degrees, minutes, and multiples of 10" are given or required.

When the angle is between 0° and 2°, or 88° and 90°, and a greater degree of accuracy is desired than that given by the table, interpolation may be employed; but for these angles interpolation does not always give true results, and it is better to use Table IV.

Find log tan 0° 2' 47", and log cos 89° 37′ 20′′.

Page 21.
Page 23.

log tan 0° 2′ 47′′ log sin 0° 2′ 47′′ = 6.90829 — 10. log cos 89° 37′ 20′′ = 7.81911 — 10.

Page 23. log cot 89° 38′ 30′′ = 7.79617 — 10
Therefore, log tan 89° 38′ 30′′= 2.20383

Find the angle for which log tan=6.92090 — 10.
Page 21. The nearest log tan is 6.92110 — 10.
The corresponding angle for which is 0° 2′ 52′′.

Find the angle for which log cos=7.70240 — 10.
Page 22. The nearest log cos is 7.70261

10.

The corresponding angle for which is 89° 42′ 40′′.

Find the angle for which log cot=2.37368.

This log cot is not contained in the table.

The colog cot = 7.62632 — 10 = log tan.

The log tan in the table nearest to this is (page 22) 7.62510 — 10, and the angle corresponding to this value of log tan is 0° 14′ 30′′.

29. If an angle x is between 90° and 360°, it follows, from formulas established in Trigonometry, that,

The letter n is placed (according to custom) after the logarithms of those functions which are negative in value.

The above formulas show, without further explanation, how to find by means of Table III. the logarithms of the functions of any angle between 90° and 360°.

Thus, log sin 137° 45′ 22′′ = log sin 42° 14′ 38′′ 9.82756 -10.

log cos 137° 45′ 22′′ logn cos 42° 14′ 38′′ = 9.86940„ — 10.

log tan 137° 45′ 22′′ = logʼn tan 42° 14′ 38′′ — 9.95815, 10.
log cot 137° 45′ 22′′ logn cot 42° 14′ 38′′ 0.04185n.
log sin 209° 32′ 50′′ = logʼn sin 29° 32′ 50′′ = 9.69297n — 10.
log cos 330° 27′ 10′′ log cos 29° 32′ 50′′ = 9.93949 10.

Conversely, to a given logarithm of a trigonometric function there correspond between 0° and 360° four angles, one angle in each quadrant, and so related that if x denote the acute angle, the other three angles are 180° x, 180° +x, and 360° X.

If besides the given logarithm it is known whether the function is positive or negative, the ambiguity is confined to two quadrants, therefore to two angles.

Thus, if the log tan = 9.47451 — 10, the angles are 16° 36′ 17′′ in Quadrant I. and 196° 36′ 17′′ in Quadrant III.; but if the log tan 9.47451-10, the angles are 163° 23′ 43′′ in Quadrant II. and 343° 23′ 43′′ in Quadrant IV.

To remove all ambiguity, further conditions are required, or a knowledge of the special circumstances connected with the problem in question.

TABLE IV.

30. This table (page 50) must be used when great accuracy is desired in working with angles between 0° and 2°, or between 88° and 90°.

The values of S and T are such that when the angle a is expressed in seconds,

Slog sin a-log a",

T= log tan a log a".

Hence follow the formulas given on page 50.

The values of S and T are printed with the characteristic 10 too large, and in using them -10 must always be annexed.