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Ex. 2. Required the number answering to the logarithm 3.233568.
The given logarithm is
The next less tabular logarithm of 1712, is 3.233504
Tab. Diff. 253) 64.00 (25
Hence the number sought is 1712.25, marking four places of integers for the characteristic 3.
TABLE OF LOGARITHMIC SINES.
XXXIII. In this table are arranged the logarithms of the numerical values of the sines, cosines, tangents, and cotangents, of all the arcs or angles of the quadrant, divided to minutes, and calculated for a radius of 10,000,000,000. The logarithm of this radius is 10. In the first and last horizontal line, of each page, are written the degrees whose logarithmic sines, &c. are expressed on the page. The vertical columns on the left and right, are columns of minutes.
To find, in the table, the logarithmic sine, cosine, tangent, or cotangent of any given arc or angle.
1. If the angle be less than 45°, look in the first horizontal line of the different pages, until the number of degrees be found; then descend along the column of minutes, on the left of the page, till you reach the number showing the minutes; then pass along the horizontal line till you come into the column designated, sine, cosine, tangent, or cotangent, as the case may be: the number so indicated, is the logarithm sought. Thus, the sine, cosine, tangent, and cotangent of 19° 55', are found on page 37, opposite 55, and are, respectively, 9.532312, 9.973215, 9.559097, 10.440903.
2. If the angle be greater than 45°, search along the bottom line of the different pages, till the number of degrees are found; then ascend along the column of minutes, on the right hand side of the page, till you reach the number expressing the minutes; then pass along the horizontal line into the columns designated tang, cotang., sine, cosine, as the case may be; the number so pointed out is the logarithm required.
It will be seen, that the column designated sine at the top of the page, is designated cosine at the bottom; the one designated tang., by cotang., and the one designated cotang., by tang.
The angle found by taking the degrees at the top of the page, and the minutes from the first vertical column on the left, is the complement of the angle, found by taking the corresponding degrees at the bottom of the page, and the minutes traced up in the right hand column to the same horizontal line. being apparent, the reason is manifest, why the columns designated sine, cosine, tang., and cotang., when the degrees are pointed out at the top of the page, and the minutes counted downwards, ought to be changed, respectively, into cosine, sine, cotang., and tang., when the degrees are shown at the bottom of the page, and the minutes counted upwards.
If the angle be greater than 90°, we have only to subtract it from 180°, and take the sine, cosine, tangent, or cotangent of the remainder.
The secants and cosecants are omitted in the table, being easily found from the cosines and sines.
For, sec.= ; or, taking the logarithms, log. sec.-2 log. R—log. cos.-20—log. cos. ; that is, the logarithmic secant is found by substracting the logarithmic cosine from 20. And cosec. = or log. cosec.=2 log. R―log. sine =20—log.
sine; that is, the logarithmic cosecant is found by subtracting the logarithmic sine from 20.
It has been shown that R2-tang. × cotang.; therefore, 2 log. R=log. tang. + log. cotang.; or 20=log. tang. + log. cotang.
The column of the table, next to the column of sines, and on the right of it, is designated by the letter D. This column is calculated in the following manner. Opening the table at any page, as 42, the sine of 24° is found to be 9.609313; of 24° 1', 9.609597: their difference is 284; this being divided by 60, the number of seconds in a minute, gives 4.73, which is entered in the column D, omitting the decimal point. Now, supposing the increase of the logarithmic sine to be proportional to the increase of the arc, and it is nearly so for 60", it follows, that 473 (the last two places being regarded as decimals) is the increase of the sine for 1". Similarly, if the arc be 24° 20', the increase of the sine for 1", is 465, the last two places being decimals. The same remarks are equally applicable in respect of the column D, after the column cosine, and of the column D, between the tangents and cotangents. The column D, between the tangents and cotangents, answers
to either of these columns; since of the same arc, the log tang.+log. cotang 20. Therefore, having two arcs, a and b, log. tang blog. cotang blog. tang a+log. cotang a; or, log. tang b-log. tang a=log. cotang a-log. cotang b.
Now, if it were required to find the logarithmic sine of an arc expressed in degrees, minutes, and seconds, we have only to find the degrees and minutes as before; then multiply the corresponding tabular number by the seconds, cut off two places to the right hand for decimals, and then add the product to the number first found, for the sine of the given arc. Thus, if we wish the sine of 40° 26' 28".
The sine 40° 26'
Product 69.16, to be added
Gives for the sine of 40° 26′ 28′′
The tangent of an arc, in which there are seconds, is found in a manner entirely similar. In regard to the cosine and cotangent, it must be remembered, that they increase while the arcs decrease, and decrease while the arcs are increased, consequently, the proportional numbers found for the seconds must be subtracted, not added.
Ex. To find the cosine 3° 40′ 40′′.
Product 5.20, which being subtracted = 5.20
Gives for the cosine of 3° 40′ 40′′ 9.999104.80
To find the degrees, minutes, and seconds answering to any given logarithmic sine, cosine, tangent, or cotangent.
Search in the table, and in the proper column, until the number be found; the degrees are shown either at the top or bottom of the page, and the minutes in the side columns, either at the left or right. But if the number cannot be exactly found in the table, take the degrees and minutes answering to the nearest less logarithm, the logarithm itself, and also the corresponding tabular difference. Subtract the logarithm taken, from the
given logarithm, annex two ciphers, and then divide the remainder by the tabular difference: the quotient is seconds, and is to be connected with the degrees and minutes before found; to be added for the sine and tangent, and subtracted for the cosine and cotangent.
Ex. 1. To find the arc answering to the sine 9.880054 Sine 49° 20′, next less in the table, 9.879963
Tab. Diff. 181)9100 (50′′ Hence the arc 49° 20′ 50′′ corresponds to the given sine 9.880054.
Ex. 2. To find the arc corresponding to cotang. 10.008688. Cotang 44° 26', next less in the table 10.008591
Tab. Diff. 421)9700(23" Hence, 44° 26'-23"-44° 25′ 37′′ is the arc corresponding to the given cotangent 10.008688.
In every right angled triangle, radius is to the sine of either of the acute angles, as the hypothenuse to the opposite side: and radius is to the cosine of either of the acute angles, as the hypothenuse to the adjacent side.
Let ABC be the proposed triangle, right-angled at A: from the point C as a centre, with a radius CD equal to the radius of the tables, describe the arc DE, which will measure the angle C ; on CD let fall the perpendicular EF, which will be the sine of the angle C, and CF will be its co
sine. The triangles CBA, CEF, are similar, and give the proportion,
CE : EF : : CB : BA: hence
But we also have,
CE CF CB: CA: hence
Cor. If the radius R=1, we shall have,
Hence, in every right angled triangle, the perpendicular is equal to the hypothenuse multiplied by the sine of the angle at the base; and the base is equal to the hypothenuse multiplied by the cosine of the angle at the base; the radius being equal to unity.
In every right angled triangle, radius is to the tangent of either of the acute angles, as the side adjacent to the side opposite.
Let CAB be the proposed triangle.
With any radius, as CD, describe the arc DE, and draw the tangent DG.
From the similar triangles CDG, CAB, we shall have,
CD: DG:: CA: AB: hence,
Cor. 1. If the radius R=1,
Hence, the perpendicular of a right angled triangle is equal to the base multiplied by the tangent of the angle at the base, the radius being unity.
Cor. 2. Since the tangent of an arc is equal to the cotangent of its complement (Art. VI.), the cotangent of B may be substituted in the proportion for tang C, which will give
R: cot B:: CA: AB.
In every rectilineal triangle, the sines of the angles are to each other as the opposite sides.