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complement is less; and, on the other hand, if the angle is less than 45°, its complement is greater. Hence, every cosine, cotangent, and cosecant of an angle greater than 45°, has its equal, among the sines, tangents, and secants of angles less than 45°, and v. v.

Now, to bring the trigonometrical tables within a small compass, the same column is made to answer for the sines of a number of angles above 45°, and for the cosines of an equal number below 45°.

Thus 9.23967 is the log. sine of 10°, and the cosine of 80°, the sine of 20°, and the cosine of 70°, &c.

9.53405

The tangents and secants are arranged in a similar man

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105. To find the Sine, Cosine, Tangent, &c. of any number of degrees and minutes.

If the given angle is less than 45°, look for the degrees at the top of the table, and the minutes on the left; then, opposite to the minutes, and under the word sine at the head of the column, will be found the sine; under the word tangent, will be found the tangent, &c.

The log.sin of 43° 25′ is 9.83715 The tan of 17° 20' is 9.49430 9.47411 of 8° 46' 9.18812 9.97982 The cot of 17° 20' 10.50570 9.99490 of 8° 46' 10.81188

of 17° 20'

The cos

of 17° 20'
of 8° 46'

The first figure is the index; and the other figures are the decimal part of the logarithm.

106. If the given angle is between 45° and 90°; look for the degrees at the bottom of the table, and the minutes on the right; then, opposite to the minutes, and over the word sine at the foot of the column, will be found the sine; over the word tangent, will be found the tangent, &c.

Particular care must be taken, when the angle is less than 45°, to look for the title of the column, at the top, and for the minutes on the left; but when the angle is between 45° and 90°, to look for the title of the column, at the bottom and for the minutes, on the right.

The log. sine of 81° 21' is 9.99503

The cosine of 72° 10'

The tangent of 54° 40′
The cotangent of 63° 22′

9.48607

10.14941

9.70026

107. If the given angle is greater than 90°, look for the sine, tangent, &c. of its supplement. (Art. 98, 99.)

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108. To find the sine, cosine, tangent, &c. of any number of degrees, minutes, and SECONDS.

In the common tables, the sine, tangent, &c. are given only to every minute of a degree.* But they may be found to seconds, by taking proportional parts of the difference of the numbers as they stand in the tables. For, within a single minute, the variations in the sine, tangent, &c. are nearly proportional to the variations in the angle. Hence,

To find the sine, tangent, &c. to seconds: Take out the number corresponding to the given degree and minute; and also that corresponding to the next greater minute, and find their difference. Then state this proportion;

As 60, to the given number of seconds;

So is the difference found, to the correction for the seconds. This correction, in the case of sines, tangents, and secants, is to be added to the number answering to the given degree and minute; but for cosines, cotangents, and cosecants, the correction is to be subtracted;

For, as the sines increase, the cosines decrease.

Ex. 1. What is the logarithmic sine of 14° 43′ 10′′?

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Here it is evident, that the sine of the required angle is greater than that of 14° 43', but less than that of 14° 44'. And as the difference corresponding to a whole minute or

In the very valuable tables of Michael Taylor, the sines and tangents are given to every second.

60" is 48; the difference for 10" must be a proportional part of 48. That is,

60" 10"::48: 8

the correction to be added to the sine of 14° 43'.

Therefore the sine of 14° 43′ 10′′ is 9.40498.

2. What is the logarithmic cosine of 32° 16′ 45′′?

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from the cosine of 32° 16'.·

Therefore the cosine of 32° 16′ 45′′ is 9.92709.

The tangent of

24° 15′ 18′′ is 9.65376

The cotangent of 31° 50' 5" is 10.20700

The sine of

The cosine of

58° 14' 32" is 9.92956

55° 10′ 26′′ is 9.75670

If the given number of seconds be any even part of 60, as,,, &c. the correction may be found, by taking a like part of the difference of the numbers in the tables, without stating a proportion in form.

109. To find the degrees and minutes belonging to any given sine, tangent, &c.

This is reversing the method of finding the sine, tangent, &c. (Art. 105, 6, 7.)

Look in the column of the same name, for the sine, tangent, &c. which is nearest to the given one; and if the title be at the head of the column, take the degrees at the top of the table, and the minutes on the left; but if the title be at the foot of the column, take the degrees at the bottom, and the minutes on the right.

Ex. 1. What is the number of degrees and minutes belonging to the logarithmic sine 9.62363?

The nearest sine in the tables is 9.62865. The title of sine is at the head of the column in which these numbers are

found. The degrees at the top of the page are 25, and the minutes on the left are 10. The angle required is, therefore

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110. To find the degrees, minutes, and SECONDS belonging to any given sine, tangent, &c.

This is reversing the method of finding the sine, tangent, &c. to seconds. (Art. 108.)

First find the difference between the sine, tangent, &c. next greater than the given one, and that which is next less; then the difference between this less number and the given one; then

As the difference first found, is to the other difference;

So are 60 seconds, to the number of seconds, which, in the case of sines, tangents, and secants, are to be added to the degrees and minutes belonging to the least of the two numbers taken from the tables; but for cosines, cotangents, and cosecants, are to be subtracted.

Ex. 1. What are the degrees, minutes, and seconds, belonging to the logarithmic sine 9.40498 ?

Sine next greater 14° 44′ 9.40538

Given sine 9.40498
Next less

9.40490

8

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Difference

Then 48 8::60": 10",

which added to 14° 43', gives

14° 43′ 10" for the answer.

2. What is the angle belonging to the cosine 9.09773 ?

Cosine next greater 82° 48′ 9.09807 Given cosine 9.09773

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Then 100

66::60" 40", which subtracted from 82°

49', gives 82° 48' 20" for the answer.

It must be observed here, as in all other cases, that of the two angles, the less has the greater cosine.

The angle belonging to

the sin 9.20621 is 9° 15' 6" the tan 10.43434 is 69° 48′ 16′′ the cos 9.98157 16° 34' 30" the cot 10.33554

24° 47' 16"

Method of Supplying the Secants and Cosecants.

111. In some trigonometrical tables, the secants and cosecants are not inserted. But they may be easily obtained from the sines and cosines. For, by art. 93, proportion 3d,

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That is, the product of the cosine and secant, is equal to the square of radius. But, in logarithms, addition takes the place of multiplication; and, in the tables of logarithmic sines, tangents, &c. the radius is 10. (Art. 103.) Therefore, in these tables,

cos+sec=20. Or sec-20-cos.

Again, by art, 93, proportion 6,

sin x cosec=R2.

Therefore, in the tables,

sin+cosec=20.

Or cosec=20-sin.

Hence,

112. To obtain the secant, subtract the cosine from 20; and to obtain the cosecant, subtract the sine from 20.

These subtractions are most easily performed, by taking the right hand figure from 10, and the others from 9, as in finding the arithmetical complement of a logarithm; (Art. 55.) observing however, to add 10 to the index of the secant or cosecant. In fact, the secant is the arithmetical complement of the cosine, with 10 added to the index.

For the secant
And the arith. comp. of cos

=20-cos.
10-cos. (Art. 54.)

So also the cosecant is the arithmetical complement of the sine, with 10 added to the index. The tables of secants and cosecants are, therefore, of use, in furnishing the arithmetical

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