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

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 9.62863?

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 25° 10'.

The angle belonging to

the sine 9.87993 is 49° 20' the cos 9.97351 is 19° 48' the tan 9.97955 43° 39' the cotan 9.75791 60° 12' the sec 10.65396 77° 11' the cosec 10.49066

18° 51'

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

Next less 14° 43' 9.40490

Given sine 9.40498

Next less 9.40490

Difference

Then, 488

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 ?

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, R2.

cos X sec

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 R3.

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

=

-20-cos.

And the arith. comp. of 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 complement of the sine and cosine, in the following simple

manner:

113. For the arithmetical complement of the sine, subtract 10 from the index of the cosecant; and for the arithmetical complement of the cosine, subtract 10 from the index of the secant.

By this, we may save the trouble of taking each of the figures from 9.

SECTION III.

SOLUTIONS OF RIGHT ANGLED TRIANGLES.

ART. 114. In a triangle there are six parts, three sides, and three angles. In every trigonometrical calculation, it is necessary that some of these should be known, to enable us to find the others. The number of parts which must be given, is THREE, one of which must be a SIDE.

If only two parts be given, they will be either two sides, a side and an angle, or two angles; neither of which will limit the triangle to a particular form and size.

a

If two sides only be given, they may make any angle with each other; and may, therefore, be the sides of a thousand different triangles. Thus, the two lines a and b may belong either to the triangle ABC, or ABC', or ABC". So that it will be impossible, from knowing two of the sides of a triangle, to determine the other parts.

Or, if a side and an angle only be given, the triangle will be indeterminate. Thus, if the side AB and the angle at A be given; they may be parts either of the triangle ABC, or ABC', or ABC".

C"

Lastly, if two angles, or even if all the angles be given, they will not determine the length of the sides. For the triangles ABC, A'B'C', A"B"C",

and a hundred others which might be drawn, with sides parallel to these, will all have the same angles. So that one of the parts given must always be a side. If this and any other

B'
B

two parts, either sides or angles, be known, the other three may be found, as will be shown, in this and the following section.

115. Triangles are either right angled or oblique angled. The calculations of the former are the most simple, and those which we have the most frequent occasion to make. A great portion of the problems in the mensuration of heights and distances, in surveying, navigation and astronomy, are solved by rectangular trigonometry. Any triangle whatever may be divided into two right angled triangles, by drawing a perpendicular from one of the angles to the opposite side.

116. One of the six parts in a right angled triangle, is always given, viz. the right angle. This is a constant quantity; while the other angles and the sides are variable. It is also to be observed, that, if one of the acute angles is given, the other is known of course. For one is the complement of the other. (Art. 76, 77.) So that, in a right angled triangle, subtracting one of the acute angles from 90° gives the