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Corol. Hence several remarkable properties easily follow from these definitions; as,

Ist, That an arc and its supplement have the same sine, tangent, and secant; but the two latter, the tangent and secant, are accounted negative when the arc is greater than a quadrant or 90 degrees.

2d, When the arc is 0, or nothing, the sine and tangent are nothing, but the secant is then the radius CA, the least it can be. As the arc increases from 0, the sines, tangents, and secants, all proceed increasing, till the arc becomes a whole quadrant AD, and then the sine is the greatest it can be, being the radius CD of the circle; and both the tangent and secant are infinite.

3d, Of any arc AB, the versed sine AF, and cosine BK, or CF, together make up the radius CA of the circle.-The radius CA, the tangent AH, and the secant CH, form a right-angled triangle CAH. So also do the radius, sine, and cosine, form another right-angled triangle CBF or CBK. As also the radius, cotangent, and cosecant, another right-angled triangle CDL. And all these right-angled triangles are simi

lar to each other.

11. The sine, tangent, or secant of an angle, is the sine, tangent, or secant of the arc by which the angle is measured, or of the degrees, &c. in the same arc or angle.

12. The method of constructing the scales of chords, sines, tangents, and secants, usually engraven on instruments, for practice, is exhibited in the annexed figure. 13. A Trigonometrical Canon, is a table showing the length of the sine, tangent, and secant, to every degree and minute of the quadrant, with respect to the radius, which is expressed by unity or 1, with any number of ciphers. The logarithms of these sines, tangents, and secants, are also ranged in the B 2 tables;

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tables; and these are most commonly used, as they perform the calculations by only addition and subtraction, instead of the multiplication and division by the natural sines, &c, according to the nature of logarithms. Such a table of log. sines and tangents, as well as the logs. of common numbers, are placed at the end of this volume, and the description and use of them are as follow; viz. of the sines and tangents; and the other table, of common logs. has been already explained in the first volume of this Course.

Description of the Table of Log. Sines and Tangents.

In the first column of the table are contained all the arcs, or angles, for every minute in the quadrant, viz. from l' to 45°, descending from top to bottom by the left-hand side, and then returning back by the right-hand side, ascending from bottom to top, from 45° to 90°; the degrees being set at top or bottom, and the minutes in the column. Then the sines, cosines, tangents, cotangents, of the degrees and minutes, are placed on the same lines with them, and in the annexed columns, according to their several respective names or titles, which are at the top of the columns for the degrees at the top, but at the bottom of the columns for the degrees at the bottom of the leaves. The secants and cosecants are omitted in this table, because they are so easily found from the sines and cosines; for, of every arc or angle, the sine and cosecant together make up 20 or double the radius, and the cosine and secant together make up the same 20 also. Therefore, if a secant is wanted, we have only to subtract the cosine from 20; or, to find the cosecant, take the sine from 20. And the best way to perform these subtractions, because it may be done at sight, is to begin at the left hand, and take every figure from 9, but the last or right hand figure from 10, prefixing 1, for 10, before the first figure of the remainder.

PROBLEM I.

To compute the Natural Sine and Cosine of a Given Arc.

THIS problem is resolved after various ways. One of these is as follows, viz. by means of the ratio between the diameter and circumference of a circle, together with the known series for the sine and cosine, hereafter demonstrated. Thus, the

semicircumference of the circle, whose radius is 1, being 3.141592653589793 &c, the proportion will therefore be, as the number of degrees or minutes in the semicircle, is to the degrees or minutes in the proposed arc,

so is 3.14159265 &c, to the length of the said arc.

This length of the arc being denoted by the letter a; and its sine and cosine by s ande; then will these two be expressed by the two following series, viz.

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EXAM. 1. If it be required to find the sine and cosine of one minute. Then, the number of minutes in 180° being 10800, it will be first, as 10800: 1 :: 3.14159265 &c : *000290888208665 = the length of an arc of one minute. Therefore, in this case,

a

and a3

=0002908882

= '000000000004 &c.

the diff. is s =0002908882 the sine of 1 minute. Also, from

1.

take a2 = 0·0000000423079 &c,

leave c

*9999999577 the cosine of 1 minute.

EXAM. 2. For the sine and cosine of 5 degrees.

Here, as 180° : 5° :: 3.14159265 &c :·08726646 = a the length of 5 degrees. Hence a = '08726646

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these collected give s = 08715574 the sine of 5°.

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After the same manner, the sine and cosine of any other arc may be computed. But the greater the arc is, the slower

the

the series will converge, in which case a greater number of terms must be taken, to bring out the conclusion to the same degree of exactness.

Or, having found the sine, the cosine will be found from it, by the property of the right-angled triangle CBF, viz, the cosine CF CB2 BF2, or c=√/ 1 − s2.

There are also other methods of constructing the canon of sines and cosines, which, for brevity's sake, are here omitted.

PROBLEM II.

To compute the Tangents and Secants.

THE sines and cosines being known, or found by the foregoing problem; the tangents and secants will be easily found, from the principle of similar triangles, in the following manner:

In the first figure, where, of the arc AB, BF is the sine, CF or BK the cosine, AH the tangent, CH the secant, DL the cotangent, and CL the cosecant, the radius being CA or CB or CD; the three similar triangles CFB, CAH, CDL, give the following proportions:

1st, CF: FB:: CA: AH; whence the tangent is known, being a fourth proportional to the cosine, sine, and radius. 2d, CF: CB:: CA: CH; whence the secant is known, being a third proportional to the cosine and radius.

3d, BF: FC:: CD: DL; whence the cotangent is known, being a fourth proportional to the sine, cosine, and radius. 4th, BF BC: CD: CL; whence the cosecant is known, being a third proportional to the sine and radius.

As for the log. sines, tangents, and secants, in the tables, they are only the logarithms of the natural sines, tangents, and secants, calculated as above,

HAVING given an idea of the calculation and use of sines, tangents, and secants, we may now proceed to resolve the several cases of Trigonometry; previous to which, however, it may be proper to add a few preparatory notes and observations, as below.

Note 1. There are usually three methods of resolving triangles, or the cases of trigonometry; namely, Geometrical Construction, Arithmetical Computation, and Instrumental Operation.

In the first Method, The triangle is constructed, by making the parts of the given magnitudes, namely, the sides from a scale of equal parts, and the angles from a scale of chords,

or

or by some other instrument. Then measuring the unknown parts by the same scales or instruments, the solution will be obtained near the truth.

In the Second Method, Having stated the terms of the proportion according to the proper rule or theorem, resolve it like any other proportion, in which a fourth term is to be found from three given terms, by multiplying the second and third together, and dividing the product by the first, in working with the natural numbers; or, in working with the logarithms, add the logs. of the second and third terms together, and from the sum take the log. of the first term; then the natural number answering to the remainder is the fourth term sought.

In the Third Method, Or Instrumentally, as suppose by the log. lines on one side of the common two-foot scales; Extend the Compasses from the first term, to the second or third, which happens to be of the same kind with it; then that extent will reach from the other term to the fourth term, as required, taking both extents towards the same end of the scale.

Note 2. Every triangle has six parts, viz. three sides and three angles. And in every triangle, or case in trigonometry, there must be given three of these parts, to find the other three. Also, of the three parts that are given, one of them at least must be a side; because, with the same angles, the sides may be greater or less in any proportion.

Note 3. All the cases in trigonometry, may be comprised in three varieties only; viz.

of

1st, When a side and its opposite angle are given.
2d,. When two sides and the contained angle are given.
3d, When the three sides are given.

For there cannot possibly be more than these three varieties cases; for each of which it will therefore be proper to give a separate theorem, as follows:

THEOREM I,

When a Side and its Opposite Angle are two of the Given Parts.

THEN the unknown parts will be found by this theorem ; viz. The sides of the triangle have the same proportion to each other, as the sines of their opposite angles have.

That is, As any one side,

Is to the sine of its opposite angle;

So is any other side,

To the sine of its opposite angle.

Demonstr

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