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4. Degrees are marked at the top of the Sgure with a small, minutes with', seconds with, and so on. Thus, 57 33 12", denote 57 degrees 30 routes and 12 seconds.
5. The Complement of an arc, is skat it wants of a quadrant or pof. Thus, if Ad be a grear, then ID is the cosplement of the arc As; and, reciprocaat, 43 is the couplement of ED. So thai, f A3 be an 20 of 50%, then its cos lesbest BD sebe 40.
6. The Soclercent of an art, is what it wants of a semicircs, or 150°. Thus, ir sds be a semicircie, then BDs is the sopplement of the 21C AB; and, reciproca", AB is the supplement of the
So that, if as be an arc cf 50, then its suppleDent EDE will be 130°.
7. The Sine, or Right Side, cf an at, is the line drawn from one extremity of the arg, perpendicelar to the diameter which passes through the other extremity. Thus, BF is the sine of the arc AB, or of the supplemental arc BDE Hence the sine (BF) is half the chord BG) of the double arc (SAG).
3. The Versed Sine of an arc, is the part of the diameter intercepted between the arc and its sine. S3, as is the Tersed sire of the arc AB, and as the rersed sine of the arc EDE
9. The Tangent of an arc, is a line touching the arce in one extremity of that art, continued from thence to meet a line drawn from the centre through the other extremis; which last line is called the Secant of the same arc. Thes, 13 is the tangent, and ch the secan:, of the arc AB. Also, El is the tangent, and cı the secans, of the supplemental are EDE. And this latter tangent and secart are equal to the foriner, but are accounted negative, as being drawn in an opposite or contrary direction to the former.
10. The Cosine, Cotangent, and Cosecant, of an art, are the sine, tanzent, and secant of the complement of that arc, the Co being only a contraction of the rend compement. Thus, the aros AB, BD, being the complements of each other, the sine, tangent, or secant of the one of these, is the cosine, cotangent, or cosecant of the other. So, BF, the sine of AB, is the cosine of BD; and bk, the sine of ED, is the cosine of as: in like manner, AH, the tangent of Ae, is the cotangent of BD; and DL, the tangent of D2, is the cotangent of AB: also, CH, the secant of AB, i le cosecant of BD; and cl, the secant of BD, is the co: 0f .
Coroi. Hence several remarkable properties easily follow from these definitions; as,
1st, 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 rightangled 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 col. And all these right-angled triangles are similar 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 con
70 structing 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 Vers. Sin of these sines, tangents, and secants, are also ranged in the B 2
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 459 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 ór double the radius, and the cosine and secant together make up 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,
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 simicircle,
This length of the arc being denoted by the letter a ; and its sine and cosine by s and c; then will these two be expressed by the two following series, viz. a3 as
at a" El +
+ &c. 24 720
EXAM. 1. If it be required to find the sine and cosine of 1 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 = '0002908882
= 0.0000000423079 &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
these collected give s = .08715574 the sine of 50
And, for the cosine, 1=l'
ja? 00380771 +7404 = •00000241
these collected give c =
.99619470 the cosine of 5o.
After the same manner, the sine and cosine of any other arc may be computed. But the greater the arc is, the slower
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 = NCB’ - BF’, or c =vi j?.
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 sinęs 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, çol, 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,