« PreviousContinue »
1. PLANE TRIGONOMETRY treats of the relations and calculations of the sides and angles of plane triangles.
2. The circumference of every circle (as before observed in Geom. Def. 56) is supposed to be divided into 360 equal parts, called Degrees; also each degree into 60 Minutes, and each minute into 60 Seconds, and so on. Hence a semicircle contains 180 degrees, and a quadrant 90 degrees. 3. The Measure of an angle (Def. 57, Geom.) is an arc of circle contained between the two lines which form any that angle, the angular point being the centre; and it is estimated by the number of degrees contained in that arc.
Hence, a right angle, being measured by a quadrant, or quarter of the circle, is an angle of 90 degrees; and the sum of the three angles of every triangle, or two right angles, is equal to 180 degrees. Therefore, in a right-angled triangle, taking one of the acute angles from 90 degrees, leaves the other acute angle; and the sum of the two angles, in any triangle, taken from 180 degrees, leaves the third angle; or one angle being taken from 180 degrees, leaves the sum of the other two angles.
4. Degrees are marked at the top of the figure with a small, minutes with, seconds with and so on. Thus, 57 30 12", denote 57 degrees 30 minutes and 12 seconds. 5. The Complement of an arc, is what it wants of a quadrant or 90o. Thus, if AD be a quadrant, then ED is the complement of the arc AB; and, reciprocally, AB is the complement of ED. So that, if AB be an arc of 50°, then its complement BD will be 40.
6. The Supplement of an arc, is what it wants of a semicircle, or 180°.
Thus, if ADE be a semicircle, then BDE is the supplement of the arc AB; and, reciprocally, AB is the supplement of the arc BDE. So that, if AB be an arc of 50, then its supplement EDE will be 1309.
7. The Sine, or Right Sine, of an arc, is the line drawn from one extremity of the arc, perpendicular 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 (BAG).
8. The Versed Sine of an arc, is the part of the diameter intercepted between the arc and its sine. So, AF is the versed sine of the arc AB, and EF the versed sine of the arc EDB..
9. The Tangent of an arc, is a line touching the circle in one extremity of that arc, continued from thence to meet a line drawn from the centre through the other extremity; which last line is called the Secant of the same arc. Thus, AH is the tangent, and CH the secant, of the arc AB. Also, EI is the tangent, and cr the secant, of the supplemental arc EDE. And this latter tangent and secant are equal to the former, but are accounted negative, as being drawn in an opposite or contrary direction to the former.
10. The Cosine, Cotangent, and Cosecant, of an arc, are the sine, tangent, and secant of the complement of that arc, the Co being only a contraction of the word comple ment. Thus, the arcs 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 AB: in like manner, AH, the tangent of AE, is the cotangent of BD; and DL, the tangent of DE, is the cotangent of AB: also, CH, the secant of AB, is the cosecant of BD; and CL, the secant of BD, is the cot of me.
Corol. 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 CDL. 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 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;
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 45o, 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.
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,
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 and c; then will these two be expressed by the two following series, viz.
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,
the diff. is s=0002908862 the sine of 1 minute.
take a2 = 0·0000000423079 &c,
leaves 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 5°
After the same manner, the sine and cosine of arc may be computed. But the greater the arc is, the slower