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1. PLANE TRIGONOMETRY is the art by which, when any three parts of a plane triangle, except the three angles, are given, the others are determined.
2. The periphery of every circle is supposed to be divided into 360 equal parts, called degrees; each de. gree into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds, &c.
3. The measure of an angle is the arc of a circle, contained between the two lines that form the angle, the angular point being the centre ; thus the angle ABC, Fig. 30, is measured by the arc DE, and contains the same number of degrees that the arc does. The measure of a right angle is therefore 90 degrees; for DH, Fig. 31, which measures the right angle DCH is one fourth part of the circumference, or 90 degrees.
Note.—The degrees, minutes, seconds, &c. contained in any arc, or angle, are written in this manner, 50° 18 35"; which signifies that the given arc or angle contains 50 degrees, 18 minutes and 35 seconds.
4. The complement of an arc, or of an angle, is what it wants of 90°; and the supplement of an arc, or of an 5. The chord of an arc, is a line drawn from one extremity of the arc to the other : thus the line B E is the chord of the arc BAE or BDE, Fig. 31.
6. The sine of an arc, is a right line drawn from one extremity of the arc, perpendicular to the diameter passing through the other extremity: thus BF is the sine of the arc AB or BD, Fig. 31.
7. The cosine of an arc, is that part of the diameter which is intercepted between the sine and the centre : thus C F is the cosine of the arc AB, and is equal to BI, the sine of its complement HB, Fig. 31.
8. The versed sine of an arc, is that part of the diameter which is intercepted between the sine and the are: thus AF is the versed sine of AB; and DF of DB, Fig. 31.
9. The tangent of an arc, is a right line touching the circle in one end of the arc, being perpendicular to the diameter passing through that end, and is terminated by a right line drawn from the centre through the other end: thus AG is the tangent of the arc AB, Fig. 31.
10. The secant of an arc, is the right line drawn from the centre and terminating the tangent: thus C G is the secant of AB. Fig. 31.
11. The cotangent of an arc, is the tangent of the complement of that arc; thus H K is the cotangent of AB. Fig. 31.
12. The cosecant of an arc, is the secant of the complement of that arc; thus C K is the cosecant of AB. 13. The sine, cosine, &c. of an angle is the same as the sine, cosine, &c. of the arc that measures the angle.
To construct the lines of chords, sines, tangents, and secants, to any radius. Fig. 32.
Describe a semicircle with any convenient radius CB; from the centre C draw CD perpendicular to A B and produce it to F; draw B E parallel to CF and join AD.
Divide the arc AD into nine equal parts, as A, 10; 10, 20; &c. and with one foot of the dividers in A, transfer the distances A, 10; A, 20; &c. to the right line AD; then will AD be a line of chords constructed to every ten degrees.
Divide B D into nine equal parts, and from the points of division 10, 20, 30, &c. draw lines parallel to CB, and meeting CD in 10, 20, 30, &c. and CD will be a line of sines.
From the centre C, through the divisions of the arc BD, draw lines meeting BE, in 10, 20, 30, &c. and BE will be a line of tangents.
With one foot of the dividers in C transfer the dis. tances from C to 10, 20, &c. in the line BE, to the line CF which will then be a line of secants.
By dividing the arcs AD and B D each into 90 equal parts, and proceeding as above, the lines of chords, sines, &c. may be constructed to every degree of the
At a given point A, in a given right line AB, to malce an angle of any proposed number of degrees, suppose 38 degrees. Fig. 33.
With the centre A, and a radius equal to 60 degrees, taken from a scale of chords, describe an arc, cutting A B in m'; from the same scale of chords, take 39 degrees and apply it to the arc from m to.n, and from A through n draw the line AC, then will the angle A contain 38 degrees.
Note.-Angles of more than 90 degrees are usually taken off at twice.
To measure a given angle A. Fig. 34,
Describe the arc mn with the chord of 60 degrees, as in the last problem. Take the arc mn between the dividers, and that extent applied to the scale of chords, will show the degrees in the given angle.
Note.-When the distance in exceeds 90 degrees, it must be taken off at twice, as before.
OF THE TABLE OF LOGARITHMIC OR AR
TIFICIAL SINES, TANGENTS, &e.
This table contains the logarithms of the sine, tangent, &c. to every degree and minute of the quadrant, the radius being 10000000000, and consequently its logarithm
Let the radius CB, Fig. 32, be supposed to consist of 10000000000 equal parts as above, and let the quadrant D B be divided into 5400 equal arcs, each of these will therefore contain 1'; and if from the several points of division in the quadrant, right lines be drawn perpendicular to CB, the sine of every minute of the quadrant, to the radius CB will be exhibited. The lengths of these lines being computed and arranged in a table constitute what is usually termed a table of natural sines. The logarithms of those numbers taken from a table of logarithms and properly arranged form the table of logarithmic or artificial sines. In like manner the logarithmie tangents and secants are to be understood.
The method by which the sines are computed is too abstruse to be explained in this work, but a familiar exposition of this subject as well as the construction of logarithms may be seen in Simpson's Trigonometry.
To find, by the table, the sine, tangent, &c. of an arc
If the degrees in the given angle be less than 45, look for them at the top of the table, and for the minutes, in the left hand column