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1. The complement of an arc is its defect from a quadrant; its supplement is its defect from a semicircumference; and its eaplement is its defect from the whole circumference.
2. The sine of an arc is a perpendicular let fall from one of its extremities upon a diameter passing through the other. * ,
3. The versed sine of an arc is that portion of a diameter intercepted between its sine and the circumference.
4. The tangent of an arc is a perpendicular drawn at one extremity to a diameter, and limited by a diameter extending through the other.
5. The secant of an arc is a straight line which joins the centre with the termination of the tangent.
In naming the sine, tangent, or secant, of the complement of an arc, it is usual to employ the abbreviated terms of cosine, cotangent and cosecant. A farther contraction is frequently made in noting the radius and other lines connected with the circle, by retaining only the first syllable of the word, or even the mere initial letter.
Let ACFE be a circle, of which the diameters AF and CE
are at right angles; having taken any arc AB, produce the
radius OB, and draw BD, A H perpendicular to AF, and BG, CI perpendicular to CE. Of this assumed arc AB, the complement is BC, and the supplement l-i BCF; the sine is BD, the cosine C V BG or OD, the versed sine AD, G (B the coversed sine CG, and the supplementary versed sine FD ; the \ tangent of AB is AH, and its co- F O A. tangent CI; and the secant of the same arc is OH, and its cosecant OI.
Several obvious consequences flow from these definitions : —
1. Since the diameter which bisects an arc bisects also the chord at right angles, it follows that half the chord of any arc is equal to the sine of half that arc.
2. In the right-angled-triangle ODB, BD*-FOD*= OB% ; and hence the squares of the sine and cosine of an arc are together equal to the square of the radius.
3. The triangle ODB being evidently similar to OAH, OD : DB : OA : AH; that is, the cosine of an arc is to the sine, as the radius to the tangent.
4. From the similar triangles ODB and OAH, OD: OB :: OA : OH ; wherefore the radius is a mean proportional between the cosine and the secant of an arc.
5. Since BD*=AD.FD, it is evident that the sine of an arc is a mean proportional between the versed sine and the
supplementary versed sine, or between the sum and differ. ence of the radius and the cosine.
6. Hence also the chord of an arc is a mean proportional between the versed sine and the diameter ; for AB*= AD.A.F.
7. The triangles OAH and ICO being similar, AH: OA :: OC: CI; and hence the radius is a mean proportional between the tangent of an arc and its cotangent.
8. Since OD*= BG*=CG.C.E, it follows that the cosine of an arc is a mean proportional between the sum and the difference of the radius and the sine.
The circumference of the circle is commonly divided into 360 equal parts, called degrees, each of them being subdivided into 60 minutes, and these again being each distinguished into 60 seconds. It very seldom is required to carry this subdivision any farther. Degrees, minutes, seconds, or thirds, are conveniently noted by these marks,
o / II ///
Thus, 23° 27' 48" 42", signifies 23 degrees, 27 minutes, 43 seconds, and 42 thirds.
Scholium. To discern more clearly the connection of the lines derived from the circle, it will be proper to trace their successive values, while the corresponding arc is supposed to increase. Let the arc AB', on the opposite side, be made equal to AB, draw the diameter FOA, extend the diameters b'OB and boB', join BB' and bb', and at A apply the double tangent HAH'. It is evident that BE=be, or that the sine of the arc AB is equal to the sine of its supplement ABb. But BE and We, or the sines of ABFW and ABFB'B' which lie on the opposite side of the diameter, are likewise equal to BE; that is, the inverted sine of an arc is equal to the sine of that arc or of its supplement, augmented, each by a semicircumference. The arc AB, and its defect ABFB" from a whole
circumference, have both the
same cosine OE; and the supplemental arc ABb, and its defect from a whole circumference, have likewise the same cosine, although with an inverted position. AH and OH are respectively the tangent and secant not only of AB, but of the arc ABöFö', which is compounded of the original arc and a semicircumference; and the similar lines AH' and OH', on the opposite side, are at once the tangent and secant of the supplementary arc ABb, and of ABöFö'B', likewise compounded of that arc and a semicircumference. As the prolonged diameter b'OBH, therefore, turns about the centre, the sine and tangent both increase, till the arc attains 90°, when the sine becomes equal to the radius, and the tangent vanishes into unlimited extent. Between 90° and 180°, the sine again diminishes, and the tangent, re-appearing in the opposite direction, likewise contracts by successive diminutions. In the third quadrant, the sine emerges with a contrary position, and increases till it becomes equal to the radius ; while the tangent, resuming its first position, stretches out till it vanishes away. Be
tween 270° and 360°, the opposite sine again contracts,
and the tangent, re-appearing on the same side, shrinks also by degrees to a point. In the first and fourth quadrants, the cosine lies on the same side of the centre, while the secant stretches from it in the direction of the extremity of the arc; but, in the second and third quadrants, the cosine shifts to the opposite side, and the secant shoots from the centre in a direction opposite to the termination of the arc. The same phases are thus repeated at each succeeding revolution. Hence, if m denote any integral number, the sine of an arc a is equal to the sine of the arc (2m—1) 180°–a, and to opposite sines of (2m-1) 180°--a and of 2m.180°–a
the cosine and secant of an arc a are equal to the cosine
the tangent or cotangent of an arca is equal to the tangent
or cotangent of the arc (2m—1) 180°,+a, and to the opposite tangents or cotangents of the arcs (2m—1) 180—a and 2m, 180—a.
An arc may, by a simple extension of analogy, be conceived to comprehend innumerable other arcs. Thus, the arc AB, in fact, represents all the arcs which have their origin at A and their termination at B; it therefore includes not only the small arc AB, but that arc as augmented by successive revolutions, or the repeated addition of entire circumferences. Hence the sine or tangent of an arc a are the same with the sine or tangent of any arc n.360°-Ha.