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PRINCIPLES OF PLANE TRIGONOMETRY.
LEMMA I. FIG. 1.
LET ABC be a rectilineal angle; if about the point B as a centre, and at any distance BA, a circle be described, meeting BA and BC, (the straight lines including the angle ABC,) in A and C; the angle ABC will be to four right angles, as the arch* AC to the whole circumference.
Produce AB till it meet the circle again in F, and through B draw DE perpendicular to AB, meeting the circle in D and E.
By 33. 6. Elem. the angle ABC is to a right angle ABD, as the arch AC to the arch AD; and quadrupling the consequents, the angle ABC will be to four right angles, as the arch AC to four times the arch AD, or to the whole circumference.
LEMMA II. FIG. 2.
Let ABC be a plane rectilineal angle as before: About B as a centre, at any two distances BD and BA, let two circles be described, meeting BA and BC, in D, E, A, C; the arch AC will be to the whole circumference of which it is an arch, as the arch DE is to the whole circumference of which it is an arch.
By Lemma 1. the arch AC is to the whole circumference of which it is an arch, as the angle ABC is to four right angles; and by the same Lemma, the arch DE is to the whole circumference of which it is an arch, as the angle ABC is to four right angles; therefore the arch AC is to the whole circumference of which it is an arch, as the arch DE to the whole circumference of which it is an arch.
• Any part of the circumference of a circle, as AC, is called an arch or arc.
DEFINITIONS. FIG. 3.
LET ABC be a plane rectilineal angle; if about B as a centre, at any distance BA, a circle ACF be described, meeting BA and BC, in A and C; the arch AC is called the measure of the angle ABC.
The circumference of a circle is supposed to be divided into 360 equal parts called degrees, and each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds, &c. A degree is marked thus, °; a minute thus, '; a second thus, "; and so on: for example, 57° 17' 44" denotes 43 degrees, 17 minutes, 44 seconds. Hence a quadrant or fourth part of the circumference contains 90°; and as many degrees, minutes, seconds, &c. as are contained in any arch, of so many degrees, minutes, seconds, &c. is the angle, of which that arch is the measure said to be; thus a right angle is said to be an angle of 90°, because its measure, a quadrant, contains 90°. COR. Whatever be the radius of the circle of which the measure of a given angle is an arch, that arch will contain the same number of degrees, minutes, seconds, &c. as is manifest from Lemma 2.
The difference between an arch and the semi-circumference, or between an angle and 180°, is called the supplement of that arch or angle. Thus the arch CHF, (which together with AC, is equal to the semi-circumference,) is called the supplement of the arch AC; and the angle CBF, (which, together with ABC is equal to two right angles,) is called the supplement of the angle ABC.
The chord of an arch is a straight line drawn from one extremity of the arch to the other. Thus CA is the chord of arch CA.
COR. The chord of 60° is equal to the radius of the circle; for
since the angle ABC is equal to 60°, the arch AC is onesixth part of the circumference; consequently, the straight line AC is (by Cor. 15. 4.) equal to AB or BC.
A straight line CD drawn through C, one of the extremities of the arch AC perpendicular to the diameter passing
through the other extremity A, is called the sine of the arch AC or of the angle ABC, of which AC is the measure. COR. 1. The sine of a quadrant, or of a right angle, is equal
to the radius.
COR. 2. The sine of any arch is half the chord of double that
arch; for the diameter which bisects an arch also bisects
The segment DA of the diameter passing through A, one ex-
A straight line AE touching the circle at A, one extremity of
COR. The tangent of half a right angle is equal to the radius.
The straight line BE between the centre and the extremity
COR. to def. 5, 7, 8, the sine, tangent, and secant of any angle
It is manifest from def. 5, that CD is the sine of the angle