PLANE TRIGONOMETRY.

SECT. I.

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.

I.

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.

II.

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.

III.

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.

IV.

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.

V.

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 chord of that arch at right angles, (by 3. 3.) Thus CD
is half CP; thus also the sine of 30° is half the radius; for
it is half the chord of 60°.

VI.

The segment DA of the diameter passing through A, one ex-
tremity of the arch AC, between the sine CD, and the
point A, is called the Versed Sine of the arch AC, or of
the angle ABC.

VII.

A straight line AE touching the circle at A, one extremity of
the arch AC, and meeting the diameter BC, passing through
C, the other extremity, in E, is called the Tangent of the
arch AC, or of the angle ABC.

COR. The tangent of half a right angle is equal to the radius.

VIII.

The straight line BE between the centre and the extremity
of the tangent AE, is called the Secant of the arch AC, or
of the angle ABC.

COR. to def. 5, 7, 8, the sine, tangent, and secant of any angle
ABC, are likewise the sine, tangent, and secant of its supple-
ment CBF.

It is manifest from def. 5, that CD is the sine of the angle
CBF. Let CB be produced till it meet the circle again in
G; and it is manifest that AE is the tangent, and BE the
secant, of the angle ABG or EBF, from def. 7, 8.
COR. to def. 5, 6, 7, 8. The sine, versed sine, tangent, and Fig. 4.
secant, of any arch which is the measure of any given angle
ABC, is to the sine, versed sine, tangent, and secant, of
any other arch which is the measure of the same angle, as
the radius of the first arch is to the radius of the second.
Let AC, MN be measures of the angle ABC, according to
def. 1. CD the sine, DA the versed sine, AE the tangent,
and BE the secant of the arch AC, according to def. 5, 6,
7, 8, and NO the sine, OM the versed sine, MP the tan-
gent, and BP the secant of the arch MN, according to the
same definitions. Since CD, NO, AE, MP are parallel,
CD is to NO as the radius CB to the radius NB, and AE
to MP as radius AB to radius BM, and BC or BA to BD,