THE

ELEMENTS

O F

PLAIN AND SPHERICAL

TRIGONOMETRY.

Th

PLAIN TRIGONOMETRY.

HE bufinefs of trigonometry is to find the angles when the fides are given, and the fides, or ratio of the fides, when the angles are given; and to find fides and angles, when fides and angles are given. For which, it is neceffary, that, not only the periphery of the circle, but likewise certain right lines in it, be fuppofed divided into fome determinate number of parts. The ancient geometers have supposed the periphery divided into 360 parts or degrees, and every degree into 60 minutes, and every minute into 60 feconds, &c.; and every angle is said to be of fuch a number of degrees and minutes as there are in that part of the periphery measuring the angle.

I.

An arch is any part of the periphery or circumference, and is the measure of the angle at the center which it fubtends.

II.

The quadrant of a circle is one fourth part of the circumference; the difference of an arch from a quadrant or go degrees, is called the complement of that arch.

III.

A chord or fubtenfe, is a right line drawn from one part of an arch to another.

IV.

The right fine, or fine of any arch, is a right line drawn from
the vertex of an arch perpendicular to the diameter of the
circle, and is equal to half the chord of double that arch 2.
If the arch DB, (fig. for the def.) is an arch of 30 deg. DE is the
fine of 30 deg. and twice DE, equal DO, is the fubtenfe of 60
deg. The fine of 30 deg. is equal one half radius .

V.

Every fine, as DE, divides the radius into two parts, that part betwixt the center and fine, as CE, is called the cofine; and the part betwixt the fine and arch, as EB, is the versed fine of the arch DB. For the fame reason, AE is the verfed fine of the arch AD; therefore, the verfed fine may be equal, greater, or less than the radius.

VI.

The arch HD is the complement of BD to a quadrant; and FD, equal CE, is the fine of that arch or cofine of BD.

VII.

If a right line, BG, is drawn from the point B, at right angles to the diameter, and meeting the right line CG, paffing thro' the point D; then BG is the tangent of the arch BD, and CG is the fecant of that arch.

The right line HI, drawn from the point H, at right angles to CH, and meeting CG produced in I, is the tangent of the arch HD, or cotangent of BD; and CI is the fecant of HD, or cofecant of BD.

The fine totus, or greatest fine, is the radius of the circle, which is the fine of 90 deg. c.

Characters used. + Addition. -Subtraction. x Multiplication. Equality. :: Proportion.

=

fquare-root.

་་ ་་་

✓ Extraction of the

When a line is drawn over any number of quantities, thefe quantities are to be confidered as one quantity. The marks , put overany numbers, are to be read degr. min. fecond. third minutes, &c. as 23°, 17, 18", 25", &c. Likewife, R. fignifies Rad. S. Sine, Cof Cofine, T. Tang. Cot. Cotangent, Sec. Secant, and Cofec. Cofecant.

a 3. 3.

b 15, 40

C IS. 3.

SCHOLIU M.

Because the triangles CED, CBG, (fig. for the definitions,) are fimilar, CE: ED:: CB: BG, by alter. CE: CB :: ED: BG, i. e. Cof. : R:: Sine; Tangent.

Again,

d 34. 1

a def. I.

Again, CE: CD :: CB: CG, i. e Cof. : R. :: R.: Secant. And, because the triangles CDF, CED, CBG, and CHI, are fimilar, CE ED CF: FD; but CEFD; therefore ED=CF4; therefore CE is the fine of the angle CDE=DCF.

Again, EC: D:: CB: BG; altern. EC: CB: ED: BG; therefore, if EC be nearly equal to CB, ED will be nearly equal to BG; therefore, if the arch DB be a very small arch, the fine and tangent are nearly to one another in the ratio of equality.

II.

Because the chord of any arch, and its fupplement to a circle, is the fame, fo the fine, tangent, or fecant, of any arch, and its fupplement to a femicircle, is likewife the fame.

III.

If two fides of a right angled triangle be given, the other can be found by 47. El. 1.

I

PROP. I.

Na right angled triangle, if the hypothenufe be made radius, then the fides are the fines of their oppofite angles; and, if either of the fides about the right angle be made radius, the other fide is the tangent of its oppofite angle, and the hypothenufe is the fecant of that angle.

For, in the triangle ABC, if, with the center A, and diftance AC, a circle be defcribed, and AB produced till it cut that circle in D, (No 1.); then CB is the fine of the arch CD, or of the angle A, and AB is the cofine of CD, or fine of the angle C. Again, (No 2.) if AB is made radius, and the arch BD drawn, then BC is the tangent of the arch BD, or of the angle A; and AC is the fecant of that angle. Wherefore, &c.

COR. Hence, as AC, Rad. taken in any given measure, is to BC, taken in the fame measure, are fo any parts into which the radius is fuppofed to be divided, viz. 10.000000, to a number expreffing the parts in proportion to the length of the fine of the angle, that is,

AC being radius,
And
AB Rad.

And

BC Rad.

And

AC: BC:: R: S,A

AC: BAR: S,C

AB: BCR:T,A
AB: AC::R: Sec. A.
BC: BAR:T,C
BC AC::R: Sec. C.

PRO P. II.

THE fides of plain triangles are to one another as the fines of their oppofite angles.

C & I.

Let ABC be the triangle, about which defcribe a circle ABC; from the center D let fall perpendiculars upon each of a 5. 4• the fides AB, BC, AC, which will be bifected in the points E, F, and G ; but the angle BDE is equal to the angie b 3. 3CDE, and BE is the fine of the angle BDE, or of the angle BAC ;. For the fame reason, BF is the fine of the angle ACB, and GC the fine of the angle ABC: Therefore, BE is to BF as twice BE is to twice BF; that is, BC is to BA as the fine of the angle A is to the fine of the angle C.

2d. If the triangle is right angled, then BD, the Rad. is the fine of the right angle; the other two angles as before.

3d. If the triangle is obtufe angled, then, if a triangle is formed upon the fame bafe, in the oppofite fegnent in the point I, then that angle will be acute; and BE is the fine of the angle BIC, or BACf. Wherefore, &c,

d I.

e 21. I.

f fchol. I,

PRO P. III.

Nany right lined triangle, the fum of any two fides, is to their difference, as the tangent of half the fum of the angles at the bafe, is to the tangent of half their difference.

Let ABC be the triangle, the fum of any two of its fides, as AB, BC, is to the difference of thefe fides, as the tangent of half the fum of the angles BAC, ACB, at the bafe, is to the tangent of half their difference.

For, let AB be produced to H; make BH equal to BC; and cut off BI equal to BA; then AH is equal to the fum of the fides, and HI to the difference of the fides, and the angle HBC equal to the fum of the angles at the bafe", viz. the angles BAC, a 32. 1. ACB. Join HC; and, from B, let BE fall perpendicular upon HC b; then, becaufe HB is equal to BC, the angle BHC is equal b 12. I. to BCH, and the angles BEC, BEH, are equal "; therefore the c 5. 1. angles HBE, EBC, are likewife equal d; and, if BE be made ra- d 32. I. dius, then EC is the tangent of half the fum of the angles at the bafe. Draw BD parallel to AC, then the angle DBC is equal to the angle ACB. Take HF equal to DC, and join FB;c 29. 1. then

f 2. 6.

8 14. 5.

h 4. 6.

then FBD is the difference of the angles, and EBD half their
difference: Through I draw IG parallel to BD or AC; then IB
is to BA as GD is to DCf; but IB is equal to BA; therefore
GD is equal to DC 8; but AH is to HC as HI is to HG o; and,
by altern. AH is to HI as HC is to HG, the confequents being
halved, as HA is to HI, fo is HC to HG; but HF, GD, are
each equal to DC, and therefore equal to one another. Add,
or take
away, GF to or from both, then HG is equal FD;
but half FD is ED; therefore AH is to HI as EC is to ED.
Wherefore, &c.

24. 4.

b 28. 1.

C 26. I. d 13. I.

C23 I.

PRO P. IV. THE OR.

IN any triangle, the rectangle under half the fum of the fides, and excess of the fame, above any of the fides, taken as the bafe, is to the rectangle contained by the right lines, by which the half of the Jum of the fides exceeds the other two fides, as the Square of the rad. is to the fquare of the tangent of half the angle oppofite to the bafe.

Let ABC be the triangle, BC the bafe; in the triangle ABC let a circle be infcribed 2, of which let G be the center, and let fall GD, GE, GF, perpendiculars to the fides AB, BC, AC; then AD is equal to AE, BD to BF, and CE to CF; and the angles at A and B bifected by the right lines - AG, BG ; produce AB, AC, to H, L; make BH equal to FC, and CL to BF; at the points H, L, raife the perpendiculars HK, LK, meeting the right line AG, produced in K; from the point K let fall KM perpendicular to BC; and join BK, KC; then the rectangle HAD will be to the rectangle BFC as the square of AD to the fquare of DG.

For, the triangles ADG, AEG, are equiangular, and the angle ADG equal to the angle AHK, for each are right ones; therefore DG is parallel to HKb; and, fince the angles AHK, HAK, are equal to the angles KAL, ALK, for AK bifects them, AH is equal to AL; but the angles ABC, CBH, are equal to two right angles ; and DGF, DBF, equal to two right angles ; for the angles at D and F are right ones. Take the angle DBF from both, and there remains DGF equal to HBM; but HBM, HKM, are equal to two right angles, for the angles at H and M are right ones; therefore the quadrilateral figures BDGF, BHKM, are equiangular, and BG bifects the angles DBF, DGF; therefore BK will likewife bifect the angles