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PLANE TRIGONOMETRY.

LEMMA I. FIG. 1."

ET ABC be a rectilineal angle, if about the point B as a centre, and with any diftance BA, a circle be defcribed, meeting BA, BC, the straight lines including the angle ABC in A, 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, 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 confequents, the angle ABC will be to four right angles, as the arch AC to four times the arch AD, or to the whole circumference.

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LEMMA II. FIG. 2.

ET ABC be a plane rectilineal angle as before: About B as a centre with any two difances BD, BA, let two circles be described meeting BA, 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 fame Lemma 1. 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.

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DEFINITIONS. FIG. 3.

I.

ET ABC be a plane rectilineal angle; if about B as a centre, with BA any distance, a circle ACF be described meeting BA, BC, in A, C; the arch AC is called the measure of the angle ABC.

II.

The circumference of a circle is supposed to be divided into

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360 equal parts called degrees, and each degree into 60 equal parts called minutes, and each minute into 60 equal parts called feconds, &c. And as many degrees, minutes, feconds, &c. as are contained in any arch, of fo many degrees, minutes, feconds, &c. is the angle, of which that arch is the measure, faid to be.

COR. Whatever be the radius of the circle of which the meafure of a given angle is an arch, that arch will contain the fame number of degrees, minutes, feconds, &c. as is manifest from Lemma 2.

III.

Let AB be produced till it meet the circle again in F, the angle CBF, which, together with ABC, is equal to two right angles, is called the Supplement of the angle ABC.

IV.

A ftraight line CD drawn through C, one of the extremities of the arch AC perpendicular upon the diameter paffing through the other extremity A, is called the Sine of the arch AC, or of the angle ABC, of which it is the measure.

COR. The Sine of a quadrant, or of a right angle, is equal to the radius.

V.

The fegment DA of the diameter paffing through A, one extremity of the arch AC between the fine CD, and that extremity, is called the Verfed Sine of the arch AC, or angle ABC.

VI.

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

VII.

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

COR. to def. 4. 6. 7. the fine, tangent, and fecant of any angle ABC, are likewife the fine, tangent, and fecant of its fupplement CBF.

It is manifest from def. 4. that CD is the fine of the angle CBF. Let CB be produced till it meet the circle again in G; and it is manifeft that AE is the tangent, and BE the fecant, of the angle ABG or EBF, from def, 8. 7.

COR.

COR. to def. 4. 5. 6. 7. The fine, verfed fine, tangent, and fecant, of any arch which is the measure of any given angle ABC, is to the fine, verfed fine, tangent, and fecant, of any other arch which is the measure of the fame angle, as the radius of the firft is to the radius of the fecond. ·

Let AC, MN be measures of the angle ABC, according to def. 1. CD the fine, DA the verfed fine, AE the tangent, and BE the fecant of the arch AC, according to def. 4. 5. 6. 7. and NO the fine, OM the verfed fine, MP the tangent, and BP the fecant of the arch MN, according to the fame 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 AB to BM, and BC or BA to BD as BN or BM to BO; and, by converfion, DA to MO as AB to MB. Hence the corollary is manifeft; therefore, if the radius be supposed to be divided into any given number of equal parts, the fine, verfed fine, tangent, and fecant of any given angle, will each contain a given number of thefe parts; and, by trigonometrical tables, the length of the fine, verfed fine, tangent, and fecant of any angle may be found in parts of which the radius contains a given number; and, vice verfa, a number expreffing the length of the fine, verfed fine, tangent, and fecant being given, the angle of which it is the fine, verfed fine, tangent, and fecant may be found.

VIII.

The difference of an angle from a right angle is called the complement of that angle. Thus, if BH be drawn perpendicular to AB, the angle CBH will be the complement of the angle ABC, or of CBF.

Fig. 3.

IX.

Let HK be the tangent, CL or DB, which is equal to it, the fine, and BK the fecant of CBH, the complement of ABC, according to def. 4. 6. 7. HK is called the co-tangent, BD the co-fine, and BK the co-fecant of the angle ABC.

COR. I.

The radius is a mean proportional between the tangent and co-tangent.

For, fince HK, BA are parallel, the angles HKB, ABC will be equal, and the angles KHB, BAE are right; therefore

the triangles BAE, KHB are fimilar, and therefore AE is to AB, as BH or BA to HK.

COR. 2. The radius is a mean proportional between the co-fine and fecant of any angle ABC.

Since CD, AE are parallel, BD is to BC or BA, as BA to BE.

PROP. I. FIG. 5.

Na right angled plain triangle, if the hypothenuse be made radius, the fides become the fines of the angles oppofite to them; and if either fide be made radius, the remaining fide is the tangent of the angle oppofite to it, and the hypothenufe the fecant of the fame angle.

Let ABC be a right angled triangle; if the hypothenufe BC be made radius, either of the fides AC will be the fine of the angle ABC oppofite to it; and if either fide BA be made radius, the other fide AC will be the tangent of the angle ABC opposite to it, and the hypothenufe BC the fecant of the fame angle.

About B as a centre, with BC, BA for distances, let two circles CD, EA be described, meeting BA, BC in D, E: Since CAB is a right angle, BC being radius, AC is the fine of the angle ABC, by def. 4. and BA being radius, AC is the tangent, and BC the fecant of the angle ABC, by def. 6. 7.

COR. 1. Of the hypothenufc a fide and an angle of a right angled triangle, any two being given, the third is alfo given. COR. 2. Of the two fides and an angle of a right angled triangle, any two being given, the third is also given.

PRO P. II. FIG. 6. 7.

HE fides of a plain triangle are to one another, as the fines of the angles oppofite to them.

In right angled triangles, this prop. is manifest from prop. 1. for if the hypothenufe be made radius, the fides are the fines of angles oppofite to them, and the radius is the fine of a right (cor, to def. 4.) which is oppofite to the hypothenuse.

In

In any oblique angled triangle ABC, any two fides AB, AC vill be to one another as the fines of the angles ACB, ABC vhich are oppofite to them.

From C, B draw CE, BD perpendicular upon the oppofite ides AB, AC produced, if need be. Since CEB, CDB are 'ight angles, BC being radius, CE is the fine of the angle CBA, ind BD the fine of the angle ACB; but the two triangles CAE, DAB have each a right angle at D and E; and likewise the common angle CAB, therefore they are fimilar, and confequently, CA is to AB, as CE to DB; that is, the fides are as the fines of the angles oppofite to them.

COR. Hence of two fides, and two angles opposite to them, in a plain triangle, any three being given, the fourth is alfo given.

IN

PROP. III. FIG. 8.

N a plain triangle, the fum of any two fides is to their difference, as the tangent of half the sum of the angles at the bafe, to the tangent of half their difference.

Let ABC be a plain triangle, the fum of any two fides AB, AC will be to their difference as the tangent of half the sum of the angles at the bafe ABC, ACB to the tangent of half their difference.

About A as a centre, with AB the greater fide for a distance, let a circle be described, meeting AC produced in E, F, and BC in D; join DA, EB, FB: and draw FG parallel to BC, meeting EB in G.

The angle EAB (32. 1.) is equal to the fum of the angles at the bafe, and the angle EFB at the circumference is equal to the half of EAB at the centre (20. 3.); therefore EFB is half the fum of the angles at the bafe; but the angle ACB (32. 1.) is equal to the angles CAD and ADC, or ABC together; therefore FAD is the difference of the angles at the bafe, and FBD at the circumference, or BFG, on account of the parallels FG, BD, is the half of that difference; but fince the angle EBF in a femicircle is a right angle (t. of this) FB being radius, BE, BG, are the tangents of the angles EFB, BFG; but it is manifeft that EC is the fum of the fides. BA, AC, and CF their difference; and fince BC, FG are parallel (2. 6.) EC is to CF, as EB to BG; that is, the fum of the Hh3 fides