473 PLANE TRIGONOMETRY. LEMMA 1. Fig. 1. LET ABC be a rectilineal angle: if about the point B as a centre, and with any distance BA, a circle be described, 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 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 with any two distances 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 same 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. 1 DEFINITIONS. FIG. 3. I. LBT 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 mea. sure of the angle ABC. II. The circumference of a circle is supposed to be divided in to 360 equal parts called degrees, and each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds, &c. 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. 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. 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 straight line CD drawn through C, one of the extremi ties of the arch AC perpendicular upon the diameter passing 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. The segment DA of the diameter passing through A, one extremity of the arch AC, between the sine CD, and that extremity, is called the Versed Sine of the arch AC, or angle ABC, VI. A straight line AE touching the circle at A, one extremity of the arch AC, and meeting the diameter BC passing through the other extremity Cin E, is called the Tangent of the arch AC, or of the angle ABC. VII. The straight 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 sine, tangent, and secant of any angle ABC, are likewise the sine, tangent, and secant of its supplement CBF. 1 It is manifest from def. 4. that CD is the sine of the angle CBF. Let CB be produced till it meet the cirele again in G; and it is manifest that AE is the tangent, and BE the secant, of the angle ABG or EBF, from def. 6. 7. Cor. to def. 4. 5. 6. 7. 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 is to the radius of the second. Let AC, MN be measures of the angle ABC, according to def. d. CD the sine, DA the versed sine, AE the tangent, and BE the secant, of the arch AC, according to def. 4. 5. 6. 7. and NO the sine, OM the versed sine, MP the tangent, 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 AB to BM, and BC or BA to BD, as BN or BM to BO; and, by conversion, DA to MO as AB to MB. Hence the corollary is manifest; therefore, if the radius be supposed to be divided into any given number of equal parts, the sine, versed sine, tangent, and secant of any given angle, will each contain a given number of these parts; and, by trigonometrical tables, the length of the sine, versed sine, tangent, and secant of any angle may be found in parts of which the radius contains a given number; and, vice versa, a number expressing the length of the sine, versed sine, tangent, and secant, being given, the angle of which it is the sine, versed sine, tangent, and secant, may be found. Fig. 3. The difference of an angle from a right angle, is called the complement of that angle. Thus, if BĦ be drawn per- IX. the sine, and BK the secant of CBH, the complement gle ABC. Cor. I. The radius is a mean proportional between the tangent and cotangent. VIII. For, since HK, BA are parallel, the angles HKB, ABC, will be equal, and the angles KHB, BAE are right; therefore the triangles BAE, KHB are similar, and there fore AE is to AB, as BH or BA to HK. Cor. 2. The radius is a mean proportional between the co sine and secant of any angle ABC. Since CD, AE, are parallel, BD is to BC or BA, as BA to BE. PROP. I. Fig. 5. In a right angled plane triangle: if the hypothemuse be made radius, the sides become the sines of the angles opposite to them; and if either side be made radius, the remaining side is the tangent of the angle opposite to it, and the hypothenuse the secant of the same angle. Let ABC be a right angled triangle: if the hypothenuse BC be made radius, either of the sides AC will be the side of the angle ABC opposite to it; and if either side BA be made radius, the other side AC will be the tangent of the angle ABC opposite to it, and the hypothenuse BC the secant of the same 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 sine of the angle ABC, by def. 4. and BA being radius, AC is the tangent, and BC the secant, of the angle ABC, by def. 6.7. Cor. 1. Of the hypothenuse, a side, and an angle of a right angled triangle, any two being given, the third is also given. Cor. 2. Of the two sides and an angle of a right angled triangle, any two being given, the third is also given. PROP. II. Fig. 6. 7. The sides of a plane triangle are to one another, as the sines of the angles opposite to them. In right angled triangles, this Prop. is manifest from Prop. 1. for if the hypothenuse be made radius, the sides are the sines of the angles opposite to them, and the radius is the sine of a right angle (cor. to def. 4.) which is opposite to the hypothenuse. In any oblique angled triangle ABC, ány two sides AB, AC, will be to one another as the sines of the angles ACB, ABC, which are opposite to them. From C, B, draw CE, BD, perpendicular upon the opposite sides AB, AC, produced if need be. Since CEB, CDB are right angles, BC being radius, CE is the sine of the angle CBA, and BD the sine 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 similar, and consequently, CA is to AB, as CE to DB; that is, the sides are as the sines of the angles opposite to them. CoR. Hence of two sides, and two angles opposite to them, in a plane triangle, any three being given, the fourth is also given. PROP. III. Fig. 8. In a plane triangle, the sum of any two sides is to their difference, as the tangent of half the sum of the angles at the base, to the tangent of half their difference. Let ABC be a plane triangle, the sum of any two sides AB, AC will be to their difference as the tangent of half the sum of the angles at the base ABC, ACB to the tangent of half their difference. About A as a centre, with AB the greater side 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 sum of the angles at the base, and the angle EFB at the circumference is equal to the half of EAB at the centre (20. 3.); therefore EFB is half the sum of the angles at the base; 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 base, and FBD at the circumference; or BFG, on account of the parallels FG, BD, is the half of that difference: but since the angle EBF in a semicircle is a right angle (1. of this), FB being radius, BE, BG are the tangents of the angles EFB, BFG; but it is manifest that EC is the sum of the sides BA, AC, and CF their difference; and since BC, FG are parallel (2. 6.), EC is to CF, as EB to BG; that is, the sum of the sides is to their difference, ag |