Produce AB till it meet the circle in E, and draw DBF perpendicular to AE. Then, because ABC, ABD are two angles at the centre of the circle ACF, the angle ABC is to the angle ABD as the arch AC to the arch AD, (33. 6.); and therefore also, the angle ABC is to four times the angle ABD as the arch AC to four times the arch AD (4. 5.). D But ABD is a right angle, and therefore four times the arch AD is equal to the whole circumference ACF; therefore, the angle ABC is F to four right angles as the arch AC to the whole circumference ACF. ( COR. Equal angles at the centres of different circles stand on arches which have the same ratio to their circumferences. For, if the angle ABC, at the centre of the circles ACE, GHK, stand on the arches AC, GH, AC is to the whole circumference of the circle ACE, as the angle ABC to four right angles; and the arch HG is to the whole circumference of the circle GHK in the same ratio. There fore, &c. DEFINITIONS. I. IF two straight lines intersect one another in the centre of a circle, thé arch of the circumference intercepted between them is called the Measure of the angle which they contain. Thus the arch AC is the measure of the angle ABC. II. If the circumference of a circle be divided into 360 equal parts, each of these parts is called a Degree; and if a degree be divided into 60 equal parts, each of these is called a minute; and if a Minute be divided into 60 equal parts, each of them is called a Second, and so on. And as many degrees, minutes, seconds, &c. as are in any arch, so many degrees, minutes, seconds, &c. are said to be in the angle measured by that arch. Cor. 1. Any arch is to the whole circumference of which it is a part, as the number of degrees, and parts of a degree contained in it is to the number 360. And any angle is to four right angles as the number of degrees and parts of a degree in the arch, which is the measure of that angle, is to 360. Cor. 2. Hence also, the arches which measure the same angle, whatever be the radii with which they are described, contain the same number of degrees, and parts of a degree. For the number of degrees and parts of a degree contained in each of these arches has the same ratio to the number 360, that the angle which they measure has to four right angles (Cor. Lem. 1.). The degrees, minutes, seconds, &c. contained in any arch or angle, are usually written as in this example, 49°. 36'. 24". 42"; that is, 49 degrees, 36 minutes, 24 seconds, and 42 thirds. III. Two angles, which are together equal to two right angles, or two arches which are together equal to a semicircle, are called the Supplements of one another. IV. 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 AВС, of which AC is the measure. The segment DA of the diameter passing through A, one extremity 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. VI. A straight line AE touching the circle at A, one extremity of the arch AC, and meeting the diameter BC, which passes through C the other extremity, 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. VII. 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. 4, 6, 7, the sine, tangent, and secant of any angle ABC, are likewise the sine, tangent, and secant of its supplement CBF. It is manifest, from Def. 4, that CD is the sine of the angle CBF. Let CB be produced till it meet the circle again in I; and it is also manifest, that AE is the tangent, and BE the secant, of the angle ABI, or CBF, from Def. 6, 7. COR. to Def. 4, 5, 6, 7. The sine versed sine, tangent, and secant of an 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. B OMD A Let AC, MN be measures of the angle ABC, according to Def. 1.; CB the sine, DA the versed sine, AE the tangent, and BE the secant of the arch AC, according to Def. 4, 5, 6, 7; 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: NO:: rad. CB: rad. NB, and AE: MP:: rad. AB : rad. BM, also BE: BP :: AB: BM; likewise because BC: BD :: BN: BO, that is, BA: BD :: BM: BO, by conversion and alternation, AD: MO:: AB: MB. Hence the corollary is manifest. And therefore, if tables be constructed, exhibiting in numbers the sines, tangents, secants, and versed sines of certain angles to a given radius, they will exhibit the ratios of the sines, tangents, &c. of the same angles to any radius whatsoever. In such tables, which are called Trigonometrical Tables, the radius is either supposed 1, or some number in the series 10, 100, 1000, &c. The use and construction of these tables are about to be explained. VIII. The difference between any angle and a right angle, or between any arch and a quadrant, is called the Complement of that angle, or of that arch. Thus, if BH be the complement of the obtuse angle FBC is the angle HBC, its excess above a right angle; and the complement of the arch B D I FC iş HC, IX. The sine, tangent, or secant of the complement of any angle is called the Cosine, Cotangent, or Cosecant of that angle. Thus, let CL or DB, which is equal to CL, be the sine of the angle CBH; HK the tangent, and BK the secant of the same angle; CL or BD is the cosine, HK the cotangent, and BK the cosecant of the angle ABC. Cor. 1. The radius is a mean proportional between the tangent and the cotangent of any angle ABC; that is, tan. ABCXcot. ABC = R2. For, since HK, BA are parallel, the angles HKB, ABC are equal, and KHB, BAE are right angles; therefore the triangles BAE, ΚΗΒ are similar, and therefore AE is to AB, as BH or BA to HK. COR. 2. The radius is a mean proportional between the cosine and secant of any angle ABC; or cos. ABC X sec. ABC=R2. Since CD, AE are parallel, BD is to BC or BA, as BA to BE. PROP. I. In a right angled plane triangle, as the hypotenuse to either of the sides, so the radius to the sine of the angle opposite to that side: and as either of the sides is to the other side, so is the radius to the tangent of the angle opposite to that side. Let ABC be a right angled plane triangle, of which BC is the hypotenuse. From the centre C, with any radius CD, describe the arch DE; draw DF at right angles to CE, and from E draw EG touching the circle in E, and meeting CB in G; DF is the sine, and EG the tangent of the arch DE, or of the angle C. The two triangles DFC, BAC are equiangular, because the angles, DFC, BAC are right angles, and B the angle at C is common. There Also, because EG touches the circle in E, CEG is a right angle, and therefore equal to the angle BAC; and since the angle at C is common to the triangles CBA, CGE, these triangles are equiangular, wherefore CA: AB:: CE: EG; but CE is the radius, and EG the tangent of the angle C; therefore, CA: AB:: R: tan. C. Cor. 1. As the radius to the secant of the angle C, so the side adjacent to that angle to the hypotenuse. For CG is the secant of Ff the angle C (def. 7.), and the triangles CGE, CBA being equiangular, CA: CB:: CE: CG, that is, CA: CB :: R: sec. C. Cor. 2. If the analogies in this proposition, and in the above corollary be arithmetically expressed, making the radius=1, they give AB AB sin. C= tan. C= sec. C BC' AC' because B is the complement of C, cos. B= Cor. 3. In every triangle, if a perpendicular be drawn, from any of the angles on the opposite side, the segments of that side are to one another as the tangents of the parts into which the opposite angle is divided by the perpendicular. For, if in the triangle ABC, AD be drawn perpendicular to the base BC, each of the triangles CAD, ABD being right angled, AD: DC:: R: tan. CAD, B and AD: DB:: R: tan. DAB; therefore, ex æquo, DC: DB::tan. CAD: tan. BAD. SCHOLIUM. DC The proposition, just demonstrated, is most easily remembered, by stating it thus: If in a right angled triangle the hypotenuse be made the radius, the sides become the sines of the opposite angles; and if one of the sides be made the radius, the other side becomes the tangent of the opposite angle, and the hypotenuse the secant of it. PROP. II. The sides of a plane triangle are to one another as the sines of the oppo site angles. From A any angle in the triangle ABC, let AD be drawn perpendicular to BC. And because the triangle ABD is right ✔ angled at D, AB: AD :: R: sin. B; and for the same reason, AC:AD::R: sin. Ç, and inversely, AD: AC :: sin. C: R; therefore, ex æquo inversely, AB:AC:: sin. C: sin. B. In the same manner, it may be demonstrated, that AB: BC:: sin. C: sin. A. Therefore B &c. Q. E. D. |