Again, let the arc ABHb be the supplement of AB; draw the radius Cb, and produce it till it meet the circle in E, and the line TAt in t; demit bf and EF perpendicular to the diameter AD; then bf is the sine of the arc ABHb, or of the angle ACb, and EF is the sine of the arc ABbDLE, or of the angle which is measured by that arc. 17. DEF. 5. The versed sine of an arc is that part of the diameter passing through the beginning of the arc which is intercepted between the beginning of the arc and the right sine. Thus, AF is the versed sine of the arc AB, or of the angle ACB; and Af is the versed sine of the arc AHb, or of the angle ACb. 18. DEF. 6. The tangent of an arc is a straight line touching the circle in the beginning of the arc, produced thence till it meets the radius (produced) drawn through the end of the arc. Thus, AT is the tangent of the arc AB, or of the angle ACB; and At is the tangent of the arc AHb, or of the angle ACb. 19. DEF. 7. The secant of an arc is a straight line drawn from the centre through the end of the arc, and produced till it meets the tangent. Thus, CT is the secant of the arc AB, or of the angle ACB; and Ct is the secant of the arc AHь, or of the angle ACb. 20. DEF. 8. The cosine of an arc is the part of the diame ter passing through the beginning of the arc which is intercepted between the centre and the sine. Thus, CF is the cosine of the arc AB, or of the angle ACB; and Cf is the cosine of the arc AHb, or of the angle ACb. 21. DEF. 9. The cotangent of an arc is a line touching the circle in the end of the first quadrant, produced thence till it meets the radius (produced) drawn through the end of the arc. Thus, draw HK, touching the circle in H, and meeting the radius produced in K; then HK is the cotangent of the arc AB, or of the angle ACB. 22. DEF. 10. The cosecant of an arc is a straight line drawn from the centre through the end of the arc, and produced till it meets the cotangent. Thus, CK is the cosecant of the arc AB, or of the angle ACB. Scholium to Definitions 8, 9, 10. 23. It is manifest that the cotangent and cosecant are referred to the diameter HL passing through the end of the quadrant, in like manner as the tangent and secant are referred to the diameter AD passing through the beginning of the quad rant. It appears also, that the cosine, cotangent, and cosecant of an arc under 90 degrees, or of an angle less than a right angle, are respectively equal to the sine, tangent, and secant of the complement of that arc or angle. Draw BI perpendicular to the diameter HL, then CF-BI. But BI, HK, CK, are respectively the sine, tangent, and secant of the arc HB, reckoned from Has its beginning. Now the arc HB is the complement of the arc AB, and the angle HCB is the complement of the angle ACB. Therefore the cosine, cotangent, and cosecant of any arc or angle, are respectively equal to the sine, tangent, and secant of the complement of that arc or angle, whence they derive their names. Of the Properties and Relations of Trigonometrical Lines. 24. The sine, cosine, tangent, secant, &c. of any angle ACB, in a circle whose radius is AC (fig. 1), will be to the sine, cosine, &c. of the same angle ACB, in any other circle whose radius is AC (fig. 2), respectively as the radius of the former circle is to the radius of the latter. For the several right-angled triangles corresponding to one another in fig. 1 and 2, having one acute angle equal in each (the angle ACB in fig. 1 equal to ACB in fig. 2), are equiangular; therefore BF (1): BF (2) :: BC (1): BC (2). Similar analogies obtain in the case of all the other corresponding triangles, in each of which one corresponding side is radius. 25. Hence, if the radius of any circle be divided into 10,000,000 equal parts, and the length of the sine, tangent, or secant, &c. of any angle in such parts be given, the length of the sine, tangent, or secant of the same angle to any other given radius may be found by the common rule of proportion. A table exhibiting the length of the sine, tangent, and secant of every degree and minute of the first quadrant in such parts, of which 10,000,000 make the radius, is called a trigonometrical canon; and it will always be, as the tabular radius is to any other given radius, so is the tabular sine, &c. of any angle to the sine, &c. of the same angle to the given radius. 26. THE chord of 60 degrees is equal to radius. For then the angles ABC, BAC being equal (5. 1), each of them is 60 degrees = angle ACB; therefore the triangle ACB being equiangular, is also equilateral (Cor. 6. 1); therefore the chord AB=radius AC. 27. The sine of 90 degrees, or a quadrant, or a right angle, is equal to radius. This is manifest from an inspection of the figure. 28. The tangent of 45 degrees is radius. For then the angle ACT being half a right angle, the other acute angle ATC must also be half a right angle; therefore AC=AT (6.1). 29. The secant of 0 (or at the beginning of the circle) is radius. 30. The cosine of no degrees is radius, and the cosine of 90 degrees is 0. 31. The versed sine of 90 degrees is radius, and the versed sine of 180 degrees is the diameter. 32. Universally, the versed sine is always either the sum or the difference of the cosine and radius, namely, the sum in the two middle quadrants HD and DL, and the difference in the two extreme quadrants AH and LA. 33. As the arc increases from 90 degrees to 180 degrees, its sine, tangent, and secant decrease. Thus, as Ab increases, it is evident that the sine bf, the tangent At, and the secant Ct must decrease till the point b coincides with D. Consequently, if there be two arcs between 90° and 180°, the greater arc will have the less sine, tangent, and secant. 34. Let the arcs Ab and AB be supplements to each other, namely, Ab greater, and AB less than a quadrant; then will their sines bf and BF be equal. Because Ab+AB=180°=Ab+bD, the arc AB=6D, or the angle_bCD=BCA. But the radii BC, ¿C are equal. Hence the right-angled triangles ¿Cƒ, BCF are equal (26. 1), and bf=BF. 35. In like manner, Cf, the cosine of the arc AHb, is equal to CF, the cosine of the arc AB; but is negative, because it falls on the other side of the centre C, whence the cosines have their origin. 36. Again, At the tangent, and Ct the secant of Ab, are respectively equal to AT the tangent, and CT the secant of AB. The right-angled triangles TCA, CA, having the angle BCA or TCA equal to the angle ¿CD or tCA, and CA common, are equal (26. 1), therefore At-AT, and Ct=CT. But the tangent and secant, being now produced in a contrary direction, will be negative. 37. In like manner, the sine, cosine, tangent, and secant of any arc terminating in the third quadrant DL, will be respectively the same as those of an arc equal to the excess of the proposed arc above a semicircle. Thus, the sine of the arc AHDB is ßf, and is equal to BF, the sine of the arc AB=Ds. And so of the other lines. 38. The sine, cosine, tangent, and secant of an arc terminating in the fourth quadrant LA, will be the same as those of an arc equal to the supplement of the proposed arc to a whole circle. Thus, the sine of the arc AHDLË, is EF, and is equal to BF, the sine of the arc AB=AE, the supplement of AHDLE to a whole circle. And so of the other lines. 39. The versed sine Af of an arc Ab, above one quadrant, but under two quadrants, is equal to the difference between the versed sine of its supplement and the diameter; that is, Af AD-AF. For CF-Cf, therefore Af=DF=AD-AF. 40. The versed sine of an arc above two quadrants is (not merely equal to, but) the same as the versed sine of its supplement to a whole circle. Thus, the versed sine of the arc AHD, and also of the arc AELs (or rather its equal ABHb) is Af: and the versed sine of the arc AHDLE, and of the arc AB AE, is in both cases AF. Scholium. From these propositions it follows, that a table of sines, tangents, secants, and versed sines, computed for every degree and minute of the first quadrant, will serve for the whole circle. 41. The right-angled triangles BCF, TCA, CKH, having the several acute angles BCF, TCA, CKH equal (29. 1), are equiangular; hence the following analogies are deduced. 1. Cosine CF: sine BF :: radius CA : tangent TA. 2. Cosine CF: radius CB :: radius CA=CB: secant CT. 3. Sine BF: radius CB :: radius CH=CB: cosecant CK. 4. Sine BF: cosine CF :: radius CH : cotangent HK. 5. Tangent TA : radius CA :: radius CH=CA : cot. HK. Hence it appears that the radius is a mean proportional between the cosine and secant, between the sine and cosecant, and between the tangent and cotangent. 42. The tangents and cotangents of any two arcs of a circle are reciprocally proportional; that is, the tangent of the first arc tangent of the second :: cot. of the second arc : cot. of the first. Let T and C denote the tangent and cotangent of the first arc, t and c the tangent and cotangent of the second; then T×C =radius (41), and txc-radius2; therefore TxC=txc, therefore Tt:c: C (16. 6). B In the same manner it may be shown, that the cosines and secants of two arcs, and also the sines and cosecants, are reciprocally proportional. 43. The sine of any arc is equal to half the chord of double that arc. For the radius CA, perpendicular to BE, bisects the chord BE in F (3. 3), and also the arc BAE subtended by it (26. 3), because the angle BCA=ECA; therefore BF, the sine of the arc BA, is half the chord BE, which subtends the arc BAE, the double of BA. 44. Conversely. The chord of any arc is double the sine of half that arc. For the radius CA, perpendicular to the chord BE, bisects BE in F (3. 3), and also the arc BAE subtended by it (26.3), because the angles ACB and ACE are equal; therefore the chord BE is double the sine BF of the arc AB-half the arc BAE. 45. The sine of 30 degrees, the cosine of 60 degrees, and the versed sine of 60 degrees, are each equal to half the radius. The chord of 60 degrees is equal to the radius (26), therefore the size of 30 degrees, being equal to half the chord of 60 degrees (43), is equal to half the radius. Again, the sine of an arc being equal to the cosine of its complement (23), the cosine of 60 degrees is equal to the sine of 30 degrees, and therefore is equal to half the radius. Thirdly, the versed sine of an arc less than a quadrant being equal to the difference between the radius and the cosine, the versed sine of 60 degrees-radius-cosine of 60 degrees= radius-half radius=half radius. 46. On the diameter AD describe a semicircle ABD; draw AB the chord, and BF the sine, of the arc AB; draw the radius CLM perpendicular to the chord in L, and cutting the circle in M; then will the radius CLM bisect the chord AB in L (3.3), and the arc AMB in M (26. 3). Hence AL will be the sine, and CL the cosine of the arc AM. Lastly, join the points D, B, then the triangle ABD will be right-angled B |