And as A, in trigonometry, must always be less than 180°, A will always be less than 90°, and consequently its sine, tangent, &c. will have the same sign as in the table (K. 100.) GENERAL PROPERTIES OF THE sines, tangents, chords, &c. OF SINGLE ARCS. PROPOSITION VIII. (Plate I. Fig. 1.) (P) The chord of any arc is a mean proportional between the diameter and the versed sine of that arc. The angle bгB being in a semicircle, is a right angle (Euclid 31 of III.) And FG, by the definition of a sine, is perpendicular to B, therefore the triangles bGF, and BGF, are similar to each other, and to the whole triangle bFB, (Euclid 8 of VI.) Hence, bB: BF::BF BG; but bв is the diameter, BF the chord of the arc BiF, and BG is the versed sine of the arc BiF. Q. E. D. (Q) COROL. The sine of any arc is a mean proportional between the versed sine of that arc, and the versed sine of its supplement. For the Q. E. D. FbGBFG, consequently bG GF::GF: BG. PROPOSITION IX. (Plate I. Fig. 1.) (R) The square of the sine of any arc added to the square of its versed sine, is equal to the square of the chord of that arc; or to four times the square of the sine of half that arc. For GF2+GB2=BF2 (Euclid 47 of Ï.) Draw ci parallel to br, and it will cut BF at right angles in z; but if it cut it at right angles, it will bisect it (Euclid 3 of III.), therefore BZ-ZF, and BF=2BZ, therefore BF2=4BZ3. Q. E. D. (S) The tangent of any arc is a mean proportional between the sum and difference of the radius and the secant; viz. rad.+ sec. tang. tang. sec.-rad. Join FC, and produce it to h and to T, so as to meet the tangent BT of the arc BF. Then hт × FT=TB2 (Euclid 36 of III.) that is hT: TB::TB: FT. (Euclid 17 of VI.) But hr=rad.+ secant CT, and FT=secant cr-rad.; therefore rad.+sec.: tang. tang. sec.-rad. Q. E. D. PROPOSITION XI. (Plate I. Fig. 1.) (T) The right angled triangles FGC, TBC, CAK, CEF, are equiangular and similar. For FGC, TBC are right angles, and the angle FCB= ≤ TCB, therefore the remaining angle CFG=/ CTB. = The triangle CEF triangle CGF, for the CFG = FCE (Euclid I. and 29), and ▲ FEC= CGF, being each of them right angles, and the side cr is common to both the triangles, therefore they are equal (Euclid 26 of I.) Again, AK is parallel to EF, by the definition of a tangent and sine; therefore the triangle CAK is equiangular with the triangle CEF; and consequently with CGF and CBT. (A) AK tangent::co-tangent: radius. BT : CA. :EF=CG:: ск : CF. co-tangent cosine: co-secant radius. (B) By comparison. Cosine sine :: co-tangent: radius. (U and Z, above.) : (C) Tangent x co-tangent=radius square. (Z, above.) Therefore the tangent of any arc x its co-tangent the tangent of any other arc x its co-tangent. (D) Sine x co-secant radius square. (Y, above.) Therefore the sine of any arc x its co-secant sine of any other arc x its co-secant. (E) Co-sine x secant radius square. (X, above.) Therefore the co-sine of any arc x its secant co-sine of any other arc its secant. (F) Square radius=square sine + square co-sine. For, CF2GF2+CG2 (Euclid 47 of I.) (G) Square radius=square secant-square tangent=square co-secant-square co-tangent. For, CB2CT2-TB2 (Euclid 47 of I.), and Ac2=CK2 — AK2. (H) Square radius-co-sine x secant (X. 103.)=sine x co-secant (Y. 103.) tangent x co-tangent (Z. 103.) (I) Sine tangent x co-sine square rad. co-sec. radius (Y. 103.)= (U.103.): = (B. 103.), &c. (K) And, generally, if any arc*, rad=radius, cos=cosine, tang tangent, cot=cotangent, sec secant, cosec=co-secant, vers versed sine, vers sup=versed sine of the supplement, or superversed sine, the following formula will be easily deduced, where the sign × is represented by a point (.). (L) I. Sine A= √rad2-cos2 A= COS A. tang A___tang A. rad sec rad2 rad tang A. rad rad /sec2 A-rad2 Sec A. COS A_tang A. cot a ✓vers sup A. vers a = √(2 rad vers A)-vers2 a. (M) II. Cos A = rad √rad2 — sine2 A= rad. sine a vers A vers sup A rad rad2 ✓rad2+tang2 A rad2 = sec A rad. cot A tang A sine A. cot A rad√(2rad. versa)—vers2▲ _rad✓(2rad.verssupa)—vers sup2à Cosec A * Emerson's Trigonometry, 2d edit. Prop. I. Scholium. √(2 rad . vers ▲)-vers3 A √(2rad. vers sup ▲)—vers sup2 A (P) V. Sec Arad + tang A= rad rad, tang A sine A. sine A. cosec A COS A COS A rad - vers A vers sup A-rad' rad2 (Q) VI. Cosec A = √rad+cot A=sine rad1 rad. sec A tang A rad. cot A COS A rad2 rad3 tang A. cot A √rad2 - cos A tang A rad+tang A /sec2 A-rad2 (R) VII. Vers A=2 rad-vers sup Arad-cos Arad (T) If the co-versed sine be wanted, it may be found by subtracting the sine from radius; that is, co-vers a = rad sine A; also the chord = √ vers2 ▲ + sine2 ✓rad-cos A2+(rad2—cos2a)=√2rad · (rad —cos a). (U) Besides the preceding formulæ, others may be deduced, thus, sineoA +cos2A=rad2 (F. 103) rad2 +tang2 A = sec2 A, and rad2+cot2 a cosec2 A (G. 104.) Now because tang a = rad. sine a hence rad2+tanga COS A =rad+ (N. 104.) tang2 A= rad. sine2 A GENERAL PROPERTIES OF SINES, TANGENTS, &c. OF DOUBLE ARCS AND OF HALF arcs. PROPOSITION XII. (Plate I. Fig. 1.) (W) The right-angled triangles bgf, bgf, brb, and Ózв, are equiangular and similar; and cz, the co-sine of the arc Bi, is equal to the half of be, the chord of the supplement of double the arc Bi. For bGF, BGF, and bгB have already been shewn to be equiangular (P. 102.), and the triangles CZB, FGB, have the angle at B common to both of them; also the angle bFB, being an angle in a semicircle, is a right angle; and since cz is parallel to br, by construction (R. 102.), the angle czB is likewise a right angle. Now BZ CZ:: BF: Fb, but BZBF (R. 102.), therefore cz=16F. Hence the following proportions. CB BZ (sine arc Bi):: BF (=2BZ): BG : radius sine of an arc:: double that sine: versed sine of double that arc. CB BZbF (2cz): GF radius sine of an arc:: double its cosine: sine double the arc. : CB: cz::br (2cz): bG radius cosine of an arc:: double its cosine : versed sine supplement of double the arc. CB: CZ:: BF (2BZ): GF (X) (Y) (Z) (A). radius cosine of an arc::double the sine sine of double the arc. : bf (=2bz): bg::bf (=2cz): GF (B) double the sine of an arc : versed sine double arc:: double cosine: sine of double the arc. (C) (D) BF (=2BZ) GF::bF (=2cz): bG double the sine of an arc sine double the arc:: double the cosine : versed sine of the supplement of double the arc. BZ: CZ:: GB GF sine of an arc its cosine::versed sine double arc : sine double arc. |