we shall have sin 27°30′=R(√1⁄2—√), and cos 27° 30′=R √(1 + √1). * XXI The values of sin a and cos la may also be obtained in terms of sin a; which will be useful on many occasions. These values are : sin {a=√(R®+R sin a)—¦ √ (Ra—R sin a), cos a=√(R2+R sin a)+ √(R2-R sin a). Accordingly, by squaring the first, we shall have sin2 }a= (R2+R sin a)+4(R3—R sin a)—¦ √ (R1—R3 sin2 a)=}R2— R cos a; in like manner, we should have cos2 a=}R2+}R cos a, values which agree with those already found for sin a and cosa. It must be observed, however, that if cos a were negative, the radical (R-R sin a) would require to be taken with a contrary sign in the values of sin a and cosa; and thus the one value would be changed into the other. XXII. The formulas of Art. 19, furnish a great number of consequences; among which it will be enough to mention those of most frequent use. The four which follow, sin a cos b= R sin (a+b)+1 R sin(a—b) serve to change a product of several sines or cosines into linear sines or cosines, that is, into sines and cosines multiplied only by constant quantities. * Let (x-2)√=√({R2—}R cos ́a)= sin a; then will x2—2xz+z2 R2-R cos a. Assume 22+22-R2; then will 2xz R. cos a, and eliminating z,gives x2+ =R2: hence -R2x2+}R1=}R2(R2 R2. cos2 a 4x2 R2 cos? a)=R2. sin2 a; and x2- =±1⁄2R. sin a; or x=√R(R+ sin a)• 2 Hence z=±√R(R‡ sin a). Therefore a√√R(R+ sin a F√√R(RF sin a); or, or, (x—*)√}=}√R(R± sin a) ‡1√R(RF sin a) ; sin a={√R2+R sin a—1√/R2—R sin a. In a similar manner, the value for cos la is obtained. XXIII. If in these formulas we put a+b=p, a—b=q, cos p+cos q=cos § (p+q) cos § (p−q) new formulas, which are often employed in trigonometrical calculations for reducing two terms to a single one. XXIV. Finally, from these latter formulas, by dividing, tang (p+q) sin p―sin qcos} (p+q)sin}(p—q) ̄ ̄tang} (p— sin p+sin q__sin} (p+q)_tang} (p+q) = cos p+cos q cost (p+q) R Ꭱ sin p-sin q sin (p−q) tang} (p—q) cos p+cos q cost (p-q) sin p-sin q cot (p+q) cot (p+q) cos q-cos psin}} (p+q) ̄ R ht cos p+cos q cost (p+q) cost (p-q) cot} (p+q) Formulas which are the expression of so many theorems. From the first, it follows that the sum of the sines of two arcs is to the difference of these sines, as the tangent of half the sum of the arcs is to the tangent of half their difference. XXV. Making b=a, or q=0, in the formulas of the three preceding Articles, we shall have the following results: but 20 cos2 a= R2+ R cos 2a сяго R2-R cos 2a 2 cos2 p R R 2 sin2 p 2 sin ip cos p R XXVI. In order likewise to develope some formulas relative to tangents, let us consider the expression cot P -5 6. tangp values of sin (a+b) and cos (a+b), we shall find R (sina cosb+sin b cos a) tang (a+b)= 19. cos a cos b―sin b sin a cos a tang a cos b tang b Now we have sin a , and sinb= Ꭱ substitute these values, dividing all the terms by cos a cos b; we shall have R2 (tang a+tang b) tang (a+b)= R2-tang a tang b which is the value of the tangent of the sum of two arcs, expressed by the tangents of each of these arcs. For the tangent of their difference, we should in like manner find Suppose b=a; for the duplication of the arcs, we shall have the formula Suppose b=2 a; for their triplication, we shall have the formula in which, substituting the value of tang 2 a, we shall have 3 R2 tang a-tang3 a tang 3 a R2-3 tang2 a ON THE CONSTRUCTION OF TABLES. XXVII. The tables in common use, for the purposes of trigonometrical calculations, are tables which show the values of the logarithms of the sines, cosines, tangents, cotangents, &c. for all the degrees and minutes of the quadrant, calculated to a given radius. XXVIII. If the radius of the circle is taken equal to 1, and the lengths of the lines representing the sines, cosines, tangents, cotangents, &c. for every minute of the quadrant be ascertained, and written in a table, this would be the table usually called a table of natural sines, cosines, &c. XXIX. If such a table were known, it would be easy to calculate a table of sines, &c. to any other radius; since, in different circles, the sines, cosines, &c. of arcs containing the same number of degrees, are to each other as their radii. Also, if the natural, or numeral sines, cosines, &c. were known, it would be easy to calculate from them the logarithmic ones. XXX. Let us glance for a moment at one of the methods of calculating a table of natural sines. The radius of a circle being 1, the semi-circumference is known to be 3.14159265358979. This being divided successively by 180 and 60, or at once by 10800, gives .000290SSS2086657, for the arc of 1 minute. Of so small an arc the sine, chord, and arc, differ almost imperceptibly from the ratio of equality; so that the first ten of the preceding figures, that is, .0002908882 may be regarded as the sine of 1'; and in fact the sign given in the tables which run to seven places of figures is .0002909. By Art. 16, we have for any arc, cos = √(1 — sin2). This theorem gives, in the present case, cos 1': .9999999577. Then, by Art. 22, we Thus may the work be continued to any extent, the whole difficulty consisting in the multiplication of each successive result by the quantity 2 cos 1' 1.9999999154. * Or, the sines of 1' and 2' being determined, the work might be continued by the last proposition, thus: In like manner, the computer might proceed for the sines of degrees, &c. thus: *Multiply together the first and second formulas of Art. XIX. substitute for cos2b, R2--sin2b and recollect that the difference of the squares of two quantities is equal to the product of their sum and difference. |