Dividing (5) by (6), we have hence, sin. p+sin. qsin. (p+q) cos. (p-2), sin. p+sin. q sin. p-sin. q sin. ≥ (p+q). sin. 1 (p−q) ___ tang. } (p+q) cos. (p + q) cos. (p-q) tang. (p-q) Numerical Values of Sines, Tangents, &c. 2 (9.) § 17. Having deduced a few of the most simple relations of the trigonometrical functions of angles, we will now proceed to determine their numerical values. Sines of very small angles are obviously very nearly equal to the arcs which measure these angles. We have found (86) that the length of an arc of 1' is equal to 0.0002908882, &c. If we regard this as the sine of 1', from which, as will be hereafter shown, it does not differ even in its eleventh decimal figure, we may find the cosine of 1' as follows (§ 8): 2 cos. 1'=√1-sin. 1'=0.9999999577, &c. Having thus found the sine and cosine of 1', we may continue our work for larger angles by the aid of equations (1) and (3), ($16), which give sin. (a+b) = 2 sin. a cos. b—sin. (a—b). cos. (a+b) = 2 cos. a cos. b-cos. (a—b). Putting b=1′ and a=1', 2', 3', 4', &c., successively, we shall obtain sin. 2′ = 2 sin. 1' cos. 1'-sin. O'. Using the above value of sine and cosine of 1', executing the work above indicated, and preserving only nine decimal places, we find The work might be continued in this way to any desired аз CD = sin. x = αo +α1 x+α2 x2 +αz Ñ3 +αş X1 +α5 x3 +,&c. 4 (1.) (2.) When x = 0, the sine of a becomes zero, which requires a to = 0. Since the sine of a negative arc (§12) is the same as minus the sine of the same arc taken positively, it follows that the expression (1) for the sine of a must be of such a form as to change the algebraic signs of all its terms when - is written for+x, consequently it cannot contain any even powers of x. Therefore a2 = 0; α1 =0, &c. Again, when x = 0, the cosine of a becomes 1, so that b1 = 1. x =1. And, since the cosine of a negative arc is precisely the same as the cosine of the same arc taken positively, it follows that the expression for the cosine of x must be of such a form as not to change the algebraic signs of any of its terms when -x is written for+x, consequently it cannot contain any odd powers of x. Therefore b1 = 0; b2 = 0; b=0, &c. 3 Equations (1) and (2) must therefore assume the following forms: sin. x =α1x+ Az x3 +α5 x2 +αq x2 +, If we suppose the arc x to be exceedingly small, we may obtain a very close approximate value for the sine of x, in (3); by retaining only the first power of x, we thus find sin. x = α1 x, very nearly. But in very small arcs the sine and the arc are very nearly equal; hence, when x is very small, we must have sin. x = x, very nearly. Consequently a1 = 1. Using this value of a, equations (3) and (4) become sin. x=x+αz X3 + α5 x5 + Aq x2 +, &c. cos. x=1+b2 x2 + b1 x2 + bç x2 +, &c. 4 6 (5.) (6.) It now remains to find the other coefficients, a3, A5, A7, &C., ba, ba, be, &c. For this purpose we proceed as follows: In (5) write 2 x for x, and we obtain sin. 2x=2x+8 αz x3 +32 α5 x5 +128 α, x2 +, &c. (7.) Dividing (7) by 2, we have sin. 2x=x+4 αz x2 +16 α5 25 +64 αy x2+, &c. (8.) Taking the product of (5) and (6), we find The left-hand members of (8) and (9) being equal [§ 15, eq. (5)], their right-hand members must also be equal; hence equating the coefficients of like powers of x, we obtain Equations (5) and (6), when squared, become respectively Since the left-hand member of (13) is equal to zero [§ 8, eq. 5 of B], the right-hand member must also equal zero; hence the coefficients of the different powers of a must equal zero, which leads to the following conditions: Equations (10) and (14) give immediately these results : sin. x = x Hence, equations (5) and (6) become 207 1×2×31×2×3×4×5 1×2×3×4×5×6 x 7+, &c. (15.) (16.) If we divide the right-hand member of (15) by the right-hand member of (16), we shall obtain an infinite series, giving the value of tang. x, as follows: Other series might be found by combining (15) and (16) for the cotan. x, sec. x, and cosec. x. The are of 1' has already been found [§ 6] to be equal to 0.0002908882086, &c. Putting this value for x, in the above series, we find If we confine ourselves to thirteen decimals, we shall obtain From this we see that the sine of 1' does not differ from its corresponding arc until after we pass the 11th decimal place, as was asserted under § 17. For ordinary purposes, seven decimal places is as accurate as is desired. Having already (817) computed the sine and cosine of each minute as far as 5', we will now, by the aid of equations (15) and (16), continue the computation, limiting ourselves to seven decimals, and shall obtain as follows: |