(73.) Expressions for the sine and cosine of a double arc. If, in the formulas of the preceding article, we make b=a the first and second will become Making radius equal to unity, and substituting the values of sin. a, cos. a. &c., from Art. 28, we obtain (74.) Expressions for the sine and cosine of half a given arc. If we put a for a in the preceding equations, we obtain 2 sin. a cos. a sin. a= cos. α= R cos. 2a-sin. a Ꭱ We may also find the sine and cosine of a in terms of a. Since the sum of the squares of the sine and cosine is equal to the square of radius, we have cos. 2a+sin. 2a=R2. And, from the preceding equation, cos. 'a-sin. a=R cos. a. If we subtract one of these from the other, we have (75.) Expressions for the products of sines and cosines. By adding and subtracting the formulas of Art. 72, we obtain If, in these formulas, we make a+b=A, and a−b=B; that is, a=1(A+B), and b=1(A−B), we shall have (76.) Dividing formula (1) by (2), and considering that sin. A+sin. B sin. (A+B) cos. (A-B)___tang. (A+B) sin. A—sin. B ̄sin. (A−B) cos. (A+B) ̄tang. (A−B) that is, The sum of the sines of two arcs is to their difference, as the tangent of half the sum of those arcs is to the tangent of half their difference. Dividing formula (3) by (4), and considering that COS. cot. sin. R The sum of the cosines of two arcs is to their difference, as the cotangent of half the sum of those arcs is to the tangent of half their difference. From the first formula of Art. 74, by substituting A+B for 2, we have Dividing formula (1), Art. 75, by this, we obtain sin. A+sin. B__sin. (A+B) cos. sin. (A+B) (A−B) __cos. (A—B) that is, The sum of the sines of two arcs is to the sine of their sum, as the cosine of half the difference of those arcs is to the cosine of half their sum. If we divide equation (1), Art. 72, by equation (3), we shall have sin. (a+b)__sin. a cos. b+cos. a sin. b a By dividing both numerator and denominator of the second tang. sin. member by cos. a cos. b, and substituting for COS. tain sin. (a+b)___tang. a+tang. b that is, The sine of the sum of two arcs is to the sine of their difference, as the sum of the tangents of those arcs is to the difference of the tangents. From equation (3), Art. 72, by dividing each member by cos a cos. b, we obtain The sine of the difference of two arcs is to the product of their cosines, as the difference of their tangents is to the square of radius. (77.) Expressions for the tangents of arcs. If we take the expression tang. (a+b)= R sin. (a+b) (Art. cos. (a+b) 28), and substitute for sin. (a+b) and cos. (a+b) their values given in Art. 72, we shall find tang. (a+b)= R (sin. a cos. b+cos. a sin. b) cos. a cos. b-sin. a sin. b If we substitute these values in the preceding equation, and divide all the terms by cos. a cos. b, we shall have 2 R2 (tang. a+tang. b) tang. (a+b)=R3—tang. a tang. b (78.) When the three sides of a triangle are given, the angles may be found by the formula (S-b) (S-c) sin. A=RV bc where S represents half the sum of the sides a, b, and c. Demonstration. Let ABC be any triangle; then (Geom., Prop. 12, B. IV.), Let a, b, c represent the sides opposite the angles A, B, C; By Art. 74, we have 2 sin. 2A=R2—R cos. A. Substituting for cos. A its value given above, we obtain Put S=(a+b+c), and we obtain, after reduction, Ex. 1. What are the angles of a plane triangle whose sides are 432, 543, and 654? Here S=814.5; S-b=382.5; S-c=271.5. Angle A=83° 25′ 13′′. In a similar manner we find the angle B=41° 0′ 39′′, the angle C=55° 34′ 8′′. and Ex. 2. What are the angles of a plane triangle whose sides are 245, 219, and 91? (79.) On the computation of a table of sines, cosines, &c. In computing a table of sines and cosines, we begin with finding the sine and cosine of one minute, and thence deduce the sines and cosines of larger arcs. The sine of so small an angle as one minute is nearly equal to the corresponding arc. The radius being taken as unity, the semicircumference is |