because PAC is the supplement of A. (Chap. II. Sect. 3.) Substituting this, we obtain, as before, a2 = b2 + c2 2bc cos A. From the equations (1) we could obtain the values of the cosines of the angles of a triangle, and therefore of the angles themselves, if the sides were given in numbers; but as these expressions are badly adapted for calculation, we shall proceed to obtain others which offer greater facility. If it be required to express the sine of an angle in terms of the sides, we proceed as follows: therefore sin A1 cos2A, = - sin3A = (1 + cos A) (1 – cos A). We shall investigate the values of these factors separately, by equation (1): But as the difference of the squares of two quantities is equal to the product of the sum and diffe Substituting, as in the last case, for the difference of the squares in the numerator the product of the difference and sum, Combining (2) and (3) with the equation sin3A = (1 + cos A) (1 - cos A), we obtain (a+b+c) (b + c − a) (a+c−b) (a+b−c) 462 c2 (3) (4) By the following assumptions we are enabled to throw this expression into a form which is better adapted to calculation:-If s be the semiperimeter of the triangle, then 28 = a + b + c ; by subtracting 2c from each side of this equation, we obtain For the factors in the numerator of equation (4), substitute these values, and we obtain Taking the square root of this expression for sin A, and of similar expressions for sin3B and sin2C, we obtain s—c)), - - c)} sin A = 2√ {8 (8 - a) (s – b) (8 − 0 = bc sin B {s (s ac sin C 2√(8 (sa) (8 − b) (8 − c)} = ab (5) If these equations be divided by a, b, and c respectively, we obtain on the right-hand side the expression 2√ {8 (8-a) (sb) (s - c)} {s abc This expression is said to be symmetrical with regard to a, b, c, because these quantities are similarly involved. From the left-hand members we obtain or abc sin A: sin B : sin C. This corresponds with the result otherwise obtained in Prop. I. If for the value of 1 + cos A obtained in equation (2) we substitute its equal, 2 cos2A (Chap. IV. (23)); and for the factors (a+b+c) and (b + c − a) their values 28 and 2 (sa), we obtain Taking the square root of this and of similar expressions for cos B and cos C, we find If we treat equation (3) in the same way, substituting 2 sin2 for 1-cos A (Chap. IV. (24)), we If we divide the first of equations (7) by the first of equations (6), we obtain the value of tan A (Chap. II. (4)), and similarly of tan B and tan C, as follows: tan A √ { (8-6) (8-0)}, = b) c) (s 8 (8-b) tan B = √ (8) b)}. (8 8 (8 - c) For the factors in the numerator of equation (4), substitute these values, and we obtain Taking the square root of this expression for sin A, and of similar expressions for sin B and sin2C, we obtain sin C = 2V (s(s- a) (s − b) (s − c)} ab (5) If these equations be divided by a, b, and e respectively, we obtain on the right-hand side the ex pression 2√(s(8-a) (8-6) (sc)} abc This expression is said to be symmetrical with regard to a, b, c, because these quantities are similarly involved. From the left-hand members we obtain abe: sin A: sin Bin C. This corresponds with the result otherwise obtaine |