45. Formulæ for the Transformation of Sums into Products. - From the four fundamental formulæ of Arts. 43 and 44 we have, by addition and subtraction, the following: These formulæ are useful in proving identities by transforming products into terms of first degree. They enable us, when read from right to left, to replace the product of a sine or a cosine into a sine or a cosine by half the sum or half the difference of two such ratios. .. x = (A+B), and y = (A — B). Substituting these values in the above formulæ, and putting, for the sake of uniformity of notation, x, y instead of A, B, we get sinxsin y2sin(x + y) cos(xy).. sinx - sin y 2 cos(x + y) sin † (x − y) . = cosx+cos y = 2 cos 1⁄2 (x + y) cos † (x − y) . (5) (6) cos y cos x 2sin(x + y) sin(x-y).. (8) = The formulæ are of great importance in mathematical investigations (especially in computations by logarithms); they enable us to express the sum or the difference of two sines or two cosines in the form of a product. The student is recommended to become familiar with them, and to com mit the following enunciations to memory: Of any two angles, the Sum of the sines Diff." 66 66 = 2 sin sum. cos diff. 2 cossum. sin diff. For, = 2 sin sum.sin diff. EXAMPLES. sin 5x cos 3x (sin 8x + sin 2x). = sin 5x cos 3x={sin (5x + 3x) + sin (5x-3x)} 66 2 sin cos = 2 sin 20 cos 3 = sin (20+3)+ sin (20-34). sin 60° + sin 30°=2 sin 45° cos 15°. sin 3x + sin x =2 cos 6 a sin 2 α. = 2 sin 2 x cos x. =2 cos 2 x sin x. sin 4x + sin 2x = 2 sin 3x cos x. 46. Useful Formulæ. The following formulæ, which are of frequent use, may be deduced by taking the quotient of each pair of the formulæ (5) to (8) of Art. 45 as follows: 2 sin(x + y) cos(x − y) 2 cos(x+y) sin † (x − y) = tan(x + y) cot≥ (x − y) The following may be proved by the student in a similar 6. cos x + cos y = cot i̟ (x + y) cot † (x − y). cos y cos 47. The Tangent of the Sum and Difference of Two Angles. - Expressions for the value of tan(x+y), tan (x − y), etc., may be established geometrically. It is simpler, however, to deduce them from the formulæ already established, as follows: Dividing the first of the 'x, y' formulæ by the second, we have, by Art. 23, Dividing both terms of the fraction by cos x cos y, 5. sin (x+y) sin (x − y) = sin2x - sin2y 6. cos (x + y) cos (x − y) = cos2x - sin2y tan x+tan y = 10. If tan x = and tan (x − y) = }. and tany, prove that tan (x + y) = §, 11. Prove that tan 15° = 2 – √3. 12. If tan x=, and tany, prove that tan(x+y)=1. What is (x+y) in this case? 48. Formulæ for the Sum of Three or More Angles. — Let x, y, z be any three angles; we have by Art. 43, sin (x+y+2)= sin (x + y) cos z + cos (x + y) sinz = sin x cos y cos z + cos x sin y cos z +cosx cos y sinz - sin x siny sinz Dividing (1) by (2), and reducing by dividing both terms. of the fraction by cos x cos y cos z, we get 1. Prove that sinx + siny + sinz - sin (x + y + z) = 4 sin(x+y) sin (y + z) sin (z+x). By (6) of Art. 44 we have sinxsin (x + y + z)=-2 cos(2x+y+ z) sin (y + z), and siny+sinz 2 sin (y + z) cos (y-z). = .. sin+siny + sinz - sin (x + y + z) =2sin(y+z) cos † (y—z) — 2 cos(2x+y+z) sin(y+z) =2sin(y + z){cos (y — 2) — cos (2x + y + z) } =2sin(y + z) 2 sin(x + y) sin (x + z) =4 sin(x+y) sin (y + z) sin † (z+x). Prove the following: 2. cosx+cos y + cos z + cos (x + y + z) = 4 cos(y + z) cos (z+x) cos(x + y). |