This process need be carried only as far as 30°. For sin (30°+x) + sin (30° — x) = 2 sin 30° cos x = cos x, cos (30°+x)—cos (30°-x)=-2 sin 30° sin x=— sin x, ..sin (30°x)cos xsin (30°-x), cos (30°+x)=- sin x + cos (30° — x). Moreover the sines and cosines need be calculated only to 45°, since sin (45° +x) = cos (45° — x), cos (45° +x)=sin (45° — x). In using this method the multiplication by cos 1', which occurs at each step, can be simplified by noting that cos 1'0.9999999—1— 0.0000001. Simpson's method is superseded in actual practice by much more rapid and convenient processes in which we employ the expansions of the trigonometric functions in infinite series. EXERCISE XXVI. 1. Compute the sine and cosine of 6' to seven decimal places. 2. In the formula (1) let y=1°. Assuming sin 1°—=0.017454+, cos 1°=0.999848+, compute the sines and cosines from degree to degree as far as 4°. § 46. DE MOIVRE'S THEOREM. Expressions of the form cos x + i sin x, when i=√√—1, play an important part in modern analysis. Given two such expressions. their product is cosx+isinx, cos +isin y, (cos x + i sin x) (cos y+i sin y) cos x cos y — sin x sin y+i (cos x sin y + sin x cos y) = cos(x + y) + i sin (x+y). Hence, the product of two expressions of the form cos x +i sinx, cos y+isin y is an expression of the same form in which x or y is replaced by x+y. In other words, the angle which enters into such a product is the sum of the angles of the factors. If x and y are equal, we have at once from the preceding (cosx+i sin x)= cos 2 x+i sin 2x; and again (cosx+isin)3=(cosx+isinx)2 (cosx+isinx) Similarly x)2 =(cos 2x + i sin 2x) (cos x + i sin x) cos 3x+isin 3x 4 (cosx+i sin x)=cos 4x+ i sin 4x, and in general if n is a positive integer Hence (cosx+isinæ)n=cos nx +isin n nx. (1) To raise the expression cos x + i sin x to the nth power when n is a positive integer, we have only to multiply the angle x by n. Again, if n is a positive integer as before, Since, however, x may be increased by any integral multiple of 2 without changing cos x + i sin x, it follows that all the From (1) and (2) it follows at once that if m and n are positive integers m (x+2kπ)+isin—(x+2 kπ) (k=0,1,2,.......n−1). (3) is a negative fraction, or Hence m (cos x + i sin x) ̄" = { cos (− x) + i sin (— x) }; Consequently if n is a positive or negative integer or fraction (cosx+isin) =cos[n (+2km)]+isin[n(x+2)], By aid of De Moivre's Theorem we may express sin no and cos no, when n is an integer, in terms of sin 0 and cos 0. Thus cos n0+i sin n0 = (cos 0+ i sin 0)” Equating now the real parts and the imaginary parts separately, we obtain 1. Find the six 6th roots of −1; of +1. 2. Find the three cube roots of i. 3. Find the four 4th roots of i. 4. Express sin 40 and cos 40 in terms of sin and cos 0. § 47. EXPANSION OF SIN x, Cos x, AND TAN ≈ IN Let one radian be denoted simply by 1, and let cos(x)+isin (-x)cos x-isin x-k-x. By taking the sum and difference of these two equations, and dividing the sum by 2 and the difference by 2i, we have |