sin(pi*x)^2=1

$$sin^{2}{left (pi x right )} = 1$$

Дано уравнение
$$sin^{2}{left (pi x right )} = 1$$
преобразуем
$$- cos^{2}{left (pi x right )} = 0$$
$$sin^{2}{left (pi x right )} – 1 = 0$$
Сделаем замену
$$w = sin{left (pi x right )}$$
Это уравнение вида

a*w^2 + b*w + c = 0

Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$w_{1} = frac{sqrt{D} – b}{2 a}$$
$$w_{2} = frac{- sqrt{D} – b}{2 a}$$
где D = b^2 – 4*a*c – это дискриминант.
Т.к.
$$a = 1$$
$$b = 0$$
$$c = -1$$
, то

D = b^2 – 4 * a * c =

(0)^2 – 4 * (1) * (-1) = 4

Т.к. D > 0, то уравнение имеет два корня.

w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b – sqrt(D)) / (2*a)

или
$$w_{1} = 1$$
$$w_{2} = -1$$
делаем обратную замену
$$sin{left (pi x right )} = w$$
Дано уравнение
$$sin{left (pi x right )} = w$$
– это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$pi x = 2 pi n + {asin}{left (w right )}$$
$$pi x = 2 pi n – {asin}{left (w right )} + pi$$
Или
$$pi x = 2 pi n + {asin}{left (w right )}$$
$$pi x = 2 pi n – {asin}{left (w right )} + pi$$
, где n – любое целое число
Разделим обе части полученного ур-ния на
$$pi$$
подставляем w:
$$x_{1} = frac{1}{pi} left(2 pi n + {asin}{left (w_{1} right )}right)$$
$$x_{1} = frac{1}{pi} left(2 pi n + {asin}{left (1 right )}right)$$
$$x_{1} = frac{1}{pi} left(2 pi n + frac{pi}{2}right)$$
$$x_{2} = frac{1}{pi} left(2 pi n + {asin}{left (w_{2} right )}right)$$
$$x_{2} = frac{1}{pi} left(2 pi n + {asin}{left (-1 right )}right)$$
$$x_{2} = frac{1}{pi} left(2 pi n – frac{pi}{2}right)$$
$$x_{3} = frac{1}{pi} left(2 pi n – {asin}{left (w_{1} right )} + piright)$$
$$x_{3} = frac{1}{pi} left(2 pi n – {asin}{left (1 right )} + piright)$$
$$x_{3} = frac{1}{pi} left(2 pi n + frac{pi}{2}right)$$
$$x_{4} = frac{1}{pi} left(2 pi n – {asin}{left (w_{2} right )} + piright)$$
$$x_{4} = frac{1}{pi} left(2 pi n – {asin}{left (-1 right )} + piright)$$
$$x_{4} = frac{1}{pi} left(2 pi n + frac{3 pi}{2}right)$$

$$x_{1} = – frac{1}{2}$$

x2 = 1/2

$$x_{2} = frac{1}{2}$$

x3 = 3/2

$$x_{3} = frac{3}{2}$$

x1 = -51.5000000000000

x2 = -71.5000000000000

x3 = -23.5000000000000

x4 = 88.5000000000000

x5 = 64.5000000000000

x6 = -55.5000000000000

x7 = 4.50000000000000

x8 = 86.5000000000000

x9 = 66.5000000000000

x10 = -25.5000000000000

x11 = 58.5000000000000

x12 = 34.5000000000000

x13 = 82.5000000000000

x14 = 50.5000000000000

x15 = -35.5000000000000

x16 = 96.5000000000000

x17 = -49.5000000000000

x18 = -87.5000000000000

x19 = -47.5000000000000

x20 = 14.5000000000000

x21 = -65.5000000000000

x22 = -11.5000000000000

x23 = 8.50000000000000

x24 = -57.5000000000000

x25 = -37.5000000000000

x26 = -61.5000000000000

x27 = 10.5000000000000

x28 = -95.5000000000000

x29 = 46.5000000000000

x30 = 98.5000000000000

x31 = 72.5000000000000

x32 = -5.50000000000000

x33 = 90.5000000000000

x34 = -93.5000000000000

x35 = 22.5000000000000

x36 = 74.5000000000000

x37 = 60.5000000000000

x38 = 26.5000000000000

x39 = 52.5000000000000

x40 = -97.5000000000000

x41 = -41.5000000000000

x42 = 70.5000000000000

x43 = -7.50000000000000

x44 = 30.5000000000000

x45 = -9.50000000000000

x46 = -75.5000000000000

x47 = 20.5000000000000

x48 = -69.5000000000000

x49 = -21.5000000000000

x50 = 12.5000000000000

x51 = -19.5000000000000

x52 = -77.5000000000000

x53 = 32.5000000000000

x54 = -29.5000000000000

x55 = -99.5000000000000

x56 = -81.5000000000000

x57 = 68.5000000000000

x58 = 84.5000000000000

x59 = -79.5000000000000

x60 = -89.5000000000000

x61 = 100.500000000000

x62 = 6.50000000000000

x63 = 78.5000000000000

x64 = 28.5000000000000

x65 = -33.5000000000000

x66 = -3.50000000000000

x67 = -27.5000000000000

x68 = -39.5000000000000

x69 = 18.5000000000000

x70 = 56.5000000000000

x71 = -53.5000000000000

x72 = 2.50000000000000

x73 = 80.5000000000000

x74 = 42.5000000000000

x75 = 92.5000000000000

x76 = -1.50000000000000

x77 = 40.5000000000000

x78 = 76.5000000000000

x79 = 24.5000000000000

x80 = 54.5000000000000

x81 = -73.5000000000000

x82 = -45.5000000000000

x83 = -59.5000000000000

x84 = 0.500000000000000

x85 = 48.5000000000000

x86 = -43.5000000000000

x87 = -15.5000000000000

x88 = -85.5000000000000

x89 = -67.5000000000000

x90 = 38.5000000000000

x91 = -13.5000000000000

x92 = -63.5000000000000

x93 = -31.5000000000000

x94 = -83.5000000000000

x95 = 36.5000000000000

x96 = 44.5000000000000

x97 = -91.5000000000000

x98 = -17.5000000000000

x99 = 16.5000000000000

x100 = 62.5000000000000

x101 = 94.5000000000000