sin(x)(sin(x)+1)=0 (уравнение)

Найду корень уравнения: sin(x)(sin(x)+1)=0

Вы ввели

sin(x)*(sin(x) + 1) = 0

$$\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)} = 0$$

Подробное решение

Дано уравнение
$$\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)} = 0$$
преобразуем
$$\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)} = 0$$
$$\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)} = 0$$
Сделаем замену
$$w = \sin{\left(x \right)}$$
Раскроем выражение в уравнении
$$w \left(w + 1\right) = 0$$
Получаем квадратное уравнение
$$w^{2} + w = 0$$
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.
$$a = 1$$
$$b = 1$$
$$c = 0$$
, то

D = b^2 - 4 * a * c = 

(1)^2 - 4 * (1) * (0) = 1

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

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

w2 = (-b - sqrt(D)) / (2*a)

или
$$w_{1} = 0$$
$$w_{2} = -1$$
делаем обратную замену
$$\sin{\left(x \right)} = w$$
Дано уравнение
$$\sin{\left(x \right)} = w$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Или
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, где n - любое целое число
подставляем w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$

Быстрый ответ

x1 = 0

$$x_{1} = 0$$

     -pi 
x2 = ----
      2  

$$x_{2} = - \frac{\pi}{2}$$

x3 = pi

$$x_{3} = \pi$$

     3*pi
x4 = ----
      2  

$$x_{4} = \frac{3 \pi}{2}$$

Численный ответ

x1 = -75.398223686155

x2 = 34.5575191894877

x3 = -45.5530935853668

x4 = -84.8230016469244

x5 = 25.1327412287183

x6 = -94.2477796076938

x7 = 23.5619450880496

x8 = -31.4159265358979

x9 = -26.7035379972524

x10 = -51.8362786901812

x11 = -89.5353901159531

x12 = -50.2654824574367

x13 = 50.2654824574367

x14 = 3.14159265358979

x15 = -34.5575191894877

x16 = -43.9822971502571

x17 = 31.4159265358979

x18 = -83.2522054990561

x19 = 42.4115007309741

x20 = 94.2477796076938

x21 = 86.3937978896245

x22 = 36.1283157927187

x23 = 40.8407044966673

x24 = 10.9955738002317

x25 = 73.8274274758159

x26 = -39.2699083493443

x27 = 54.9778715673102

x28 = -91.106186954104

x29 = -58.1194640010631

x30 = -28.2743338823081

x31 = 65.9734457253857

x32 = 61.261057240759

x33 = -21.9911485751286

x34 = -62.8318530717959

x35 = -15.707963267949

x36 = -6.28318530717959

x37 = 54.9778709836

x38 = -72.2566310325652

x39 = -53.4070751110265

x40 = -89.5353907430632

x41 = 84.8230016469244

x42 = -56.5486677646163

x43 = 21.9911485751286

x44 = -3.14159265358979

x45 = -97.3893722612836

x46 = -32.9867224730462

x47 = -37.6991118430775

x48 = -9.42477796076938

x49 = 17.2787594994858

x50 = -81.6814089933346

x51 = -87.9645943005142

x52 = 72.2566310325652

x53 = 15.707963267949

x54 = -18.8495559215388

x55 = -51.8362785033387

x56 = 91.106186954104

x57 = -1.57079642735461

x58 = 61.2610565484827

x59 = -32.9867232359983

x60 = -26.7035375777291

x61 = -20.4203520633206

x62 = 10.9955745314484

x63 = 80.1106131505346

x64 = 98.9601685293103

x65 = 6.28318530717959

x66 = 98.9601686349023

x67 = 4.71238880126794

x68 = 29.8451303173708

x69 = 75.398223686155

x70 = -12.5663706143592

x71 = -64.4026492153213

x72 = 56.5486677646163

x73 = 47.1238898038469

x74 = 69.1150383789755

x75 = -70.6858346750384

x76 = -14.1371668415708

x77 = -76.969019611197

x78 = 59.6902604182061

x79 = -40.8407044966673

x80 = 78.5398163397448

x81 = -59.6902604182061

x82 = -7.85398150061792

x83 = -76.9690201700248

x84 = 100.530964914873

x85 = 92.6769831047852

x86 = 28.2743338823081

x87 = 81.6814089933346

x88 = 43.9822971502571

x89 = 37.6991118430775

x90 = -95.8185758682081

x91 = 67.5442422387326

x92 = 12.5663706143592

x93 = -100.530964914873

x94 = 0.0

x95 = -65.9734457253857

x96 = 17.2787599718548

x97 = 48.6946859526732

x98 = -47.1238898038469

x99 = 87.9645943005142

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