Sin^2x-4sinx=0 (уравнение)

Найду корень уравнения: Sin^2x-4sinx=0

Вы ввели

   2                  
sin (x) - 4*sin(x) = 0

$$\sin^{2}{\left(x \right)} - 4 \sin{\left(x \right)} = 0$$

Подробное решение

Дано уравнение
$$\sin^{2}{\left(x \right)} - 4 \sin{\left(x \right)} = 0$$
преобразуем
$$\left(\sin{\left(x \right)} - 4\right) \sin{\left(x \right)} = 0$$
$$\left(\sin^{2}{\left(x \right)} - 4 \sin{\left(x \right)}\right) + 0 = 0$$
Сделаем замену
$$w = \sin{\left(x \right)}$$
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.
$$a = 1$$
$$b = -4$$
$$c = 0$$
, то

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

(-4)^2 - 4 * (1) * (0) = 16

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

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

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

или
$$w_{1} = 4$$
Упростить
$$w_{2} = 0$$
Упростить
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(4 \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(4 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{2} = 2 \pi n$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(4 \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(4 \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi$$

Быстрый ответ

x1 = 0

$$x_{1} = 0$$

x2 = pi

$$x_{2} = \pi$$

x3 = pi - re(asin(4)) - I*im(asin(4))

$$x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}$$

x4 = I*im(asin(4)) + re(asin(4))

$$x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}$$

Сумма и произведение корней

сумма

0 + 0 + pi + pi - re(asin(4)) - I*im(asin(4)) + I*im(asin(4)) + re(asin(4))

$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right) - \left(- 2 \pi + \operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right)$$

=

2*pi

$$2 \pi$$

произведение

1*0*pi*(pi - re(asin(4)) - I*im(asin(4)))*(I*im(asin(4)) + re(asin(4)))

$$1 \cdot 0 \pi \left(- \operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right)$$

=

0

$$0$$

Численный ответ

x1 = -100.530964914873

x2 = -157.07963267949

x3 = -78.5398163397448

x4 = 97.3893722612836

x5 = 18.8495559215388

x6 = -94.2477796076938

x7 = -65.9734457253857

x8 = -56.5486677646163

x9 = 56.5486677646163

x10 = 6.28318530717959

x11 = -69.1150383789755

x12 = 78.5398163397448

x13 = -3.14159265358979

x14 = -21.9911485751286

x15 = 81.6814089933346

x16 = -91.106186954104

x17 = -62.8318530717959

x18 = -31.4159265358979

x19 = 0.0

x20 = 6270.61893656523

x21 = 62.8318530717959

x22 = 84.8230016469244

x23 = 72.2566310325652

x24 = 91.106186954104

x25 = -18.8495559215388

x26 = -9.42477796076938

x27 = 28.2743338823081

x28 = 471.238898038469

x29 = 43.9822971502571

x30 = -6.28318530717959

x31 = -34.5575191894877

x32 = -59.6902604182061

x33 = -53.4070751110265

x34 = 87.9645943005142

x35 = 69.1150383789755

x36 = -84.8230016469244

x37 = -75.398223686155

x38 = 75.398223686155

x39 = 3.14159265358979

x40 = -298.45130209103

x41 = -37.6991118430775

x42 = -115544.636206379

x43 = 21.9911485751286

x44 = 94.2477796076938

x45 = -87.9645943005142

x46 = -25.1327412287183

x47 = 100.530964914873

x48 = 50.2654824574367

x49 = 1369.73439696515

x50 = -43.9822971502571

x51 = 65.9734457253857

x52 = 9.42477796076938

x53 = 40.8407044966673

x54 = 37.6991118430775

x55 = -28.2743338823081

x56 = 53.4070751110265

x57 = -40.8407044966673

x58 = 47.1238898038469

x59 = -50.2654824574367

x60 = -97.3893722612836

x61 = 15.707963267949

x62 = 25.1327412287183

x63 = -12.5663706143592

x64 = 31.4159265358979

x65 = 12.5663706143592

x66 = 59.6902604182061

x67 = -15.707963267949

x68 = -81.6814089933346

x69 = -72.2566310325652

x70 = 34.5575191894877

x71 = -47.1238898038469

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