3sin^2x-5sinx-2=0 (уравнение)

Найду корень уравнения: 3sin^2x-5sinx-2=0

Вы ввели

     2                      
3*sin (x) - 5*sin(x) - 2 = 0

$$3 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)} - 2 = 0$$

Подробное решение

Дано уравнение
$$3 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)} - 2 = 0$$
преобразуем
$$3 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)} - 2 = 0$$
$$\left(3 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)} - 2\right) + 0 = 0$$
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.
$$a = 3$$
$$b = -5$$
$$c = -2$$
, то

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

(-5)^2 - 4 * (3) * (-2) = 49

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

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

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

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

Быстрый ответ

x1 = pi + asin(1/3)

$$x_{1} = \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi$$

x2 = -asin(1/3)

$$x_{2} = - \operatorname{asin}{\left(\frac{1}{3} \right)}$$

x3 = pi - re(asin(2)) - I*im(asin(2))

$$x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}$$

x4 = I*im(asin(2)) + re(asin(2))

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

Сумма и произведение корней

сумма

0 + pi + asin(1/3) - asin(1/3) + pi - re(asin(2)) - I*im(asin(2)) + I*im(asin(2)) + re(asin(2))

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

=

2*pi

$$2 \pi$$

произведение

1*(pi + asin(1/3))*-asin(1/3)*(pi - re(asin(2)) - I*im(asin(2)))*(I*im(asin(2)) + re(asin(2)))

$$1 \left(\operatorname{asin}{\left(\frac{1}{3} \right)} + \pi\right) \left(- \operatorname{asin}{\left(\frac{1}{3} \right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)$$

=

(pi + asin(1/3))*(I*im(asin(2)) + re(asin(2)))*(-pi + I*im(asin(2)) + re(asin(2)))*asin(1/3)

$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \left(\operatorname{asin}{\left(\frac{1}{3} \right)} + \pi\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \operatorname{asin}{\left(\frac{1}{3} \right)}$$

Численный ответ

x1 = 85.1628385563785

x2 = -94.5876165171479

x3 = -15.3681263584948

x4 = 9.7646148702235

x5 = -25.4725781381725

x6 = -63.17168998125

x7 = 53.7469120204806

x8 = -103.332720659009

x9 = 47.463726713301

x10 = 12.2265337049051

x11 = -46.7840528943928

x12 = -69.4548752884296

x13 = 75.0583867767009

x14 = -6.62302221663371

x15 = -166.164573730805

x16 = 78.879653249199

x17 = -56.8885046740704

x18 = 5.94334839772546

x19 = 37.3592749336234

x20 = -82.0212459027887

x21 = -0.339836909454122

x22 = 97.7292091707377

x23 = 3.48142956304392

x24 = 24.7929043192642

x25 = 34.8973560989418

x26 = 68.7752014695213

x27 = -88.3044312099683

x28 = -2.80175574413567

x29 = 60.0300973276602

x30 = 16.0478001774031

x31 = -27.934496972854

x32 = 31.0760896264438

x33 = 91.4460238635581

x34 = 41.1805414061214

x35 = -34.2176822800336

x36 = -40.5008675872132

x37 = 175.589351691574

x38 = -78.1999794302907

x39 = 28.6141707917623

x40 = -6117020.52838282

x41 = 66.3132826348398

x42 = -38.0389487525316

x43 = -19.1893928309929

x44 = -31.7557634453521

x45 = -90.7663500446499

x46 = 81.3415720838805

x47 = 62.4920161623417

x48 = -65.6336088159315

x49 = 18.5097190120846

x50 = 100.191128005419

x51 = -21.6513116656744

x52 = 2512.93428596238

x53 = 72.5964679420194

x54 = -75.7380605956092

x55 = -71.9167941231111

x56 = 49.9256455479826

x57 = -50.6053193668908

x58 = -53.0672382015724

x59 = 87.6247573910601

x60 = -84.4831647374703

x61 = -1194.14504527358

x62 = -44.3221340597112

x63 = 56.2088308551622

x64 = -9.08494105131526

x65 = -59.350423508752

x66 = 22.3309854845827

x67 = 93.9079426982397

x68 = 4335.05802504446

x69 = -12.9062075238133

x70 = -97.0495353518295

x71 = 43.642460240803

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