2sin^2x-5sinx+2=0 (уравнение)

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

Вы ввели

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

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

Подробное решение

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

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

(-5)^2 - 4 * (2) * (2) = 9

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
$$w_{2} = \frac{1}{2}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{6}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{5 \pi}{6}$$

Сумма и произведение корней

сумма

    pi   5*pi                                                                 
0 + -- + ---- + pi - re(asin(2)) - I*im(asin(2)) + I*im(asin(2)) + re(asin(2))
    6     6                                                                   

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

=

2*pi

$$2 \pi$$

произведение

  pi 5*pi                                                                 
1*--*----*(pi - re(asin(2)) - I*im(asin(2)))*(I*im(asin(2)) + re(asin(2)))
  6   6                                                                   

$$\frac{5 \pi}{6} \cdot 1 \frac{\pi}{6} \left(- \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)$$

=

     2                                                                  
-5*pi *(I*im(asin(2)) + re(asin(2)))*(-pi + I*im(asin(2)) + re(asin(2)))
------------------------------------------------------------------------
                                   36                                   

$$- \frac{5 \pi^{2} \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)}{36}$$

Быстрый ответ

     pi
x1 = --
     6 

$$x_{1} = \frac{\pi}{6}$$

     5*pi
x2 = ----
      6  

$$x_{2} = \frac{5 \pi}{6}$$

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))

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

Численный ответ

x1 = -62.3082542961976

x2 = 804.771318094585

x3 = 6.80678408277789

x4 = -49.7418836818384

x5 = -5.75958653158129

x6 = -37.1755130674792

x7 = -66.497044500984

x8 = -81.1578102177363

x9 = 94.7713783832921

x10 = 52.8834763354282

x11 = -87.4409955249159

x12 = -60.2138591938044

x13 = 7056.54069873827

x14 = -74.8746249105567

x15 = 82.2050077689329

x16 = 101.054563690472

x17 = 65.4498469497874

x18 = 534.594349885863

x19 = 13.0899693899575

x20 = 21.4675497995303

x21 = -47.6474885794452

x22 = 40.317105721069

x23 = 50.789081233035

x24 = -72.7802298081635

x25 = -204.727121258935

x26 = -12.0427718387609

x27 = -93.7241808320955

x28 = 34.0339204138894

x29 = 25.6563400043166

x30 = -9.94837673636768

x31 = 0.523598775598299

x32 = 15.1843644923507

x33 = -79.0634151153431

x34 = 69.6386371545737

x35 = 44.5058959258554

x36 = -2190.21367832768

x37 = 75.9218224617533

x38 = -85.3466004225227

x39 = 59.1666616426078

x40 = 88.4881930761125

x41 = 57.0722665402146

x42 = 71.733032256967

x43 = -22.5147473507269

x44 = -35.081117965086

x45 = -56.025068989018

x46 = -100.007366139275

x47 = -30.8923277602996

x48 = 63.3554518473942

x49 = 2.61799387799149

x50 = 78.0162175641465

x51 = -97.9129710368819

x52 = -16.2315620435473

x53 = 46.6002910282486

x54 = 38.2227106186758

x55 = -91.6297857297023

x56 = -3.66519142918809

x57 = -28.7979326579064

x58 = -43.4586983746588

x59 = 2113.76825709033

x60 = -53.9306738866248

x61 = 96.8657734856853

x62 = 84.2994028713261

x63 = 8.90117918517108

x64 = -68.5914396033772

x65 = -41.3643032722656

x66 = -18.3259571459405

x67 = 31.9395253114962

x68 = 19.3731546971371

x69 = 27.7507351067098

x70 = -24.60914245312

x71 = 90.5825881785057

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