sin^2x-4sinx+3=0 (уравнение)

Найду корень уравнения: sin^2x-4sinx+3=0

Вы ввели

   2                      
sin (x) - 4*sin(x) + 3 = 0

$$\sin^{2}{\left(x \right)} - 4 \sin{\left(x \right)} + 3 = 0$$

Подробное решение

Дано уравнение
$$\sin^{2}{\left(x \right)} - 4 \sin{\left(x \right)} + 3 = 0$$
преобразуем
$$\sin^{2}{\left(x \right)} - 4 \sin{\left(x \right)} + 3 = 0$$
$$\left(\sin^{2}{\left(x \right)} - 4 \sin{\left(x \right)} + 3\right) + 0 = 0$$
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.
$$a = 1$$
$$b = -4$$
$$c = 3$$
, то

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

(-4)^2 - 4 * (1) * (3) = 4

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

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

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

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

Быстрый ответ

     pi
x1 = --
     2 

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

x2 = pi - re(asin(3)) - I*im(asin(3))

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

x3 = I*im(asin(3)) + re(asin(3))

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

Сумма и произведение корней

сумма

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

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

=

3*pi
----
 2  

$$\frac{3 \pi}{2}$$

произведение

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

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

=

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

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

Численный ответ

x1 = -61.2610559872498

x2 = 45.5530927298952

x3 = 70.6858344615412

x4 = -36.1283154155162

x5 = 32.9867227366993

x6 = -67.5442411623096

x7 = -4.71238957496506

x8 = -67.5442421735662

x9 = -17.2787598056407

x10 = 51.8362779053669

x11 = 89.5353909174858

x12 = 32.9867234780811

x13 = 45.553093779068

x14 = -67.5442428487442

x15 = -36.1283164076433

x16 = 83.2522057754448

x17 = 89.5353898897507

x18 = -36.1283146476205

x19 = 89.5353908642258

x20 = 64.4026485529699

x21 = -80.110613563712

x22 = -61.2610570183256

x23 = 20.4203521459434

x24 = -48.6946857070249

x25 = -29.845131147252

x26 = -98.9601690341974

x27 = 26.7035373016312

x28 = -10.995573682783

x29 = 76.9690206389115

x30 = 70.6858343660166

x31 = 95.8185767586848

x32 = -98.9601680045927

x33 = -80.1106117856929

x34 = -86.3937977232411

x35 = 20.4203531508632

x36 = -73.8274272796538

x37 = -23.5619450138407

x38 = 95.8185760666683

x39 = -29.8451292549847

x40 = 45.55309375725

x41 = -23.561944005588

x42 = -23.5619457164261

x43 = 26.7035383321189

x44 = 76.9690196086123

x45 = 7.85398074872064

x46 = -27236.0375103869

x47 = -42.4115004076971

x48 = 64.4026503070327

x49 = -73.8274282962485

x50 = -73.8274264057273

x51 = -86.3937987489281

x52 = -17.2787588275021

x53 = -86.3937975081328

x54 = 58.1194650449015

x55 = 7.85398248919332

x56 = -86.3937978894049

x57 = 70.6858354916056

x58 = -80.1106125768075

x59 = -4.71238854623148

x60 = -92.6769838965717

x61 = 1.5707955701585

x62 = -10.9955747127236

x63 = 64.4026493058065

x64 = 32.9867224479628

x65 = -61.2610568815392

x66 = -29.8451300946379

x67 = 26.7035372771885

x68 = -17.2787598583239

x69 = 51.8362789061518

x70 = 14.1371660023601

x71 = 58.1194631525574

x72 = -48.6946867357837

x73 = 58.119464383798

x74 = 1.57079659705183

x75 = 20.4203514175156

x76 = 102.101760303312

x77 = 1.5707966882103

x78 = 95.8185750622832

x79 = 14.137167895233

x80 = -54.9778718734445

x81 = -54.9778708436759

x82 = -92.6769828677856

x83 = 39.2699086145926

x84 = -42.4115015894519

x85 = 83.2522047467435

x86 = 14.1371671144011

x87 = 39.269907585879

x88 = 51.8362796246605

x89 = 7.85398174544364

x90 = -42.4115005631109

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