sin^2x-3sinx+2=0 (уравнение)

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

Вы ввели

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

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

Подробное решение

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

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

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

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

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

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

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

Сумма и произведение корней

сумма

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

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

=

3*pi
----
 2  

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

произведение

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

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

=

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

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

Быстрый ответ

     pi
x1 = --
     2 

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

x2 = pi - re(asin(2)) - I*im(asin(2))

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

x3 = I*im(asin(2)) + re(asin(2))

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

Численный ответ

x1 = -67.5442421763137

x2 = -73.8274260609448

x3 = 83.2522058456645

x4 = -48.6946856448184

x5 = -23.561945016053

x6 = 26.7035385469741

x7 = -29.8451300938139

x8 = 20.4203521441984

x9 = 32.9867236138576

x10 = -10.9955735516589

x11 = 1.57079700398873

x12 = 64.4026481915252

x13 = 51.8362789090115

x14 = -98.9601691037059

x15 = -4.71238848059836

x16 = -4.71238970180774

x17 = -92.6769845303487

x18 = 76.9690195526133

x19 = -86.3937989639545

x20 = -10.9955747752993

x21 = -61.2610571936019

x22 = 14.1371682454946

x23 = -17.2787601164358

x24 = 95.8185771224127

x25 = -86.3937977050157

x26 = -36.1283142806347

x27 = 95.8185760701987

x28 = -42.4115005430641

x29 = -48.6946868672216

x30 = -80.1106114181945

x31 = 102.101759965899

x32 = 26.703537282924

x33 = -23.5619437177603

x34 = 45.5530925300164

x35 = 58.1194653976648

x36 = 70.6858340517028

x37 = -54.9778707171509

x38 = -29.8451314931042

x39 = 7.85398046563447

x40 = 45.553094091839

x41 = 1.57079661901596

x42 = -36.1283166952282

x43 = -92.676982808917

x44 = 58.1194643770702

x45 = -54.9778719394428

x46 = 64.4026493044641

x47 = 89.5353911752829

x48 = -92.6769840326577

x49 = -17.2787586177095

x50 = -73.8274286445858

x51 = -325.154840065363

x52 = 51.8362776268483

x53 = -29.8451289073854

x54 = -73.8274272794653

x55 = 32.9867223887206

x56 = 83.2522046289214

x57 = 1.57079536523077

x58 = -61.2610570407565

x59 = 95.8185747883961

x60 = -117.80972560988

x61 = 14.1371656591617

x62 = 45.5530937812277

x63 = 64.4026506037314

x64 = 20.4203534431639

x65 = 83.2522058481918

x66 = -42.4115017994301

x67 = -36.1283154137715

x68 = 14.1371671181822

x69 = 7.85398174770883

x70 = -23.561946075942

x71 = 76.9690207793905

x72 = 70.6858357115182

x73 = -67.544243206816

x74 = 1.57079785005069

x75 = -98.9601678826108

x76 = 58.1194628121746

x77 = -67.544240879025

x78 = 51.8362799897705

x79 = 20.4203510568788

x80 = 7.85398285538609

x81 = -42.4115000881114

x82 = -17.2787598788452

x83 = 39.2699086837397

x84 = -61.2610557825211

x85 = -86.3937971842945

x86 = 39.2699074635758

x87 = 89.5353909435736

x88 = 26.7035369653861

x89 = 70.6858344445529

x90 = -80.1106125755117

x91 = 89.5353896949152

x92 = -80.1106138557219

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