sin^2(x)+sin(x)-2=0 (уравнение)

Найду корень уравнения: sin^2(x)+sin(x)-2=0

Вы ввели

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

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

Подробное решение

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

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

(1)^2 - 4 * (1) * (-2) = 9

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

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

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

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

Быстрый ответ

     pi
x1 = --
     2 

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

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

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

Сумма и произведение корней

сумма

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

$$\left(\left(0 + \frac{\pi}{2}\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)\right) - \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)$$

=

3*pi
----
 2  

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

произведение

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

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

Численный ответ

x1 = -29.845130096071

x2 = 51.8362789010648

x3 = -98.9601682955883

x4 = 26.7035373376707

x5 = 89.5353908670432

x6 = -48.6946863938179

x7 = -17.2787595558177

x8 = 1.57079638821583

x9 = 39.269908467403

x10 = -92.6769829904857

x11 = -80.1106125790894

x12 = -98.9601688818699

x13 = 32.9867231423258

x14 = -4.71238925012177

x15 = 70.6858346623542

x16 = -17.2787598185623

x17 = -92.6769835365305

x18 = -2247.80954299513

x19 = 64.40264930819

x20 = -934.623813817357

x21 = -14895.8615670421

x22 = 20.4203525182045

x23 = 76.969020286866

x24 = -48.6946858375778

x25 = 51.8362787709979

x26 = 89.5353902626862

x27 = -10.9955740029714

x28 = -61.26105697461

x29 = -42.4115007093557

x30 = -117.809724108567

x31 = -42.4115006012209

x32 = 76.9690197275097

x33 = -61.2610566723778

x34 = -61.2610571370788

x35 = 7.85398174142849

x36 = 95.8185760603641

x37 = -54.9778711489614

x38 = 1.57079655506356

x39 = 58.1194644186892

x40 = -4.71238868494521

x41 = 45.5530937111524

x42 = -86.3937977576988

x43 = 39.2699078944659

x44 = -73.8274272799751

x45 = -186.924763176023

x46 = 32.9867225747057

x47 = 45.5530931580187

x48 = 14.1371671066768

x49 = -36.1283154185983

x50 = -86.3937977995351

x51 = 102.101761823663

x52 = -54.9778717299973

x53 = -67.5442421686302

x54 = 83.2522050398652

x55 = 70.6858344940848

x56 = 83.2522056194568

x57 = 51.8362787447895

x58 = 45.5530934923341

x59 = 20.4203521490533

x60 = 89.5353906021911

x61 = -23.5619450098824

x62 = -10.9955745778008

x63 = 26.7035375559949

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