cos(cosx)=1 (уравнение)

Найду корень уравнения: cos(cosx)=1

Вы ввели

cos(cos(x)) = 1

$$\cos{\left(\cos{\left(x \right)} \right)} = 1$$

Подробное решение

Дано уравнение
$$\cos{\left(\cos{\left(x \right)} \right)} = 1$$
преобразуем
$$\cos{\left(\cos{\left(x \right)} \right)} - 1 = 0$$
$$\cos{\left(\cos{\left(x \right)} \right)} - 1 = 0$$
Сделаем замену
$$w = \cos{\left(\cos{\left(x \right)} \right)}$$
Переносим свободные слагаемые (без w)
из левой части в правую, получим:
$$w = 1$$
Получим ответ: w = 1
делаем обратную замену
$$\cos{\left(\cos{\left(x \right)} \right)} = w$$
подставляем w:

Быстрый ответ

     pi
x1 = --
     2 

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

     3*pi
x2 = ----
      2  

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

x3 = 2*pi - I*im(acos(2*pi))

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

x4 = I*im(acos(2*pi)) + re(acos(2*pi))

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

Сумма и произведение корней

сумма

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

$$\left(\left(\left(0 + \frac{\pi}{2}\right) + \frac{3 \pi}{2}\right) + \left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(2 \pi \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(2 \pi \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(2 \pi \right)}\right)}\right)$$

=

4*pi + re(acos(2*pi))

$$\operatorname{re}{\left(\operatorname{acos}{\left(2 \pi \right)}\right)} + 4 \pi$$

произведение

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

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

=

    2                                                              
3*pi *(2*pi - I*im(acos(2*pi)))*(I*im(acos(2*pi)) + re(acos(2*pi)))
-------------------------------------------------------------------
                                 4                                 

$$\frac{3 \pi^{2} \cdot \left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(2 \pi \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(2 \pi \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(2 \pi \right)}\right)}\right)}{4}$$

Численный ответ

x1 = 10.9955740626336

x2 = 7.85398173979667

x3 = -76.9690199449714

x4 = -51.8362786898897

x5 = -54.9778713855636

x6 = 83.2522055478591

x7 = -17.2787598017848

x8 = 64.4026493091674

x9 = 98.9601684814773

x10 = 26.7035373528248

x11 = -1.57079642899459

x12 = -375.420320933289

x13 = 45.5530936923417

x14 = -26.7035373207748

x15 = -92.6769830394677

x16 = 54.9778712102147

x17 = 89.5353914792263

x18 = 4.71238877548286

x19 = 54.9778713870579

x20 = 80.1106131450394

x21 = -14.1371668399677

x22 = 58.1194644457362

x23 = -86.3937977717082

x24 = -7.85398149919389

x25 = -98.960168505672

x26 = -48.6946859664211

x27 = -76.9690202300489

x28 = 39.2699082143081

x29 = 42.4115007297162

x30 = 73.827427478448

x31 = 32.9867226363296

x32 = -89.5353907456861

x33 = 10.9955742835351

x34 = 92.6769830852837

x35 = -29.8451300966564

x36 = 17.2787598271878

x37 = 36.1283155949472

x38 = 20.4203521503294

x39 = -39.2699077617593

x40 = 98.9601683585175

x41 = -95.8185758681553

x42 = -36.1283154198515

x43 = 70.6858345077264

x44 = -10.99557427116

x45 = -61.2610569560919

x46 = -4.71238874671378

x47 = -64.4026491941832

x48 = -92.676983043851

x49 = 89.5353908464231

x50 = -32.9867230841506

x51 = 17.2787596652732

x52 = -32.9867228275787

x53 = 54.9778715197242

x54 = 95.8185760578041

x55 = -54.9778716572167

x56 = 86.393797888685

x57 = -10.9955745108633

x58 = -20.4203520393509

x59 = 51.8362788989982

x60 = 4.71238926207409

x61 = -70.6858344693493

x62 = -73.8274272801059

x63 = -42.4115006167305

x64 = -83.2522055331285

x65 = 14.1371671032868

x66 = 83.2522053188618

x67 = 23.561945115186

x68 = -58.1194639998644

x69 = 76.969019935506

x70 = 67.544242269421

x71 = 89.5353906290246

x72 = -70.6858345045219

x73 = 29.8451303194497

x74 = 61.2610567655691

x75 = 1.5707965379548

x76 = -26.7035374252973

x77 = -98.9601688026335

x78 = -17.2787595262424

x79 = 48.6946859302397

x80 = -39.2699083789769

x81 = -45.5530935874593

x82 = 39.2699084009225

x83 = 32.9867228363358

x84 = -4.71238888133151

x85 = -67.5442421666021

x86 = 76.9690197842801

x87 = -80.1106125800177

x88 = -23.5619450082571

x89 = -48.6946858949887

x90 = 61.2610569744825

График