cos(4х)=1/3 (уравнение)

Найду корень уравнения: cos(4х)=1/3

Вы ввели

cos(4*x) = 1/3

$$\cos{\left(4 x \right)} = \frac{1}{3}$$

Подробное решение

Дано уравнение
$$\cos{\left(4 x \right)} = \frac{1}{3}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$4 x = \pi n + \operatorname{acos}{\left(\frac{1}{3} \right)}$$
$$4 x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{3} \right)}$$
Или
$$4 x = \pi n + \operatorname{acos}{\left(\frac{1}{3} \right)}$$
$$4 x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{3} \right)}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$4$$
получим ответ:
$$x_{1} = \frac{\pi n}{4} + \frac{\operatorname{acos}{\left(\frac{1}{3} \right)}}{4}$$
$$x_{2} = \frac{\pi n}{4} - \frac{\pi}{4} + \frac{\operatorname{acos}{\left(\frac{1}{3} \right)}}{4}$$

Быстрый ответ

     pi   acos(1/3)
x1 = -- - ---------
     2        4    

$$x_{1} = - \frac{\operatorname{acos}{\left(\frac{1}{3} \right)}}{4} + \frac{\pi}{2}$$

     acos(1/3)
x2 = ---------
         4    

$$x_{2} = \frac{\operatorname{acos}{\left(\frac{1}{3} \right)}}{4}$$

Сумма и произведение корней

сумма

    pi   acos(1/3)   acos(1/3)
0 + -- - --------- + ---------
    2        4           4    

$$\frac{\operatorname{acos}{\left(\frac{1}{3} \right)}}{4} + \left(0 + \left(- \frac{\operatorname{acos}{\left(\frac{1}{3} \right)}}{4} + \frac{\pi}{2}\right)\right)$$

=

pi
--
2 

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

произведение

  /pi   acos(1/3)\ acos(1/3)
1*|-- - ---------|*---------
  \2        4    /     4    

$$1 \left(- \frac{\operatorname{acos}{\left(\frac{1}{3} \right)}}{4} + \frac{\pi}{2}\right) \frac{\operatorname{acos}{\left(\frac{1}{3} \right)}}{4}$$

=

(-acos(1/3) + 2*pi)*acos(1/3)
-----------------------------
              16             

$$\frac{\left(- \operatorname{acos}{\left(\frac{1}{3} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{1}{3} \right)}}{16}$$

Численный ответ

x1 = -39.5776480242076

x2 = 89.8431304816443

x3 = 42.103760969127

x4 = 22.2988884294637

x5 = 86.0860581193841

x6 = 9.11703810643419

x7 = 45.8608333313872

x8 = -35.8205756619474

x9 = -93.9400397533586

x10 = -20.1126123939985

x11 = 63.1395929261311

x12 = -12.8741104686944

x13 = 100.223225060538

x14 = 70.3780948514352

x15 = -89.8431304816443

x16 = -97.6971121156188

x17 = 30.1528700634382

x18 = -61.5687965993362

x19 = -31.7236663902331

x20 = 4.4046491260495

x21 = 81.9891488476698

x22 = 7.54624177963929

x23 = -52.1440186385668

x24 = -86.0860581193841

x25 = -74.1351672136953

x26 = 0.307739854335194

x27 = 88.2723341548494

x28 = -13.8294270868189

x29 = -27.9665940279729

x30 = 59.9980002725413

x31 = -71.9488911782301

x32 = -57.811724237076

x33 = -5.97544545284439

x34 = 66.2811855797209

x35 = 49.9577426031015

x36 = -75.7059635404902

x37 = -10.6878344332291

x38 = 41.1484443510025

x39 = 71.9488911782301

x40 = -96.1263157888239

x41 = 20.1126123939985

x42 = 74.1351672136953

x43 = 26.395797701178

x44 = -81.9891488476698

x45 = 34.2497793351525

x46 = -38.0068516974127

x47 = -31.1081866815627

x48 = -79.8028728122045

x49 = 8.16172148830968

x50 = -17.5864994490791

x51 = 52.1440186385668

x52 = -64.0949095442556

x53 = -67.8519819065157

x54 = 78.2320764854096

x55 = 27.9665940279729

x56 = -53.7148149653617

x57 = -43.6745572959219

x58 = -65.6657058710505

x59 = -83.5599451744647

x60 = 93.9400397533586

x61 = -56.2409279102811

x62 = 44.2900370045923

x63 = 67.8519819065157

x64 = 85.1307415012596

x65 = -87.656854446179

x66 = -16.0157031222842

x67 = 25.4404810830535

x68 = -21.6834087207934

x69 = -1.87853618113009

x70 = 1.87853618113009

x71 = -78.2320764854096

x72 = 64.0949095442556

x73 = -42.103760969127

x74 = 23.8696847562586

x75 = 38.0068516974127

x76 = 31.1081866815627

x77 = 48.3869462763066

x78 = -59.9980002725413

x79 = 56.2409279102811

x80 = 5.97544545284439

x81 = 12.258630760024

x82 = -9.73251781510457

x83 = -23.8696847562586

x84 = 16.0157031222842

x85 = -49.9577426031015

x86 = -34.2497793351525

x87 = 96.1263157888239

x88 = -45.8608333313872

x89 = 92.3692434265637

x90 = -100.223225060538

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