cos(4x)=1/4 (уравнение)

Найду корень уравнения: cos(4x)=1/4

Вы ввели

cos(4*x) = 1/4

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

Подробное решение

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

Быстрый ответ

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

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

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

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

Сумма и произведение корней

сумма

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

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

=

pi
--
2 

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

произведение

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

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

=

(-acos(1/4) + 2*pi)*acos(1/4)
-----------------------------
              16             

$$\frac{\left(- \operatorname{acos}{\left(\frac{1}{4} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{1}{4} \right)}}{16}$$

Численный ответ

x1 = -45.8826224949652

x2 = -93.9182505897806

x3 = 88.2941233184274

x4 = 74.1569563772733

x5 = -16.9492305768307

x6 = -60.0197894361193

x7 = 16.0374922858622

x8 = -86.0642689558061

x9 = -82.0109380112478

x10 = 92.3474542629857

x11 = -53.7366041289397

x12 = -34.2279901715745

x13 = 60.0197894361193

x14 = -83.5817343380427

x15 = 4.38285996247149

x16 = 78.2102873218316

x17 = -75.7277527040682

x18 = -20.0908232304205

x19 = 70.3563056878571

x20 = 66.3029747432989

x21 = -78.2102873218316

x22 = 42.081971805549

x23 = 48.3651571127286

x24 = -17.6082886126571

x25 = -2135.04173713218

x26 = -56.2191387467031

x27 = 8.18351065188769

x28 = 52.1658078021448

x29 = -65.6439167074725

x30 = -89.8649196452223

x31 = 56.2191387467031

x32 = 71.927102014652

x33 = 82.9226763022163

x34 = 34.2279901715745

x35 = 82.0109380112478

x36 = 1092.94471443133

x37 = -31.7454555538111

x38 = -43.6527681323439

x39 = 23.8914739198367

x40 = -39.5994371877856

x41 = 63.1613820897091

x42 = 86.0642689558061

x43 = 96.1481049524019

x44 = -23.8914739198367

x45 = 30.1746592270162

x46 = -71.927102014652

x47 = 64.0731203806776

x48 = 27.9448048643949

x49 = 41.1702335145805

x50 = -74.1569563772733

x51 = 44.3118261681703

x52 = -49.9359534395235

x53 = 49.9359534395235

x54 = -42.081971805549

x55 = 89.8649196452223

x56 = -9.75430697868258

x57 = -21.6616195572153

x58 = 12.236841596446

x59 = 0.329529017913204

x60 = 22.3206775930418

x61 = -13.8076379232409

x62 = -27.9448048643949

x63 = -97.7189012791968

x64 = -38.0286408609907

x65 = 5.95365628926638

x66 = -51.5067497663184

x67 = 67.8737710700938

x68 = -57.789935073498

x69 = -61.5905857629142

x70 = -1.9003253447081

x71 = 100.20143589696

x72 = -64.0731203806776

x73 = -67.8737710700938

x74 = -96.1481049524019

x75 = -79.7810836486265

x76 = 31.0863975179847

x77 = 53.0775460931133

x78 = 1.9003253447081

x79 = -16.0374922858622

x80 = 26.3740085376

x81 = 93.9182505897806

x82 = -1026.97126870595

x83 = 38.0286408609907

x84 = 45.8826224949652

x85 = -35.7987864983694

x86 = -87.635065282601

x87 = -2435.06383555

x88 = 20.0908232304205

x89 = -100.20143589696

x90 = -12.236841596446

x91 = -5.95365628926638

x92 = -422.214682889914

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