cos4x=√3/2 (уравнение)

Найду корень уравнения: cos4x=√3/2

Вы ввели

             ___
           \/ 3 
cos(4*x) = -----
             2  

$$\cos{\left(4 x \right)} = \frac{\sqrt{3}}{2}$$

Подробное решение

Дано уравнение
$$\cos{\left(4 x \right)} = \frac{\sqrt{3}}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$4 x = \pi n + \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)}$$
$$4 x = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)}$$
Или
$$4 x = \pi n + \frac{\pi}{6}$$
$$4 x = \pi n - \frac{5 \pi}{6}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$4$$
получим ответ:
$$x_{1} = \frac{\pi n}{4} + \frac{\pi}{24}$$
$$x_{2} = \frac{\pi n}{4} - \frac{5 \pi}{24}$$

Быстрый ответ

     pi
x1 = --
     24

$$x_{1} = \frac{\pi}{24}$$

     11*pi
x2 = -----
       24 

$$x_{2} = \frac{11 \pi}{24}$$

Сумма и произведение корней

сумма

    pi   11*pi
0 + -- + -----
    24     24 

$$\left(0 + \frac{\pi}{24}\right) + \frac{11 \pi}{24}$$

=

pi
--
2 

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

произведение

  pi 11*pi
1*--*-----
  24   24 

$$\frac{11 \pi}{24} \cdot 1 \frac{\pi}{24}$$

=

     2
11*pi 
------
 576  

$$\frac{11 \pi^{2}}{576}$$

Численный ответ

x1 = -22.1220482690281

x2 = 48.5637864367422

x3 = -57.9885643975116

x4 = -20.2894525544341

x5 = -37.8300115369771

x6 = 59.8211601121056

x7 = -23.692844595823

x8 = -6.15228561328001

x9 = 20.2894525544341

x10 = -75.5291233800546

x11 = 35.997415822383

x12 = -53.5379748049261

x13 = 58.2503637853107

x14 = 7.98488132787406

x15 = -56.6795674585159

x16 = 36.2592152101822

x17 = -65.8425460314861

x18 = -42.2806011295626

x19 = -14.0062672472545

x20 = 26.8344372494128

x21 = -15.8388629618485

x22 = -64.2717497046912

x23 = 12.4354709204596

x24 = 92.5460835869993

x25 = 14.2680666350536

x26 = 1.70169602069447

x27 = 15.8388629618485

x28 = -87.8336946066146

x29 = 86.2628982798197

x30 = 55.108771131721

x31 = 80.2415123604393

x32 = 73.9583270532597

x33 = -97.5202719551832

x34 = 70.5549350118708

x35 = -1.70169602069447

x36 = 78.4089166458453

x37 = -72.1257313386657

x38 = -84.953901340824

x39 = -7.72308194007491

x40 = 6.15228561328001

x41 = 34.4266194955881

x42 = -59.8211601121056

x43 = -45.6839931709516

x44 = -35.997415822383

x45 = -94.1168799137942

x46 = 94.1168799137942

x47 = 95.9494756283883

x48 = 50.1345827635371

x49 = -43.8513974563575

x50 = 26.5726378616137

x51 = -100.661864608773

x52 = 81.8123086872342

x53 = 72.1257313386657

x54 = -39.400807863772

x55 = -67.6751417460801

x56 = 0.130899693899575

x57 = -50.1345827635371

x58 = 18.7186562276392

x59 = 4.58148928648512

x60 = -95.6876762405891

x61 = -21.860248881229

x62 = -73.6965276654606

x63 = 51.9671784781312

x64 = 89.6662903212087

x65 = -9.55567765466895

x66 = 44.1131968441567

x67 = -86.2628982798197

x68 = 28.1434341884086

x69 = 100.400065220974

x70 = -11350.7051571138

x71 = -31.5468262297975

x72 = -79.9797129726402

x73 = 37.8300115369771

x74 = 88.0954939944138

x75 = 45.6839931709516

x76 = -29.7142305152035

x77 = 66.1043454192852

x78 = 42.2806011295626

x79 = -83.1213056262299

x80 = 23.692844595823

x81 = 102.232660935568

x82 = -78.6707160336444

x83 = -51.705379090332

x84 = 64.2717497046912

x85 = -28.1434341884086

x86 = 22.1220482690281

x87 = -17.4096592886434

x88 = 29.9760299030026

x89 = 67.6751417460801

x90 = 56.4177680707167

x91 = -89.6662903212087

x92 = -81.8123086872342

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