sin(4x)=1/4 (уравнение)

Найду корень уравнения: sin(4x)=1/4

Вы ввели

sin(4*x) = 1/4

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

Подробное решение

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

Быстрый ответ

       asin(1/4)   pi
x1 = - --------- + --
           4       4 

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

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

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

Сумма и произведение корней

сумма

      asin(1/4)   pi   asin(1/4)
0 + - --------- + -- + ---------
          4       4        4    

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

=

pi
--
4 

$$\frac{\pi}{4}$$

произведение

  /  asin(1/4)   pi\ asin(1/4)
1*|- --------- + --|*---------
  \      4       4 /     4    

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

=

(pi - asin(1/4))*asin(1/4)
--------------------------
            16            

$$\frac{\left(\pi - \operatorname{asin}{\left(\frac{1}{4} \right)}\right) \operatorname{asin}{\left(\frac{1}{4} \right)}}{16}$$

Численный ответ

x1 = -11.8441425147472

x2 = -32.9235527989073

x3 = -15.6447932041634

x4 = -84.1007735473125

x5 = -2.41936455397786

x6 = -33.8352910898758

x7 = 66.0366157891712

x8 = 32.1381546355099

x9 = 83.9744334197414

x10 = -69.9636066061584

x11 = -62.1096249721839

x12 = 80.1737827303252

x13 = 64.4658194623763

x14 = 6.34635537096511

x15 = -36.0651454524971

x16 = 44.0454672140426

x17 = -58.0562940276257

x18 = 50.3286525212222

x19 = 91.8284150537159

x20 = -21.927978511343

x21 = 94.3109496714793

x22 = 39.3330782336579

x23 = 7.91715169776

x24 = -80.0474426027542

x25 = -35.4060874166707

x26 = 333.071991344304

x27 = 72.3198010963508

x28 = 88.0277643642997

x29 = 73.8905974231457

x30 = 61.9832848446129

x31 = 22.0543186389141

x32 = -95.7554058707032

x33 = -3.99016088077276

x34 = -98.2379404884666

x35 = -307.812909988014

x36 = -37.635941779292

x37 = -51.7731087204461

x38 = 47.8461179034588

x39 = -47.9724580310299

x40 = 68.2664701517925

x41 = 46.2753215766639

x42 = -12.5032005505737

x43 = 31.4790965996835

x44 = 42.4746708872477

x45 = 24.2841730015354

x46 = 25.8549693283303

x47 = -22.8397168023115

x48 = 98.1116003608955

x49 = 54.1293032106384

x50 = 58.1826341551967

x51 = -99.8087368152615

x52 = -18.1273278219268

x53 = 76.120451785767

x54 = -63.6804212989788

x55 = 99.6823966876904

x56 = -87.9014242367287

x57 = 10.1470060603813

x58 = 3.86382075320172

x59 = -77.8175882401329

x60 = 20.4835223121192

x61 = -40.1184763970554

x62 = 0.0631700637855197

x63 = -59.6270903544206

x64 = 36.1914855800681

x65 = 90.257618726921

x66 = 51.8994488480171

x67 = -65.9102756616001

x68 = 29.9083002728886

x69 = -91.954755181287

x70 = -55.8264396650044

x71 = 39.9921362694843

x72 = 18.0009876943558

x73 = 86.4569680375048

x74 = -44.8308653774401

x75 = -29.7819601453175

x76 = -85.6715698741074

x77 = -73.7642572955746

x78 = -54.9147013740359

x79 = 82.4036370929466

x80 = -81.6182389295491

x81 = 14.2003370049396

x82 = -43.9191270864716

x83 = -14.0739968773686

x84 = -7.79081157018896

x85 = -25.9813094559013

x86 = -41.6892727238503

x87 = 95.8817459982742

x88 = 28.3375039460937

x89 = 69.8372664785874

x90 = 11.0587443513498

x91 = 2.29302442640683

x92 = -19.6981241487217

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