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

Разделим обе части ур-ния на -1

Ур-ние превратится в
$$\sin{\left(4 x \right)} = - \frac{1}{4}$$
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

     pi   asin(1/4)
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

    pi   asin(1/4)   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

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

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

=

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

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

Численный ответ

x1 = 18.1273278219268

x2 = -99.6823966876904

x3 = -59.7534304819916

x4 = -58.1826341551967

x5 = 87.9014242367287

x6 = 84.1007735473125

x7 = 98.2379404884666

x8 = -91.8284150537159

x9 = 91.954755181287

x10 = 29.7819601453175

x11 = 6.22001524339407

x12 = -18.0009876943558

x13 = -96.5408040341006

x14 = -81.7445790571201

x15 = -14.2003370049396

x16 = 54.2556433382095

x17 = 47.9724580310299

x18 = 90.3839588544921

x19 = -47.8461179034588

x20 = -77.6912481125619

x21 = 76.246791913338

x22 = -37.762281906863

x23 = 40.1184763970554

x24 = 97.3262021974981

x25 = -15.7711333317345

x26 = -73.8905974231457

x27 = -33.7089509623048

x28 = 17883264.1309195

x29 = -3.86382075320172

x30 = 14.0739968773686

x31 = 46.401661704235

x32 = 50.2023123936512

x33 = 3.99016088077276

x34 = 72.1934609687797

x35 = -88.0277643642997

x36 = -54.1293032106384

x37 = 11.8441425147472

x38 = -85.5452297465364

x39 = 64.3394793348052

x40 = -45.6162635408375

x41 = 42.3483307596767

x42 = 36.0651454524971

x43 = -98.1116003608955

x44 = -19.5717840211507

x45 = 6258.90113417805

x46 = -41.5629325962792

x47 = 28.2111638185226

x48 = 43.9191270864716

x49 = 74.6759955865431

x50 = 24.4105131291064

x51 = 10.2733461879523

x52 = 2.41936455397786

x53 = 58.9680323185941

x54 = -66.0366157891712

x55 = 69.9636066061584

x56 = 68.3928102793635

x57 = -25.8549693283303

x58 = 58.0562940276257

x59 = -0.0631700637855197

x60 = 25.9813094559013

x61 = -63.5540811714078

x62 = -83.9744334197414

x63 = 35.4060874166707

x64 = -29.9083002728886

x65 = 21.927978511343

x66 = -23.625114965709

x67 = 65.9102756616001

x68 = 86.3306279099338

x69 = -69.8372664785874

x70 = -89.5985606910946

x71 = -55.7000995374333

x72 = 62.1096249721839

x73 = -32.1381546355099

x74 = 20.3571821845481

x75 = -61.9832848446129

x76 = 94.1846095439083

x77 = -80.1737827303252

x78 = 7.79081157018896

x79 = 80.0474426027542

x80 = -22.0543186389141

x81 = -1.63396639058042

x82 = 73.7642572955746

x83 = -76.120451785767

x84 = 51.7731087204461

x85 = 32.2644947630809

x86 = -39.9921362694843

x87 = -51.8994488480171

x88 = -67.6074121159661

x89 = -95.8817459982742

x90 = -36.1914855800681

x91 = -7.91715169776

x92 = -10.1470060603813

x93 = -11.7178023871762

x94 = -72.9788591321772

x95 = -44.0454672140426

График