sin(3x)=2/3 (уравнение)

Найду корень уравнения: sin(3x)=2/3

Вы ввели

sin(3*x) = 2/3

$$\sin{\left(3 x \right)} = \frac{2}{3}$$

Подробное решение

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

Быстрый ответ

       asin(2/3)   pi
x1 = - --------- + --
           3       3 

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

     asin(2/3)
x2 = ---------
         3    

$$x_{2} = \frac{\operatorname{asin}{\left(\frac{2}{3} \right)}}{3}$$

Сумма и произведение корней

сумма

      asin(2/3)   pi   asin(2/3)
0 + - --------- + -- + ---------
          3       3        3    

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

=

pi
--
3 

$$\frac{\pi}{3}$$

произведение

  /  asin(2/3)   pi\ asin(2/3)
1*|- --------- + --|*---------
  \      3       3 /     3    

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

=

(pi - asin(2/3))*asin(2/3)
--------------------------
            9             

$$\frac{\left(\pi - \operatorname{asin}{\left(\frac{2}{3} \right)}\right) \operatorname{asin}{\left(\frac{2}{3} \right)}}{9}$$

Численный ответ

x1 = 96.5854172621627

x2 = -26.4231813319906

x3 = -81.438166441259

x4 = 94.4910221597695

x5 = -37.4558692910019

x6 = 4.43203275686205

x7 = -70.4054784822477

x8 = -53.6503176631021

x9 = 90.3022319549831

x10 = -9.66802051284504

x11 = -50.022239905361

x12 = -57.8391078678885

x13 = 72.0133884804896

x14 = 34.3142766374121

x15 = 37.9423543951532

x16 = -83.5325615436522

x17 = -6.03994275510393

x18 = 78.2965737876692

x19 = -79.3437713388658

x20 = -43.7390545981814

x21 = 17.5591158182665

x22 = -91.910141953225

x23 = 0.243242552075655

x24 = -32.7063666391702

x25 = -39.5502643933951

x26 = 28.0310913302325

x27 = 74.1077835828828

x28 = -13.8568107176314

x29 = -18.0456009224178

x30 = 88.2078368525899

x31 = 76.202178685276

x32 = -87.7213517484386

x33 = -76.6886637894273

x34 = 21.7479060230529

x35 = -41.6446594957883

x36 = 2.33763765446885

x37 = -73.0605860316862

x38 = -99.7270099157524

x39 = 32.2198815350189

x40 = 30.1254864326257

x41 = 92.3966270573763

x42 = -96.0989321580113

x43 = 6.52642785925524

x44 = -94.0045370556181

x45 = -221.762638301603

x46 = 25.9366962278393

x47 = -29.0782888814291

x48 = -52.1166350077542

x49 = -15.9512058200246

x50 = -20.139996024811

x51 = 44.2255397023328

x52 = -10.2287329598903

x53 = -8.13433785749713

x54 = -47.9278448029678

x55 = 11.2759305110869

x56 = -89.8157468508318

x57 = 65.73020317331

x58 = 63.6358080709168

x59 = 8.62082296164844

x60 = -3.94554765271074

x61 = -1.85115255031754

x62 = 84.0190466478035

x63 = -64.1222931750681

x64 = -45.8334497005746

x65 = 86.1134417501967

x66 = 19.6535109206597

x67 = 40.0367494975464

x68 = 46.319934804726

x69 = 50.5087250095123

x70 = -85.6269566460454

x71 = -62.0278980726749

x72 = 69.9189933780964

x73 = 52.6031201119055

x74 = 201.305172381822

x75 = -66.2166882774613

x76 = -22.2343911272042

x77 = -24.3287862295974

x78 = 67.8245982757032

x79 = 54.6975152142987

x80 = 48.4143299071192

x81 = -97.6326148133592

x82 = -11.7624156152382

x83 = 81.9246515454103

x84 = 98.6798123645558

x85 = -68.3110833798545

x86 = 23.8423011254461

x87 = 42.1311445999396

x88 = -59.9335029702817

x89 = 984.608940676878

x90 = -35.3614741886087

x91 = -55.7447127654953

График