sin5x=4/5 (уравнение)

Найду корень уравнения: sin5x=4/5

Вы ввели

sin(5*x) = 4/5

$$\sin{\left(5 x \right)} = \frac{4}{5}$$

Подробное решение

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

Быстрый ответ

       asin(4/5)   pi
x1 = - --------- + --
           5       5 

$$x_{1} = - \frac{\operatorname{asin}{\left(\frac{4}{5} \right)}}{5} + \frac{\pi}{5}$$

     asin(4/5)
x2 = ---------
         5    

$$x_{2} = \frac{\operatorname{asin}{\left(\frac{4}{5} \right)}}{5}$$

Сумма и произведение корней

сумма

  asin(4/5)   pi   asin(4/5)
- --------- + -- + ---------
      5       5        5    

$$\frac{\operatorname{asin}{\left(\frac{4}{5} \right)}}{5} + \left(- \frac{\operatorname{asin}{\left(\frac{4}{5} \right)}}{5} + \frac{\pi}{5}\right)$$

=

pi
--
5 

$$\frac{\pi}{5}$$

произведение

/  asin(4/5)   pi\ asin(4/5)
|- --------- + --|*---------
\      5       5 /     5    

$$\left(- \frac{\operatorname{asin}{\left(\frac{4}{5} \right)}}{5} + \frac{\pi}{5}\right) \frac{\operatorname{asin}{\left(\frac{4}{5} \right)}}{5}$$

=

(pi - asin(4/5))*asin(4/5)
--------------------------
            25            

$$\frac{\left(\pi - \operatorname{asin}{\left(\frac{4}{5} \right)}\right) \operatorname{asin}{\left(\frac{4}{5} \right)}}{25}$$

Численный ответ

x1 = -37.5136527994772

x2 = -83.7518236290888

x3 = -2.0704146357542

x4 = -67.6729422739392

x5 = -32.2297041102162

x6 = 62.0180754974776

x7 = -53.5925341546268

x8 = -87.7791352569139

x9 = 48.1950678216825

x10 = 54.2208526853448

x11 = -57.6198457824519

x12 = -13.6375486321948

x13 = 88.1500533441145

x14 = 500.327009495095

x15 = -46.0527117860113

x16 = 7.98268185573314

x17 = -66.158904768986

x18 = -79.9819124447811

x19 = -73.6987271376015

x20 = -86.522498195478

x21 = 40.3978450095497

x22 = 20.291652026575

x23 = -90.0350089362684

x24 = 60.7614384360417

x25 = 31.8587860230156

x26 = -53.8499345981441

x27 = 51.9649790059902

x28 = 68.3012608046572

x29 = -100.088105427756

x30 = -23.6906451236821

x31 = -25.9465188030366

x32 = 94.4332386512941

x33 = 41.9118825145029

x34 = 28.0888748387078

x35 = -47.5667492909645

x36 = 16.7791412857846

x37 = 6.46864435077991

x38 = 99.4597868970378

x39 = -12.1235111272415

x40 = 18.0357783472205

x41 = -27.2031558644725

x42 = 64.2739491768321

x43 = -42.5402010452209

x44 = 38.1419713301952

x45 = 65.7879866817853

x46 = 11.7525930400409

x47 = 88.4074537876319

x48 = -7.35436332501518

x49 = 14.2658671629127

x50 = -97.8322317484012

x51 = -56.1058082774986

x52 = -94.0623205640935

x53 = -29.7164299873444

x54 = 34.1146597023701

x55 = 98.2031498356019

x56 = -11.1242745093229

x57 = 75.8410831732727

x58 = -3.58445214070743

x59 = -77.7260387654266

x60 = -194.593285478967

x61 = 2197.0444428771

x62 = -59.8757194618064

x63 = -69.9288159532937

x64 = -5.84032582006195

x65 = 72.0711719889649

x66 = -91.5490464412216

x67 = -39.7695264788317

x68 = 74.3270456683194

x69 = -22.1766076187289

x70 = -35.999615294524

x71 = 92.1773649719396

x72 = -49.8226229703191

x73 = 78.0969568526272

x74 = 55.4774897467807

x75 = -81.238549506217

x76 = 10.2385555350877

x77 = 0.185459043600322

x78 = 85.89417966476

x79 = 82.1242684804523

x80 = 98.4605502791192

x81 = -63.9030310896315

x82 = -224.495174509912

x83 = 95.9472761562474

x84 = 84.3801421598068

x85 = -76.2120012604733

x86 = 4.21277067142539

x87 = -33.7437416151694

x88 = -93.8049201205762

x89 = -9.86763744788702

x90 = -15.8934223115493

x91 = 21.8056895315282

x92 = 58.2481643131698

x93 = 44.1677561938574

x94 = 10.495955978605

x95 = 24.3189636544001

x96 = 30.3447485180623

x97 = 50.450941501037

x98 = -43.7968381066568

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