sin2x=3/4 (уравнение)

Найду корень уравнения: sin2x=3/4

Вы ввели

sin(2*x) = 3/4

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

Подробное решение

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

Быстрый ответ

     pi   asin(3/4)
x1 = -- - ---------
     2        2    

$$x_{1} = - \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2} + \frac{\pi}{2}$$

     asin(3/4)
x2 = ---------
         2    

$$x_{2} = \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2}$$

Сумма и произведение корней

сумма

    pi   asin(3/4)   asin(3/4)
0 + -- - --------- + ---------
    2        2           2    

$$\frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2} + \left(0 + \left(- \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2} + \frac{\pi}{2}\right)\right)$$

=

pi
--
2 

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

произведение

  /pi   asin(3/4)\ asin(3/4)
1*|-- - ---------|*---------
  \2        2    /     2    

$$1 \left(- \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2} + \frac{\pi}{2}\right) \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2}$$

=

(pi - asin(3/4))*asin(3/4)
--------------------------
            4             

$$\frac{\left(\pi - \operatorname{asin}{\left(\frac{3}{4} \right)}\right) \operatorname{asin}{\left(\frac{3}{4} \right)}}{4}$$

Численный ответ

x1 = 88.3886253400049

x2 = 22.4151796146193

x3 = 57.6954330519204

x4 = 6.70721634667033

x5 = -12.1423395748684

x6 = 60.1142914576968

x7 = -30.9918954964072

x8 = 38.1231428825683

x9 = -52.9830440715357

x10 = 95.3945448949979

x11 = 34.9815502289785

x12 = -61.6850877844917

x13 = -2.71756161409905

x14 = -1.99482736628564

x15 = 66.3974767648764

x16 = 48.2706550911511

x17 = 56.972698804107

x18 = 70.2618036662796

x19 = 9.84880900026012

x20 = 51.4122477447408

x21 = -39.6939392093632

x22 = 13.7131359016633

x23 = 31.8399575753887

x24 = -45.9771245165427

x25 = -65.5494146858949

x26 = 53.8311061505172

x27 = -78.1157853002541

x28 = -84.3989706074337

x29 = 4.28835794089395

x30 = 44.4063281897478

x31 = -80.5346437060305

x32 = 63.9786183591

x33 = 16.1319943074397

x34 = -83.6762363596203

x35 = 100.954995954364

x36 = 50.6895134969274

x37 = 92.2529522414082

x38 = 78.9638473792356

x39 = -81.2573779538439

x40 = -74.2514583988509

x41 = 89.1113595878184

x42 = 75.8222547256458

x43 = 85.9697669342286

x44 = -9.00074692127864

x45 = 12.9904016538499

x46 = -36.5523465557734

x47 = 19.9963212088429

x48 = -23.9859759414142

x49 = 79.686581627049

x50 = 72.680662072056

x51 = 28.6983649217989

x52 = -56.1246367251255

x53 = -49.841451417946

x54 = 41.9874697839715

x55 = -30.2691612485938

x56 = -58.5434951309019

x57 = -59.2662293787153

x58 = -71.8325999930745

x59 = -15.2839322284582

x60 = -37.2750808035868

x61 = 0.424031039490741

x62 = -34.133488149997

x63 = -43.5582661107664

x64 = -14.5611979806448

x65 = 29.4210991696123

x66 = 94.6718106471845

x67 = 73.4033963198694

x68 = 82.1054400328254

x69 = -93.8237485682031

x70 = -21.5671175356378

x71 = -8.27801267346522

x72 = -100.106933875383

x73 = -52.2603098237223

x74 = 7.42995059448374

x75 = -67.9682730916713

x76 = -17.7027906342346

x77 = -87.5405632610235

x78 = 26.2795065160225

x79 = 35.7042844767919

x80 = 107.960915509357

x81 = -27.8503028428174

x82 = -5.85915426768885

x83 = -96.2426069739794

x84 = -89.9594216667998

x85 = 97.8134033007743

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