sin(3*x-pi/4)=-sqrt(3)/2 (уравнение)

Найду корень уравнения: sin(3*x-pi/4)=-sqrt(3)/2

Вы ввели

                   ___ 
   /      pi\   -\/ 3  
sin|3*x - --| = -------
   \      4 /      2   

$$\sin{\left(3 x - \frac{\pi}{4} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$

Подробное решение

Дано уравнение
$$\sin{\left(3 x - \frac{\pi}{4} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
- это простейшее тригонометрическое ур-ние

Разделим обе части ур-ния на -1

Ур-ние превратится в
$$\cos{\left(3 x + \frac{\pi}{4} \right)} = \frac{\sqrt{3}}{2}$$
Это ур-ние преобразуется в
$$3 x + \frac{\pi}{4} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)}$$
$$3 x + \frac{\pi}{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)}$$
Или
$$3 x + \frac{\pi}{4} = \pi n + \frac{\pi}{6}$$
$$3 x + \frac{\pi}{4} = \pi n - \frac{5 \pi}{6}$$
, где n - любое целое число
Перенесём
$$\frac{\pi}{4}$$
в правую часть ур-ния
с противоположным знаком, итого:
$$3 x = \pi n - \frac{\pi}{12}$$
$$3 x = \pi n - \frac{13 \pi}{12}$$
Разделим обе части полученного ур-ния на
$$3$$
получим ответ:
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{36}$$
$$x_{2} = \frac{\pi n}{3} - \frac{13 \pi}{36}$$

Сумма и произведение корней

сумма

    pi   19*pi
0 - -- + -----
    36     36 

$$\left(- \frac{\pi}{36} + 0\right) + \frac{19 \pi}{36}$$

=

pi
--
2 

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

произведение

  -pi  19*pi
1*----*-----
   36    36 

$$\frac{19 \pi}{36} \cdot 1 \left(- \frac{\pi}{36}\right)$$

=

      2
-19*pi 
-------
  1296 

$$- \frac{19 \pi^{2}}{1296}$$

Быстрый ответ

     -pi 
x1 = ----
      36 

$$x_{1} = - \frac{\pi}{36}$$

     19*pi
x2 = -----
       36 

$$x_{2} = \frac{19 \pi}{36}$$

Численный ответ

x1 = -50.3527489200364

x2 = 94.1605131450941

x3 = 12.1300383013606

x4 = -63.2681853847944

x5 = 35.5174502780846

x6 = 24.6964089157198

x7 = -31.8522588488965

x8 = -35.691983203284

x9 = 10.0356431989674

x10 = -80.02334620394

x11 = 89.9717229403077

x12 = -41.9751685104636

x13 = 39.706240482871

x14 = -39.8807734080704

x15 = -77.9289511015468

x16 = 77.0562864755497

x17 = 33.0739893252925

x18 = 41.4515697348653

x19 = 43.8950306876574

x20 = -0.0872664625997165

x21 = -23.4746784393237

x22 = 75.3109572235553

x23 = 68.6787060659769

x24 = -60.8247244320024

x25 = -88.0518607631139

x26 = -44.0695636128568

x27 = 37.6118453804778

x28 = 96.2549082474873

x29 = -83.8630705583275

x30 = 71.1221670187689

x31 = 58.2067305540109

x32 = -90.1462558655071

x33 = 20.8566845613322

x34 = -50.7018147704353

x35 = 48.0838208924438

x36 = -8.4648468721725

x37 = -81.7686754559343

x38 = 4.10152374218667

x39 = -33.9466539512897

x40 = -54.8906049752217

x41 = -71.6457657943672

x42 = 62.3955207587973

x43 = 100.094632601875

x44 = 2.00712863979348

x45 = 50.178215994837

x46 = -75.8345559991536

x47 = -19.2858882345373

x48 = -67.4569755895808

x49 = 83.6885376331281

x50 = 20.5076187109334

x51 = -108.995811787046

x52 = 56.1123354516177

x53 = 45.9894257900506

x54 = -37.7863783056772

x55 = 66.5843109635837

x56 = 22.6020138133266

x57 = 87.8773278379145

x58 = -29.7578637465033

x59 = -92.2406509679003

x60 = -6.3704517697793

x61 = 14.2244334037538

x62 = -16.8424272817453

x63 = -85.9574656607207

x64 = -21.3802833369305

x65 = 25.0454747661186

x66 = 52.2726110972302

x67 = -73.7401608967604

x68 = -52.4471440224296

x69 = 98.0002374994816

x70 = -27.6634686441101

x71 = -4.27605666738611

x72 = 92.0661180427009

x73 = -2.18166156499291

x74 = 64.4899158611905

x75 = 27.1398698685118

x76 = -96.4294411726867

x77 = 79.4997474283417

x78 = -25.5690735417169

x79 = 60.3011256564041

x80 = -46.16395871525

x81 = 85.7829327355213

x82 = -65.3625804871876

x83 = 54.0179403492245

x84 = 8.29031394697306

x85 = 81.5941425307349

x86 = 41.8006355852642

x87 = -98.8729021254788

x88 = -94.3350460702935

x89 = 18.4132236085402

x90 = 16.318828506147

x91 = -10.9083078249646

x92 = -48.2583538176432

x93 = 6.19591884457987

x94 = -69.551370691974

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