7sin(4x-1)=5 (уравнение)

Найду корень уравнения: 7sin(4x-1)=5

Вы ввели

7*sin(4*x - 1) = 5

$$7 \sin{\left(4 x - 1 \right)} = 5$$

Подробное решение

Дано уравнение
$$7 \sin{\left(4 x - 1 \right)} = 5$$
- это простейшее тригонометрическое ур-ние

Разделим обе части ур-ния на 7

Ур-ние превратится в
$$\sin{\left(4 x - 1 \right)} = \frac{5}{7}$$
Это ур-ние преобразуется в
$$4 x - 1 = 2 \pi n + \operatorname{asin}{\left(\frac{5}{7} \right)}$$
$$4 x - 1 = 2 \pi n - \operatorname{asin}{\left(\frac{5}{7} \right)} + \pi$$
Или
$$4 x - 1 = 2 \pi n + \operatorname{asin}{\left(\frac{5}{7} \right)}$$
$$4 x - 1 = 2 \pi n - \operatorname{asin}{\left(\frac{5}{7} \right)} + \pi$$
, где n - любое целое число
Перенесём
$$-1$$
в правую часть ур-ния
с противоположным знаком, итого:
$$4 x = 2 \pi n + \operatorname{asin}{\left(\frac{5}{7} \right)} + 1$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(\frac{5}{7} \right)} + 1 + \pi$$
Разделим обе части полученного ур-ния на
$$4$$
получим ответ:
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(\frac{5}{7} \right)}}{4} + \frac{1}{4}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(\frac{5}{7} \right)}}{4} + \frac{1}{4} + \frac{\pi}{4}$$

Быстрый ответ

     1   asin(5/7)
x1 = - + ---------
     4       4    

$$x_{1} = \frac{\operatorname{asin}{\left(\frac{5}{7} \right)}}{4} + \frac{1}{4}$$

     1   asin(5/7)   pi
x2 = - - --------- + --
     4       4       4 

$$x_{2} = - \frac{\operatorname{asin}{\left(\frac{5}{7} \right)}}{4} + \frac{1}{4} + \frac{\pi}{4}$$

Сумма и произведение корней

сумма

    1   asin(5/7)   1   asin(5/7)   pi
0 + - + --------- + - - --------- + --
    4       4       4       4       4 

$$\left(0 + \left(\frac{\operatorname{asin}{\left(\frac{5}{7} \right)}}{4} + \frac{1}{4}\right)\right) + \left(- \frac{\operatorname{asin}{\left(\frac{5}{7} \right)}}{4} + \frac{1}{4} + \frac{\pi}{4}\right)$$

=

1   pi
- + --
2   4 

$$\frac{1}{2} + \frac{\pi}{4}$$

произведение

  /1   asin(5/7)\ /1   asin(5/7)   pi\
1*|- + ---------|*|- - --------- + --|
  \4       4    / \4       4       4 /

$$1 \left(\frac{\operatorname{asin}{\left(\frac{5}{7} \right)}}{4} + \frac{1}{4}\right) \left(- \frac{\operatorname{asin}{\left(\frac{5}{7} \right)}}{4} + \frac{1}{4} + \frac{\pi}{4}\right)$$

=

(1 + asin(5/7))*(1 + pi - asin(5/7))
------------------------------------
                 16                 

$$\frac{\left(\operatorname{asin}{\left(\frac{5}{7} \right)} + 1\right) \left(- \operatorname{asin}{\left(\frac{5}{7} \right)} + 1 + \pi\right)}{16}$$

Численный ответ

x1 = -57.67056335304

x2 = 10.2612753857957

x3 = -29.0086327840767

x4 = 32.2524239609242

x5 = -3.87589155535838

x6 = 54.2435725360528

x7 = -40.004207071641

x8 = -13.6882662027829

x9 = 3.97809007861611

x10 = 91.9426843791303

x11 = -63.9537486602196

x12 = 85.2719023852955

x13 = -7.01748420894817

x14 = 33.8232202877191

x15 = 24.0108456402946

x16 = -68.2785409539491

x17 = 19.6860533465651

x18 = 77.8055174379762

x19 = 82.1303097317058

x20 = -10.159076862538

x21 = -35.6794147779115

x22 = -41.9626000850911

x23 = 62.0975541700273

x24 = 2.01969706516603

x25 = 47.9603872288732

x26 = 30.2940309474742

x27 = -32.1502254376665

x28 = 25.9692386537447

x29 = 60.1391611565772

x30 = 99.7966660131048

x31 = -79.6617119281686

x32 = -24.296243803692

x33 = -2.30509522856348

x34 = 74.2763280977313

x35 = 18.1152570197702

x36 = -77.7033189147185

x37 = -21.5422478367574

x38 = -5.83428456880845

x39 = -55.71217033959

x40 = -84.3741009085533

x41 = -1321856.0760237

x42 = -83.9865042218981

x43 = -98.1236711630522

x44 = -38.4334107448461

x45 = -76.1325225879236

x46 = -54.1413740127951

x47 = 46.0019942154231

x48 = 11.8320717125906

x49 = 76.2347211111814

x50 = 22.4400493134997

x51 = -61.9953556467696

x52 = 69.9515358040018

x53 = 0.448900738371134

x54 = -49.8165817190656

x55 = -25.8670401304869

x56 = 84.0887027451558

x57 = 14.9736643661804

x58 = -99.6944674898471

x59 = 8.30288237234562

x60 = 16.1568640063201

x61 = 55.8143688628477

x62 = 67.9931427905517

x63 = 88.4134950388853

x64 = -85.9448972353482

x65 = 36.9648129413089

x66 = -27.825433143937

x67 = 44.4311978886282

x68 = 89.9842913656802

x69 = -11.7298731893329

x70 = 80.5595134049109

x71 = -91.8404858558726

x72 = 41.2896052350384

x73 = -18.0130584965124

x74 = -47.8581887056155

x75 = -65.5245449870145

x76 = 58.5683648297823

x77 = -33.7210217644614

x78 = -43.533396411886

x79 = -46.2873923788206

x80 = -87.5156935621431

x81 = -50.9997813592053

x82 = 40.1064055948987

x83 = 96.2674766728598

x84 = -62.3829523334247

x85 = 96.655073359515

x86 = -93.7988788693227

x87 = 63.280753810167

x88 = -69.849337280744

x89 = -90.2696895290777

x90 = -71.8077302941941

x91 = 66.4223464637568

x92 = -19.9714515099625

x93 = 52.2851795226027

x94 = 38.1480125814487

x95 = 98.2258696863099

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