27*81^sinx-12*9^sinx+1=0 (уравнение)

Найду корень уравнения: 27*81^sinx-12*9^sinx+1=0

Вы ввели

     sin(x)       sin(x)        
27*81       - 12*9       + 1 = 0

$$27 \cdot 81^{\sin{\left(x \right)}} - 12 \cdot 9^{\sin{\left(x \right)}} + 1 = 0$$

Подробное решение

Дано уравнение
$$27 \cdot 81^{\sin{\left(x \right)}} - 12 \cdot 9^{\sin{\left(x \right)}} + 1 = 0$$
преобразуем
$$27 \cdot 3^{4 \sin{\left(x \right)}} - 12 \cdot 3^{2 \sin{\left(x \right)}} + 1 = 0$$
$$\left(27 \cdot 81^{\sin{\left(x \right)}} - 12 \cdot 9^{\sin{\left(x \right)}} + 1\right) + 0 = 0$$
Сделаем замену
$$w = \sin{\left(x \right)}$$
$$27 \cdot 81^{w} - 12 \cdot 9^{w} + 1 = 0$$
или
$$27 \cdot 81^{w} - 12 \cdot 9^{w} + 1 = 0$$
Сделаем замену
$$v = 9^{w}$$
получим
$$27 v^{2} - 12 v + 1 = 0$$
или
$$27 v^{2} - 12 v + 1 = 0$$
Это уравнение вида

a*v^2 + b*v + c = 0

Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
где D = b^2 - 4*a*c - это дискриминант.
Т.к.
$$a = 27$$
$$b = -12$$
$$c = 1$$
, то

D = b^2 - 4 * a * c = 

(-12)^2 - 4 * (27) * (1) = 36

Т.к. D > 0, то уравнение имеет два корня.

v1 = (-b + sqrt(D)) / (2*a)

v2 = (-b - sqrt(D)) / (2*a)

или
$$v_{1} = \frac{1}{3}$$
Упростить
$$v_{2} = \frac{1}{9}$$
Упростить
делаем обратную замену
$$9^{w} = v$$
или
$$w = \frac{\log{\left(v \right)}}{\log{\left(9 \right)}}$$
Тогда, окончательный ответ
$$w_{1} = \frac{\log{\left(\frac{1}{3} \right)}}{\log{\left(9 \right)}} = - \frac{1}{2}$$
$$w_{2} = \frac{\log{\left(\frac{1}{9} \right)}}{\log{\left(9 \right)}} = -1$$
делаем обратную замену
$$\sin{\left(x \right)} = w$$
Дано уравнение
$$\sin{\left(x \right)} = w$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Или
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, где n - любое целое число
подставляем w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$

Быстрый ответ

     -pi 
x1 = ----
      2  

$$x_{1} = - \frac{\pi}{2}$$

     -pi 
x2 = ----
      6  

$$x_{2} = - \frac{\pi}{6}$$

     7*pi
x3 = ----
      6  

$$x_{3} = \frac{7 \pi}{6}$$

     3*pi
x4 = ----
      2  

$$x_{4} = \frac{3 \pi}{2}$$

              /    /1    pi*I \\     /    /1    pi*I \\
x5 = pi + I*im|asin|- - ------|| + re|asin|- - ------||
              \    \2   log(3)//     \    \2   log(3)//

$$x_{5} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)}$$

              /    /     pi*I \\     /    /     pi*I \\
x6 = pi + I*im|asin|1 - ------|| + re|asin|1 - ------||
              \    \    log(3)//     \    \    log(3)//

$$x_{6} = \operatorname{re}{\left(\operatorname{asin}{\left(1 - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(1 - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)}$$

         /    /1    pi*I \\       /    /1    pi*I \\
x7 = - re|asin|- - ------|| - I*im|asin|- - ------||
         \    \2   log(3)//       \    \2   log(3)//

$$x_{7} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)}$$

         /    /     pi*I \\       /    /     pi*I \\
x8 = - re|asin|1 - ------|| - I*im|asin|1 - ------||
         \    \    log(3)//       \    \    log(3)//

$$x_{8} = - \operatorname{re}{\left(\operatorname{asin}{\left(1 - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(1 - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)}$$

Сумма и произведение корней

сумма

    pi   pi   7*pi   3*pi            /    /1    pi*I \\     /    /1    pi*I \\            /    /     pi*I \\     /    /     pi*I \\       /    /1    pi*I \\       /    /1    pi*I \\       /    /     pi*I \\       /    /     pi*I \\
0 - -- - -- + ---- + ---- + pi + I*im|asin|- - ------|| + re|asin|- - ------|| + pi + I*im|asin|1 - ------|| + re|asin|1 - ------|| + - re|asin|- - ------|| - I*im|asin|- - ------|| + - re|asin|1 - ------|| - I*im|asin|1 - ------||
    2    6     6      2              \    \2   log(3)//     \    \2   log(3)//            \    \    log(3)//     \    \    log(3)//       \    \2   log(3)//       \    \2   log(3)//       \    \    log(3)//       \    \    log(3)//

$$\left(\left(\left(\operatorname{re}{\left(\operatorname{asin}{\left(1 - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(1 - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)}\right) + \left(\left(\left(\left(\left(- \frac{\pi}{2} + 0\right) - \frac{\pi}{6}\right) + \frac{7 \pi}{6}\right) + \frac{3 \pi}{2}\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)}\right)\right)\right) - \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)}\right)\right) - \left(\operatorname{re}{\left(\operatorname{asin}{\left(1 - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(1 - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)}\right)$$

=

4*pi

$$4 \pi$$

произведение

  -pi  -pi  7*pi 3*pi /         /    /1    pi*I \\     /    /1    pi*I \\\ /         /    /     pi*I \\     /    /     pi*I \\\ /    /    /1    pi*I \\       /    /1    pi*I \\\ /    /    /     pi*I \\       /    /     pi*I \\\
1*----*----*----*----*|pi + I*im|asin|- - ------|| + re|asin|- - ------|||*|pi + I*im|asin|1 - ------|| + re|asin|1 - ------|||*|- re|asin|- - ------|| - I*im|asin|- - ------|||*|- re|asin|1 - ------|| - I*im|asin|1 - ------|||
   2    6    6    2   \         \    \2   log(3)//     \    \2   log(3)/// \         \    \    log(3)//     \    \    log(3)/// \    \    \2   log(3)//       \    \2   log(3)/// \    \    \    log(3)//       \    \    log(3)///

$$\frac{3 \pi}{2} \frac{7 \pi}{6} - \frac{\pi}{6} \cdot 1 \left(- \frac{\pi}{2}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(1 - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(1 - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(1 - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(1 - \frac{i \pi}{\log{\left(3 \right)}} \right)}\right)}\right)$$

=

    4 /    /    /-pi*I + log(3)\\     /    /-pi*I + log(3)\\\ /    /    /-2*pi*I + log(3)\\     /    /-2*pi*I + log(3)\\\ /         /    /-pi*I + log(3)\\     /    /-pi*I + log(3)\\\ /         /    /-2*pi*I + log(3)\\     /    /-2*pi*I + log(3)\\\
7*pi *|I*im|asin|--------------|| + re|asin|--------------|||*|I*im|asin|----------------|| + re|asin|----------------|||*|pi + I*im|asin|--------------|| + re|asin|--------------|||*|pi + I*im|asin|----------------|| + re|asin|----------------|||
      \    \    \    log(3)    //     \    \    log(3)    /// \    \    \    2*log(3)    //     \    \    2*log(3)    /// \         \    \    log(3)    //     \    \    log(3)    /// \         \    \    2*log(3)    //     \    \    2*log(3)    ///
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                           48                                                                                                                          

$$\frac{7 \pi^{4} \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\log{\left(3 \right)} - 2 i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\log{\left(3 \right)} - 2 i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\log{\left(3 \right)} - i \pi}{\log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\log{\left(3 \right)} - i \pi}{\log{\left(3 \right)}} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\log{\left(3 \right)} - 2 i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\log{\left(3 \right)} - 2 i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\log{\left(3 \right)} - i \pi}{\log{\left(3 \right)}} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\log{\left(3 \right)} - i \pi}{\log{\left(3 \right)}} \right)}\right)}\right)}{48}$$

Численный ответ

x1 = 28.7979326579064

x2 = -83.2522055120905

x3 = 80.110613146131

x4 = -19.3731546971371

x5 = -1.57079643126535

x6 = -7.85398149709644

x7 = 5.75958653158129

x8 = 91.6297857297023

x9 = 36.1283157526274

x10 = 72.7802298081635

x11 = 67.544242261172

x12 = 43.4586983746588

x13 = 22.5147473507269

x14 = 47.6474885794452

x15 = 74.8746249105567

x16 = 4.71238878135175

x17 = -34.0339204138894

x18 = 29.8451303223305

x19 = -51.8362786893523

x20 = 37.1755130674792

x21 = 30.8923277602996

x22 = -52.8834763354282

x23 = 81.1578102177363

x24 = -46.6002910282486

x25 = 98.9601681124219

x26 = -21.4675497995303

x27 = -27.7507351067098

x28 = -31.9395253114962

x29 = 53.9306738866248

x30 = -0.523598775598299

x31 = 62.3082542961976

x32 = 18.3259571459405

x33 = -58.1194639980725

x34 = -78.0162175641465

x35 = 98.9601683767802

x36 = 3.66519142918809

x37 = 41.3643032722656

x38 = -94.7713783832921

x39 = -59.1666616426078

x40 = -2.61799387799149

x41 = -69.6386371545737

x42 = -89.5353907489903

x43 = -71.733032256967

x44 = 92.6769830836429

x45 = 73.8274274819368

x46 = 66.497044500984

x47 = 86.3937978872822

x48 = -20.4203520388702

x49 = -25.6563400043166

x50 = 79.0634151153431

x51 = -96.8657734856853

x52 = -40.317105721069

x53 = -88.4881930761125

x54 = -63.3554518473942

x55 = -82.2050077689329

x56 = 42.4115007279036

x57 = -65.4498469497874

x58 = -44.5058959258554

x59 = 60.2138591938044

x60 = -15.1843644923507

x61 = 49.7418836818384

x62 = 100.007366139275

x63 = -8.90117918517108

x64 = -14.1371668376557

x65 = -64.4026491910109

x66 = 97.9129710368819

x67 = 68.5914396033772

x68 = 56.025068989018

x69 = -39.2699083668319

x70 = 9.94837673636768

x71 = -6.80678408277789

x72 = -50.789081233035

x73 = -95.8185758680478

x74 = 12.0427718387609

x75 = -13.0899693899575

x76 = 48.6946859316831

x77 = -38.2227106186758

x78 = -57.0722665402146

x79 = 87.4409955249159

x80 = 35.081117965086

x81 = 93.7241808320955

x82 = -75.9218224617533

x83 = -101.054563690472

x84 = 23.5619451119578

x85 = -90.5825881785057

x86 = 85.3466004225227

x87 = -45.5530935902303

x88 = -84.2994028713261

x89 = 24.60914245312

x90 = 16.2315620435473

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