cos(-7x)=2/5 (уравнение)

Найду корень уравнения: cos(-7x)=2/5

Вы ввели

cos(-7*x) = 2/5

$$\cos{\left(- 7 x \right)} = \frac{2}{5}$$

Подробное решение

Дано уравнение
$$\cos{\left(- 7 x \right)} = \frac{2}{5}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$7 x = \pi n + \operatorname{acos}{\left(\frac{2}{5} \right)}$$
$$7 x = \pi n - \pi + \operatorname{acos}{\left(\frac{2}{5} \right)}$$
Или
$$7 x = \pi n + \operatorname{acos}{\left(\frac{2}{5} \right)}$$
$$7 x = \pi n - \pi + \operatorname{acos}{\left(\frac{2}{5} \right)}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$7$$
получим ответ:
$$x_{1} = \frac{\pi n}{7} + \frac{\operatorname{acos}{\left(\frac{2}{5} \right)}}{7}$$
$$x_{2} = \frac{\pi n}{7} - \frac{\pi}{7} + \frac{\operatorname{acos}{\left(\frac{2}{5} \right)}}{7}$$

Быстрый ответ

       acos(2/5)   2*pi
x1 = - --------- + ----
           7        7  

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

     acos(2/5)
x2 = ---------
         7    

$$x_{2} = \frac{\operatorname{acos}{\left(\frac{2}{5} \right)}}{7}$$

Сумма и произведение корней

сумма

      acos(2/5)   2*pi   acos(2/5)
0 + - --------- + ---- + ---------
          7        7         7    

$$\frac{\operatorname{acos}{\left(\frac{2}{5} \right)}}{7} + \left(0 + \left(- \frac{\operatorname{acos}{\left(\frac{2}{5} \right)}}{7} + \frac{2 \pi}{7}\right)\right)$$

=

2*pi
----
 7  

$$\frac{2 \pi}{7}$$

произведение

  /  acos(2/5)   2*pi\ acos(2/5)
1*|- --------- + ----|*---------
  \      7        7  /     7    

$$1 \left(- \frac{\operatorname{acos}{\left(\frac{2}{5} \right)}}{7} + \frac{2 \pi}{7}\right) \frac{\operatorname{acos}{\left(\frac{2}{5} \right)}}{7}$$

=

(-acos(2/5) + 2*pi)*acos(2/5)
-----------------------------
              49             

$$\frac{\left(- \operatorname{acos}{\left(\frac{2}{5} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{2}{5} \right)}}{49}$$

Численный ответ

x1 = 59.9734480143293

x2 = -99.7989783682374

x3 = -19.912765176954

x4 = 71.9734434364421

x5 = 70.1782476343907

x6 = 15.9911508640722

x7 = -50.0998711030471

x8 = 74.3350144307397

x9 = 96.2085867641347

x10 = 86.0037871440733

x11 = 40.2262941917649

x12 = 36.0695273954158

x13 = 80.0518245456729

x14 = -68.0518291235602

x15 = 18.1175693749027

x16 = -47.7383001087494

x17 = 98.003782566186

x18 = 62.0998665251598

x19 = -35.7383046866366

x20 = -87.7989829461246

x21 = -95.8773640553555

x22 = -98.003782566186

x23 = -71.9734434364421

x24 = -91.7205972590065

x25 = 52.2262896138776

x26 = 10.0391882656718

x27 = -86.0037871440733

x28 = 58.178252212278

x29 = -63.8950623272111

x30 = -51.8950669050984

x31 = -65.6902581292625

x32 = -15.9911508640722

x33 = 26.1959504841336

x34 = 15.0935529630465

x35 = 68.0518291235602

x36 = 0.16561135438963

x37 = -39.6599189995185

x38 = -55.8166812179803

x39 = -7.91276975484127

x40 = 76.1302102327911

x41 = -77.9254060348424

x42 = 6.11757395278996

x43 = -37.8647231974671

x44 = 42.0214899938162

x45 = 100.696576269263

x46 = 45.943104306698

x47 = 92.2869724512529

x48 = -33.9431088845853

x49 = -89.9254014569552

x50 = 1.96080715644094

x51 = -11.8343840677231

x52 = -94.0821682533042

x53 = -1.96080715644094

x54 = -17.7863466661235

x55 = -25.8647277753544

x56 = -73.7686392384934

x57 = -6.44879666156922

x58 = -69.8470249256115

x59 = 34.2743315933645

x60 = -27.9911462861849

x61 = -56.7142791190059

x62 = 24.0695319733031

x63 = 8.24399246362053

x64 = -3.75600295849225

x65 = -59.9734480143293

x66 = -45.943104306698

x67 = 22.2743361712518

x68 = -43.8166857958675

x69 = 32.147913082534

x70 = 27.9911462861849

x71 = 88.1302056549038

x72 = 89.9254014569552

x73 = -24.0695319733031

x74 = -61.7686438163806

x75 = -21.7079609790054

x76 = -42.0214899938162

x77 = 78.2566287436216

x78 = 30.3527172804826

x79 = 35.1719294943902

x80 = 54.0214854159289

x81 = 84.208591342022

x82 = 48.3046753009958

x83 = -29.7863420882363

x84 = 94.0821682533042

x85 = 44.1479085046467

x86 = -79.7206018368937

x87 = 50.0998711030471

x88 = 4.32237815073865

x89 = 14.1959550620209

x90 = 63.8950623272111

x91 = -54.0214854159289

x92 = 19.912765176954

x93 = -81.8470203477243

x94 = 66.2566333215089

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