cos(x)=pi/4 (уравнение)

Найду корень уравнения: cos(x)=pi/4

Вы ввели

         pi
cos(x) = --
         4 

$$\cos{\left(x \right)} = \frac{\pi}{4}$$

Подробное решение

Дано уравнение
$$\cos{\left(x \right)} = \frac{\pi}{4}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = \pi n + \operatorname{acos}{\left(\frac{\pi}{4} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(\frac{\pi}{4} \right)}$$
Или
$$x = \pi n + \operatorname{acos}{\left(\frac{\pi}{4} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(\frac{\pi}{4} \right)}$$
, где n - любое целое число

Сумма и произведение корней

сумма

          /pi\              /pi\
0 + - acos|--| + 2*pi + acos|--|
          \4 /              \4 /

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

=

2*pi

$$2 \pi$$

произведение

  /      /pi\       \     /pi\
1*|- acos|--| + 2*pi|*acos|--|
  \      \4 /       /     \4 /

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

=

/      /pi\       \     /pi\
|- acos|--| + 2*pi|*acos|--|
\      \4 /       /     \4 /

$$\left(- \operatorname{acos}{\left(\frac{\pi}{4} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{\pi}{4} \right)}$$

Быстрый ответ

           /pi\       
x1 = - acos|--| + 2*pi
           \4 /       

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

         /pi\
x2 = acos|--|
         \4 /

$$x_{2} = \operatorname{acos}{\left(\frac{\pi}{4} \right)}$$

Численный ответ

x1 = -76.0656809021834

x2 = -37.0316546270491

x3 = 2035.08458231016

x4 = -99.863507698845

x5 = 74.7307664701267

x6 = -43.3148399342287

x7 = -1124.02271276912

x8 = -25.8001984447467

x9 = -88.6320515165426

x10 = -68.4475811629471

x11 = 19.5170131375671

x12 = 5.6157280911512

x13 = -69.7824955950038

x14 = -30.7484693198695

x15 = 44.6497543662855

x16 = 68.4475811629471

x17 = 25.8001984447467

x18 = -18.1820987055104

x19 = -44.6497543662855

x20 = 32.0833837519263

x21 = 49.5980252414083

x22 = -11.8989133983308

x23 = 13.2338278303876

x24 = 87.2971370844858

x25 = -57.2161249806447

x26 = 82.348866209363

x27 = 76.0656809021834

x28 = -5.6157280911512

x29 = 94.9152368237222

x30 = 55.8812105485879

x31 = -87.2971370844858

x32 = -13219.1544290898

x33 = -50.9329396734651

x34 = -0.667457216028384

x35 = 43.3148399342287

x36 = -19.5170131375671

x37 = -74.7307664701267

x38 = 37.0316546270491

x39 = -82.348866209363

x40 = -49.5980252414083

x41 = -38.3665690591059

x42 = -93.5803223916654

x43 = 99.863507698845

x44 = 24.46528401269

x45 = 57.2161249806447

x46 = 38.3665690591059

x47 = -13.2338278303876

x48 = 6.95064252320797

x49 = -94.9152368237222

x50 = 63.4993102878243

x51 = -81.0139517773062

x52 = 18.1820987055104

x53 = -24.46528401269

x54 = -62.1643958557675

x55 = -101.198422130902

x56 = 11.8989133983308

x57 = 62.1643958557675

x58 = -41155.5312192423

x59 = -55.8812105485879

x60 = -6.95064252320797

x61 = 88.6320515165426

x62 = 81.0139517773062

x63 = 1106.50807127964

x64 = 0.667457216028384

x65 = 69.7824955950038

x66 = 50.9329396734651

x67 = 30.7484693198695

x68 = -63.4993102878243

x69 = -32.0833837519263

x70 = 93.5803223916654

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