cos(x)=3/pi (уравнение)

Найду корень уравнения: cos(x)=3/pi

Вы ввели

         3 
cos(x) = --
         pi

$$\cos{\left(x \right)} = \frac{3}{\pi}$$

Подробное решение

Дано уравнение
$$\cos{\left(x \right)} = \frac{3}{\pi}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = \pi n + \operatorname{acos}{\left(\frac{3}{\pi} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(\frac{3}{\pi} \right)}$$
Или
$$x = \pi n + \operatorname{acos}{\left(\frac{3}{\pi} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(\frac{3}{\pi} \right)}$$
, где n - любое целое число

Быстрый ответ

           /3 \       
x1 = - acos|--| + 2*pi
           \pi/       

$$x_{1} = - \operatorname{acos}{\left(\frac{3}{\pi} \right)} + 2 \pi$$

         /3 \
x2 = acos|--|
         \pi/

$$x_{2} = \operatorname{acos}{\left(\frac{3}{\pi} \right)}$$

Сумма и произведение корней

сумма

          /3 \              /3 \
0 + - acos|--| + 2*pi + acos|--|
          \pi/              \pi/

$$\operatorname{acos}{\left(\frac{3}{\pi} \right)} + \left(0 + \left(- \operatorname{acos}{\left(\frac{3}{\pi} \right)} + 2 \pi\right)\right)$$

=

2*pi

$$2 \pi$$

произведение

  /      /3 \       \     /3 \
1*|- acos|--| + 2*pi|*acos|--|
  \      \pi/       /     \pi/

$$1 \left(- \operatorname{acos}{\left(\frac{3}{\pi} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{3}{\pi} \right)}$$

=

/      /3 \       \     /3 \
|- acos|--| + 2*pi|*acos|--|
\      \pi/       /     \pi/

$$\left(- \operatorname{acos}{\left(\frac{3}{\pi} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{3}{\pi} \right)}$$

Численный ответ

x1 = 87.6632206907689

x2 = -68537.2867043247

x3 = -31.1145529261526

x4 = 49.9641088476914

x5 = -5856.23007990112

x6 = -18.5481823117935

x7 = -49.9641088476914

x8 = 37.3977382333322

x9 = -24.8313676189731

x10 = 88.2659679102595

x11 = -50.566856067182

x12 = 616.053533713345

x13 = -5.98181169743429

x14 = -63.1332266815412

x15 = 18.5481823117935

x16 = 43.6809235405118

x17 = -93.9464059979485

x18 = -25.4341148384636

x19 = 12.2649970046139

x20 = 5.98181169743429

x21 = -31.7173001456432

x22 = 75.6995972959003

x23 = -94.5491532174391

x24 = -100.229591305128

x25 = 12.8677442241045

x26 = 69.4164119887207

x27 = -75.0968500764098

x28 = 81.3800353835893

x29 = 50.566856067182

x30 = 81.9827826030799

x31 = -81.9827826030799

x32 = 100.832338524619

x33 = -81.3800353835893

x34 = -88.2659679102595

x35 = -100.832338524619

x36 = 31.7173001456432

x37 = 835.96501946463

x38 = -68.8136647692302

x39 = 56.247294154871

x40 = -56.247294154871

x41 = 62.5304794620506

x42 = 25.4341148384636

x43 = 0.301373609745291

x44 = 38.0004854528228

x45 = -56.8500413743616

x46 = -12.8677442241045

x47 = 75.0968500764098

x48 = -75.6995972959003

x49 = 6.58455891692488

x50 = 44.2836707600024

x51 = -43.6809235405118

x52 = 31.1145529261526

x53 = -62.5304794620506

x54 = -1960.05244223029

x55 = -0.301373609745291

x56 = -12.2649970046139

x57 = -44.2836707600024

x58 = -38.0004854528228

x59 = -87.6632206907689

x60 = -69.4164119887207

x61 = 93.9464059979485

x62 = -19.150929531284

x63 = 68.8136647692302

x64 = 24.8313676189731

x65 = -37.3977382333322

x66 = 94.5491532174391

x67 = 19.150929531284

x68 = -6.58455891692488

x69 = 100.229591305128

x70 = 63.1332266815412

x71 = 56.8500413743616

x72 = -301.291521134875

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