cos(x)=-п/4 (уравнение)

Найду корень уравнения: cos(x)=-п/4

Вы ввели

         -pi 
cos(x) = ----
          4  

$$\cos{\left(x \right)} = \frac{\left(-1\right) \pi}{4}$$

Подробное решение

Дано уравнение
$$\cos{\left(x \right)} = \frac{\left(-1\right) \pi}{4}$$
- 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

Быстрый ответ

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

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

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

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

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  /

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

Численный ответ

x1 = 84.155544430896

x2 = -91.7736441701324

x3 = 96.7219150452552

x4 = -72.9240882485936

x5 = 90.4387297380756

x6 = 10.0922351767978

x7 = -33.8900619734593

x8 = -35.2249764055161

x9 = 60.3577176342345

x10 = -146.987397502692

x11 = -77.8723591237164

x12 = 59.0228032021777

x13 = 72.9240882485936

x14 = 16.3754204839774

x15 = 71.5891738165369

x16 = 8.757320744741

x17 = -54.0745323270549

x18 = 85.4904588629528

x19 = -3.80904986961818

x20 = -66.640902941414

x21 = 41.5081617126957

x22 = 66.640902941414

x23 = -21.3236913591002

x24 = -10.0922351767978

x25 = 3.80904986961818

x26 = 21.3236913591002

x27 = -90.4387297380756

x28 = -28.9417910983365

x29 = 46.4564325878185

x30 = 79.2072735557732

x31 = 52.7396178949981

x32 = 28.9417910983365

x33 = -59.0228032021777

x34 = -2.47413543756141

x35 = -40.1732472806389

x36 = -41.5081617126957

x37 = 40.1732472806389

x38 = -47.7913470198753

x39 = 47.7913470198753

x40 = 77.8723591237164

x41 = 91.7736441701324

x42 = -27.6068766662798

x43 = -22.6586057911569

x44 = 98.056829477312

x45 = -84.155544430896

x46 = -52.7396178949981

x47 = -79.2072735557732

x48 = 22.6586057911569

x49 = 60264.504416597

x50 = -60.3577176342345

x51 = 15.0405060519206

x52 = -98.056829477312

x53 = 35.2249764055161

x54 = -65.3059885093573

x55 = 27.6068766662798

x56 = 2.47413543756141

x57 = -96.7219150452552

x58 = -85.4904588629528

x59 = -16.3754204839774

x60 = -71.5891738165369

x61 = 33.8900619734593

x62 = -46.4564325878185

x63 = -8.757320744741

x64 = 65.3059885093573

x65 = -15.0405060519206

x66 = 54.0745323270549

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