tan(cos(x))^(2)=1 (уравнение)

Найду корень уравнения: tan(cos(x))^(2)=1

Вы ввели

   2            
tan (cos(x)) = 1

$$\tan^{2}{\left(\cos{\left(x \right)} \right)} = 1$$

Подробное решение

Дано уравнение
$$\tan^{2}{\left(\cos{\left(x \right)} \right)} = 1$$
преобразуем
$$\tan^{2}{\left(\cos{\left(x \right)} \right)} - 1 = 0$$
$$\tan^{2}{\left(\cos{\left(x \right)} \right)} - 1 = 0$$
Сделаем замену
$$w = \tan{\left(\cos{\left(x \right)} \right)}$$
Это уравнение вида

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

Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
где D = b^2 - 4*a*c - это дискриминант.
Т.к.
$$a = 1$$
$$b = 0$$
$$c = -1$$
, то

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

(0)^2 - 4 * (1) * (-1) = 4

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

w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

или
$$w_{1} = 1$$
Упростить
$$w_{2} = -1$$
Упростить
делаем обратную замену
$$\tan{\left(\cos{\left(x \right)} \right)} = w$$
подставляем w:

Быстрый ответ

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

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

           /pi\       
x2 = - acos|--| + 2*pi
           \4 /       

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

         /-pi \
x3 = acos|----|
         \ 4  /

$$x_{3} = \operatorname{acos}{\left(- \frac{\pi}{4} \right)}$$

         /pi\
x4 = acos|--|
         \4 /

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

Сумма и произведение корней

сумма

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

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

=

4*pi

$$4 \pi$$

произведение

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

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

=

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

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

Численный ответ

x1 = -46.4564325878185

x2 = 85.4904588629528

x3 = -68.4475811629471

x4 = -2.47413543756141

x5 = -11.8989133983308

x6 = 98.056829477312

x7 = -90.4387297380756

x8 = 162.695360770641

x9 = 54.0745323270549

x10 = 5.6157280911512

x11 = -16.3754204839774

x12 = 93.5803223916654

x13 = -60.3577176342345

x14 = -65.3059885093573

x15 = -49.5980252414083

x16 = 49.5980252414083

x17 = -41.5081617126957

x18 = -10.0922351767978

x19 = 76.0656809021834

x20 = 22.6586057911569

x21 = 96.7219150452552

x22 = 55.8812105485879

x23 = -98.056829477312

x24 = -62.1643958557675

x25 = -99.863507698845

x26 = 47.7913470198753

x27 = -32.0833837519263

x28 = -120.047978052441

x29 = -38.3665690591059

x30 = -85.4904588629528

x31 = 84.155544430896

x32 = -87.2971370844858

x33 = 91.7736441701324

x34 = 71.5891738165369

x35 = 25.8001984447467

x36 = 69.7824955950038

x37 = -91.7736441701324

x38 = -5.6157280911512

x39 = 88.6320515165426

x40 = 19.5170131375671

x41 = -21.3236913591002

x42 = -27.6068766662798

x43 = 8.757320744741

x44 = -82.348866209363

x45 = 33.8900619734593

x46 = 74.7307664701267

x47 = 41.5081617126957

x48 = 62.1643958557675

x49 = -43.3148399342287

x50 = 24.46528401269

x51 = -24.46528401269

x52 = -77.8723591237164

x53 = -167.171867856287

x54 = 60.3577176342345

x55 = 40.1732472806389

x56 = 0.667457216028384

x57 = -54.0745323270549

x58 = -33.8900619734593

x59 = 27.6068766662798

x60 = 18.1820987055104

x61 = 90.4387297380756

x62 = -3.80904986961818

x63 = 30.7484693198695

x64 = -63.4993102878243

x65 = -76.0656809021834

x66 = 11.8989133983308

x67 = -25.8001984447467

x68 = 99.863507698845

x69 = -69.7824955950038

x70 = 16.3754204839774

x71 = -55.8812105485879

x72 = 38.3665690591059

x73 = 77.8723591237164

x74 = 3.80904986961818

x75 = 68.4475811629471

x76 = 2.47413543756141

x77 = -40.1732472806389

x78 = 44.6497543662855

x79 = 82.348866209363

x80 = -71.5891738165369

x81 = 52.7396178949981

x82 = -84.155544430896

x83 = -18.1820987055104

x84 = -93.5803223916654

x85 = -19.5170131375671

x86 = 32.0833837519263

x87 = 66.640902941414

x88 = -47.7913470198753

x89 = 46.4564325878185

x90 = 63.4993102878243

x91 = 10.0922351767978

График