tan2(x)=tan(x) (уравнение)

Найду корень уравнения: tan2(x)=tan(x)

Вы ввели

   2            
tan (x) = tan(x)

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

Подробное решение

Дано уравнение
$$\tan^{2}{\left(x \right)} = \tan{\left(x \right)}$$
преобразуем
$$\left(\tan{\left(x \right)} - 1\right) \tan{\left(x \right)} = 0$$
$$\tan^{2}{\left(x \right)} - \tan{\left(x \right)} = 0$$
Сделаем замену
$$w = \tan{\left(x \right)}$$
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.
$$a = 1$$
$$b = -1$$
$$c = 0$$
, то

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

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

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

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

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

или
$$w_{1} = 1$$
$$w_{2} = 0$$
делаем обратную замену
$$\tan{\left(x \right)} = w$$
Дано уравнение
$$\tan{\left(x \right)} = w$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
Или
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, где n - любое целое число
подставляем w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(1 \right)}$$
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(0 \right)}$$
$$x_{2} = \pi n$$

Быстрый ответ

x1 = 0

$$x_{1} = 0$$

     pi
x2 = --
     4 

$$x_{2} = \frac{\pi}{4}$$

Численный ответ

x1 = -75.398223686155

x2 = 16.4933614313464

x3 = 34.5575191894877

x4 = -84.8230016469244

x5 = -68.329640215578

x6 = 25.1327412287183

x7 = -94.2477796076938

x8 = 18.8495559215388

x9 = -31.4159265358979

x10 = -84.037603483527

x11 = -99.7455667514759

x12 = -50.2654824574367

x13 = 50.2654824574367

x14 = -25.1327412287183

x15 = -55.7632696012188

x16 = 3.14159265358979

x17 = -34.5575191894877

x18 = 85.6083998103219

x19 = -24.3473430653209

x20 = -43.9822971502571

x21 = 31.4159265358979

x22 = 94.2477796076938

x23 = -74.6128255227576

x24 = 40.8407044966673

x25 = -91.106186954104

x26 = 63.6172512351933

x27 = -28.2743338823081

x28 = 65.9734457253857

x29 = -21.9911485751286

x30 = -62.8318530717959

x31 = -15.707963267949

x32 = -69.1150383789755

x33 = -72.2566310325652

x34 = -53.4070751110265

x35 = 84.8230016469244

x36 = 98.174770424681

x37 = -56.5486677646163

x38 = -2.35619449019234

x39 = 21.9911485751286

x40 = -3.14159265358979

x41 = -77.7544181763474

x42 = 91.8915851175014

x43 = -97.3893722612836

x44 = -37.6991118430775

x45 = -9.42477796076938

x46 = 41.6261026600648

x47 = -81.6814089933346

x48 = -87.9645943005142

x49 = 13.3517687777566

x50 = 72.2566310325652

x51 = 19.6349540849362

x52 = -96.6039740978861

x53 = -18.8495559215388

x54 = -52.621676947629

x55 = 91.106186954104

x56 = -46.3384916404494

x57 = 6.28318530717959

x58 = 75.398223686155

x59 = -12.5663706143592

x60 = 56.5486677646163

x61 = 54.1924732744239

x62 = 47.9092879672443

x63 = -90.3207887907066

x64 = 47.1238898038469

x65 = 69.1150383789755

x66 = 35.3429173528852

x67 = -40.8407044966673

x68 = -62.0464549083984

x69 = 78.5398163397448

x70 = -59.6902604182061

x71 = 25.9181393921158

x72 = 53.4070751110265

x73 = 100.530964914873

x74 = -8.63937979737193

x75 = 28.2743338823081

x76 = 9.42477796076938

x77 = 43.9822971502571

x78 = 81.6814089933346

x79 = -11.7809724509617

x80 = 37.6991118430775

x81 = 76.1836218495525

x82 = 10.2101761241668

x83 = 62.8318530717959

x84 = -30.6305283725005

x85 = 12.5663706143592

x86 = 0.0

x87 = -33.7721210260903

x88 = -65.9734457253857

x89 = -78.5398163397448

x90 = -21.2057504117311

x91 = -40.0553063332699

x92 = 32.2013246992954

x93 = 97.3893722612836

x94 = 3.92699081698724

x95 = -47.1238898038469

x96 = 69.9004365423729

x97 = 87.9645943005142

x98 = -18.0641577581413

График