cos(x)^(2)=cos(x) (уравнение)

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

Вы ввели

   2            
cos (x) = cos(x)

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

Подробное решение

Дано уравнение
$$\cos^{2}{\left (x \right )} = \cos{\left (x \right )}$$
преобразуем
$$\left(\cos{\left (x \right )} - 1\right) \cos{\left (x \right )} = 0$$
$$\cos^{2}{\left (x \right )} - \cos{\left (x \right )} = 0$$
Сделаем замену
$$w = \cos{\left (x \right )}$$
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
$$\cos{\left (x \right )} = w$$
Дано уравнение
$$\cos{\left (x \right )} = w$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = \pi n + \operatorname{acos}{\left (w \right )}$$
$$x = \pi n + \operatorname{acos}{\left (w \right )} - \pi$$
Или
$$x = \pi n + \operatorname{acos}{\left (w \right )}$$
$$x = \pi n + \operatorname{acos}{\left (w \right )} - \pi$$
, где n - любое целое число
подставляем w:
$$x_{1} = \pi n + \operatorname{acos}{\left (w_{1} \right )}$$
$$x_{1} = \pi n + \operatorname{acos}{\left (1 \right )}$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n + \operatorname{acos}{\left (w_{2} \right )}$$
$$x_{2} = \pi n + \operatorname{acos}{\left (0 \right )}$$
$$x_{2} = \pi n + \frac{\pi}{2}$$
$$x_{3} = \pi n + \operatorname{acos}{\left (w_{1} \right )} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left (1 \right )}$$
$$x_{3} = \pi n - \pi$$
$$x_{4} = \pi n + \operatorname{acos}{\left (w_{2} \right )} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left (0 \right )}$$
$$x_{4} = \pi n - \frac{\pi}{2}$$

Быстрый ответ

x1 = 0

$$x_{1} = 0$$

     pi
x2 = --
     2 

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

     3*pi
x3 = ----
      2  

$$x_{3} = \frac{3 \pi}{2}$$

x4 = 2*pi

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

Численный ответ

x1 = 39.2699081699000

x2 = -95.8185759345000

x3 = 50.2654824464000

x4 = 76.9690200129000

x5 = -32.9867228627000

x6 = -75.3982238479000

x7 = 37.6991120060000

x8 = 36.1283155163000

x9 = -92.6769832809000

x10 = 80.1106126665000

x11 = -81.6814090378000

x12 = 10.9955742876000

x13 = 23.5619449019000

x14 = 51.8362787842000

x15 = -12.5663704470000

x16 = -94.2477794613000

x17 = 14.1371669412000

x18 = 43.9822971694000

x19 = -69.1150384972000

x20 = 12.5663704623000

x21 = -7.85398163397000

x22 = -76.9690200129000

x23 = 89.5353906273000

x24 = -23.5619449019000

x25 = 48.6946861306000

x26 = 29.8451302091000

x27 = 42.4115008235000

x28 = -54.9778714378000

x29 = -6.28318514935000

x30 = -56.5486675908000

x31 = 45.5530934771000

x32 = -70.6858347058000

x33 = 81.6814091609000

x34 = -50.2654823051000

x35 = -87.9645943589000

x36 = 100.530964774000

x37 = -80.1106126665000

x38 = -58.1194640914000

x39 = 75.3982238342000

x40 = 18.8495557729000

x41 = 32.9867228627000

x42 = 64.4026493986000

x43 = -26.7035375555000

x44 = -83.2522053201000

x45 = 26.7035375555000

x46 = 69.1150379837000

x47 = -37.6991118770000

x48 = 62.8318529132000

x49 = -67.5442420522000

x50 = -86.3937979737000

x51 = 25.1327418086000

x52 = -1.57079632679000

x53 = 0.0

x54 = -389.557489135000

x55 = 87.9645943356000

x56 = 86.3937979737000

x57 = -17.2787595947000

x58 = -10.9955742876000

x59 = -48.6946861306000

x60 = -45.5530934771000

x61 = 4.71238898038000

x62 = 7.85398163397000

x63 = -39.2699081699000

x64 = -73.8274273594000

x65 = -14.1371669412000

x66 = -98.9601685881000

x67 = 98.9601685881000

x68 = -100.530964736000

x69 = 25.1327408584000

x70 = 20.4203522483000

x71 = -29.8451302091000

x72 = 94.2477796094000

x73 = 61.2610567450000

x74 = 56.5486676180000

x75 = -89.5353906273000

x76 = -4.71238898038000

x77 = -43.9822971746000

x78 = 17.2787595947000

x79 = -42.4115008235000

x80 = 58.1194640914000

x81 = 31.4159266949000

x82 = 95.8185759345000

x83 = -36.1283155163000

x84 = 73.8274273594000

x85 = -18.8495562410000

x86 = 83.2522053201000

x87 = -31.4159266930000

x88 = -62.8318534973000

x89 = 54.9778714378000

x90 = -51.8362787842000

x91 = 92.6769832809000

x92 = -20.4203522483000

x93 = -25.1327413642000

x94 = 6.28318528430000

x95 = 70.6858347058000

x96 = 69.1150385134000

x97 = 1.57079632679000

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