cos(x)=cos(2*x) (уравнение)

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

Вы ввели

[TeX]

[pretty]

[text]

cos(x) = cos(2*x)

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

Подробное решение

[TeX]

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

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

(1)^2 - 4 * (-2) * (1) = 9

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

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

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

или
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = 1$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left (w_{1} \right )}$$
$$x_{1} = \pi n + \operatorname{acos}{\left (- \frac{1}{2} \right )}$$
$$x_{1} = \pi n + \frac{2 \pi}{3}$$
$$x_{2} = \pi n + \operatorname{acos}{\left (w_{2} \right )}$$
$$x_{2} = \pi n + \operatorname{acos}{\left (1 \right )}$$
$$x_{2} = \pi n$$
$$x_{3} = \pi n + \operatorname{acos}{\left (w_{1} \right )} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left (- \frac{1}{2} \right )}$$
$$x_{3} = \pi n - \frac{\pi}{3}$$
$$x_{4} = \pi n + \operatorname{acos}{\left (w_{2} \right )} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left (1 \right )}$$
$$x_{4} = \pi n - \pi$$

Быстрый ответ

[TeX]

[pretty]

[text]

x1 = 0

$$x_{1} = 0$$

x2 = 2*pi

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

             /          ___\
             |  1   I*\/ 3 |
x3 = -2*I*log|- - - -------|
             \  2      2   /

$$x_{3} = - 2 i \log{\left (- \frac{1}{2} - \frac{\sqrt{3} i}{2} \right )}$$

             /          ___\
             |  1   I*\/ 3 |
x4 = -2*I*log|- - + -------|
             \  2      2   /

$$x_{4} = - 2 i \log{\left (- \frac{1}{2} + \frac{\sqrt{3} i}{2} \right )}$$

             /        ___\
             |1   I*\/ 3 |
x5 = -2*I*log|- - -------|
             \2      2   /

$$x_{5} = - 2 i \log{\left (\frac{1}{2} - \frac{\sqrt{3} i}{2} \right )}$$

             /        ___\
             |1   I*\/ 3 |
x6 = -2*I*log|- + -------|
             \2      2   /

$$x_{6} = - 2 i \log{\left (\frac{1}{2} + \frac{\sqrt{3} i}{2} \right )}$$

Численный ответ

[pretty]

[text]

x1 = 12.5663700883000

x2 = 50.2654824464000

x3 = -77.4926187885000

x4 = 48.1710873550000

x5 = 58.6430628670000

x6 = -25.1327414475000

x7 = 43.9822969706000

x8 = 14.6607657168000

x9 = -71.2094334814000

x10 = 25.1327412234000

x11 = 92.1533845053000

x12 = 2.09439510239000

x13 = 69.1150383780000

x14 = -2.09439510239000

x15 = -33.5103216383000

x16 = 39.7935069455000

x17 = 77.4926187885000

x18 = -62.8318529504000

x19 = 43.9822971694000

x20 = -92.1533845053000

x21 = -62.8318537995000

x22 = -18.8495555012000

x23 = -87.9645943588000

x24 = -90.0589894029000

x25 = 100.530964769000

x26 = -56.5486675394000

x27 = -4.18879020479000

x28 = 52.3598775598000

x29 = 41.8879020479000

x30 = -31.4159267013000

x31 = -81.6814090379000

x32 = 25.1327411126000

x33 = -75.3982238576000

x34 = 12.5663704552000

x35 = 79.5870138909000

x36 = 8.37758040957000

x37 = 35.6047167407000

x38 = 54.4542726622000

x39 = -27.2271363311000

x40 = 81.6814091713000

x41 = -52.3598775598000

x42 = -73.3038285838000

x43 = -46.0766922527000

x44 = -6.28318514162000

x45 = 83.7758040957000

x46 = 69.1150383296000

x47 = 0.0

x48 = -18.8495558711000

x49 = 90.0589894029000

x50 = -10.4719755120000

x51 = 75.3982239117000

x52 = 31.4159267619000

x53 = 62.8318528532000

x54 = -37.6991118771000

x55 = -41.8879020479000

x56 = 60.7374579694000

x57 = 69.1150378321000

x58 = -100.530964691000

x59 = 87.9645943357000

x60 = -54.4542726622000

x61 = 56.5486676120000

x62 = 85.8701991981000

x63 = 96.3421747101000

x64 = -98.4365698125000

x65 = 25.1327412731000

x66 = 18.8495557025000

x67 = 33.5103216383000

x68 = -69.1150390954000

x69 = -94.2477794557000

x70 = 10.4719755120000

x71 = 46.0766922527000

x72 = 98.4365698125000

x73 = 94.2477796094000

x74 = -8.37758040957000

x75 = -50.2654822985000

x76 = -18.8495558410000

x77 = -29.3215314335000

x78 = -43.9822971746000

x79 = -79.5870138909000

x80 = -18.8495558006000

x81 = -83.7758040957000

x82 = 6.28318528427000

x83 = -39.7935069455000

x84 = 25.1327417460000

x85 = -85.8701991981000

x86 = 4.18879020479000

x87 = -96.3421747101000

x88 = -48.1710873550000

x89 = -35.6047167407000

x90 = -62.8318529623000

x91 = -12.5663703885000

x92 = 37.6991120150000

x93 = 16.7551608191000

x94 = -69.1150385968000

x95 = 69.1150384283000