sin(x)=cos(2*x) (уравнение)

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

Вы ввели

sin(x) = cos(2*x)

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

Подробное решение

Дано уравнение
$$\sin{\left (x \right )} = \cos{\left (2 x \right )}$$
преобразуем
$$\sin{\left (x \right )} - \cos{\left (2 x \right )} = 0$$
$$2 \sin^{2}{\left (x \right )} + \sin{\left (x \right )} - 1 = 0$$
Сделаем замену
$$w = \sin{\left (x \right )}$$
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
$$\sin{\left (x \right )} = w$$
Дано уравнение
$$\sin{\left (x \right )} = w$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = 2 \pi n + \operatorname{asin}{\left (w \right )}$$
$$x = 2 \pi n - \operatorname{asin}{\left (w \right )} + \pi$$
Или
$$x = 2 \pi n + \operatorname{asin}{\left (w \right )}$$
$$x = 2 \pi n - \operatorname{asin}{\left (w \right )} + \pi$$
, где n - любое целое число
подставляем w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left (w_{1} \right )}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left (\frac{1}{2} \right )}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left (w_{2} \right )}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left (-1 \right )}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left (w_{1} \right )} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left (\frac{1}{2} \right )} + \pi$$
$$x_{3} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left (w_{2} \right )} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left (-1 \right )} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$

Быстрый ответ

     -pi 
x1 = ----
      2  

$$x_{1} = - \frac{\pi}{2}$$

           /      ___\
           |I   \/ 3 |
x2 = -I*log|- - -----|
           \2     2  /

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

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

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

Численный ответ

x1 = -100.007366139000

x2 = 71.7330322570000

x3 = 98.9601683848000

x4 = 92.6769826186000

x5 = 88.4881930761000

x6 = 23.5619451122000

x7 = -60.2138591938000

x8 = -43.4586983747000

x9 = 94.7713783833000

x10 = -85.3466004225000

x11 = -24.6091424531000

x12 = -12.0427718388000

x13 = 52.8834763354000

x14 = -26.7035373476000

x15 = -41.3643032723000

x16 = 31.9395253115000

x17 = 10.9955740993000

x18 = -76.9690198122000

x19 = -39.2699083757000

x20 = 25.6563400043000

x21 = -5.75958653158000

x22 = 86.3937978887000

x23 = 63.3554518474000

x24 = -47.6474885794000

x25 = 0.523598775598000

x26 = -14.1371668400000

x27 = 19.3731546971000

x28 = -7.85398149924000

x29 = -62.3082542962000

x30 = 92.6769830872000

x31 = -95.8185758682000

x32 = 46.6002910282000

x33 = 80.1106131458000

x34 = -68.5914396034000

x35 = 73.8274274783000

x36 = 29.8451303194000

x37 = -70.6858344925000

x38 = 61.2610569380000

x39 = 75.9218224618000

x40 = -16.2315620435000

x41 = -1.57079642893000

x42 = -45.5530935874000

x43 = -97.9129710369000

x44 = 78.0162175641000

x45 = -58.1194639999000

x46 = -51.8362786899000

x47 = 4.71238877821000

x48 = -66.4970445010000

x49 = 17.2787597960000

x50 = 2.61799387799000

x51 = 10.9955740562000

x52 = 84.2994028713000

x53 = -64.4026491963000

x54 = 69.6386371546000

x55 = -49.7418836818000

x56 = -93.7241808321000

x57 = -22.5147473507000

x58 = -79.0634151153000

x59 = -87.4409955249000

x60 = -89.5353907456000

x61 = -56.0250689890000

x62 = -18.3259571459000

x63 = -3.66519142919000

x64 = 34.0339204139000

x65 = 8.90117918517000

x66 = 67.5442422660000

x67 = -9.94837673637000

x68 = 82.2050077689000

x69 = -35.0811179651000

x70 = 44.5058959259000

x71 = -32.9867230406000

x72 = 42.4115007298000

x73 = 48.6946859325000

x74 = -83.2522055293000

x75 = 38.2227106187000

x76 = 40.3171057211000

x77 = -95.8185760435000

x78 = -53.9306738866000

x79 = -32.9867234123000

x80 = -20.4203520419000

x81 = -76.9690204512000

x82 = 90.5825881785000

x83 = -76.9690201781000

x84 = 36.1283159917000

x85 = 54.9778712412000

x86 = -91.6297857297000

x87 = 27.7507351067000

x88 = 96.8657734857000

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