sin(x)+cos(2*x)=0

Дано

$$\sin{\left (x \right )} + \cos{\left (2 x \right )} = 0$$
Подробное решение
Дано уравнение
$$\sin{\left (x \right )} + \cos{\left (2 x \right )} = 0$$
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
$$x = 2 \pi n + {asin}{\left (w \right )}$$
$$x = 2 \pi n — {asin}{\left (w \right )} + \pi$$
Или
$$x = 2 \pi n + {asin}{\left (w \right )}$$
$$x = 2 \pi n — {asin}{\left (w \right )} + \pi$$
, где n — любое целое число
подставляем w:
$$x_{1} = 2 \pi n + {asin}{\left (w_{1} \right )}$$
$$x_{1} = 2 \pi n + {asin}{\left (- \frac{1}{2} \right )}$$
$$x_{1} = 2 \pi n — \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + {asin}{\left (w_{2} \right )}$$
$$x_{2} = 2 \pi n + {asin}{\left (1 \right )}$$
$$x_{2} = 2 \pi n + \frac{\pi}{2}$$
$$x_{3} = 2 \pi n — {asin}{\left (w_{1} \right )} + \pi$$
$$x_{3} = 2 \pi n — {asin}{\left (- \frac{1}{2} \right )} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n — {asin}{\left (w_{2} \right )} + \pi$$
$$x_{4} = 2 \pi n — {asin}{\left (1 \right )} + \pi$$
$$x_{4} = 2 \pi n + \frac{\pi}{2}$$

Ответ
Читайте также  e^(5*x)=0

pi
x1 = —
2

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

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

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

/ ___
|/ 3 I|
x3 = -I*log|—— — -|
2 2/

$$x_{3} = — i \log{\left (\frac{\sqrt{3}}{2} — \frac{i}{2} \right )}$$
Численный ответ

x1 = -10.9955748250000

x2 = 64.4026493092000

x3 = -67.5442421665000

x4 = -73.8274272801000

x5 = 41.3643032723000

x6 = -69.6386371546000

x7 = -21.4675497995000

x8 = -2.61799387799000

x9 = -71.7330322570000

x10 = 47.6474885794000

x11 = -44.5058959259000

x12 = 1.57079653523000

x13 = 76.9690198128000

x14 = 51.8362788989000

x15 = -0.523598775598000

x16 = 18.3259571459000

x17 = -13.0899693900000

x18 = -36.1283154199000

x19 = 26.7035373553000

x20 = -59.1666616426000

x21 = -31.9395253115000

x22 = -25.6563400043000

x23 = -75.9218224618000

x24 = 95.8185760577000

x25 = 7.85398173973000

x26 = 70.6858345098000

x27 = 5.75958653158000

x28 = -27.7507351067000

x29 = -4.71238877564000

x30 = 72.7802298082000

x31 = -40.3171057211000

x32 = 62.3082542962000

x33 = 22.5147473507000

x34 = 49.7418836818000

x35 = 58.1194644720000

x36 = -23.5619450082000

x37 = 45.5530936891000

x38 = -38.2227106187000

x39 = -80.1106125800000

x40 = 32.9867226700000

x41 = 66.4970445010000

x42 = 97.9129710369000

x43 = -61.2610569526000

x44 = 14.1371671029000

x45 = 12.0427718388000

x46 = -48.6946859199000

x47 = -42.4115006190000

x48 = -34.0339204139000

x49 = 89.5353908427000

x50 = -105.243352994000

x51 = -92.6769830654000

x52 = 16.2315620435000

x53 = 83.2522055084000

x54 = 100.007366139000

x55 = -17.2787597988000

x56 = -29.8451300967000

x57 = 9.94837673637000

x58 = -90.5825881785000

x59 = -46.6002910282000

x60 = -63.3554518474000

x61 = -78.0162175641000

x62 = 68.5914396034000

x63 = -54.9778716097000

x64 = -65.4498469498000

x65 = 56.0250689890000

x66 = -82.2050077689000

x67 = -57.0722665402000

x68 = 85.3466004225000

x69 = 30.8923277603000

x70 = -10.9955744709000

x71 = -54.9778719401000

x72 = 24.6091424531000

x73 = -98.9601689600000

x74 = 91.6297857297000

x75 = 74.8746249106000

x76 = -88.4881930761000

x77 = -98.9601687457000

x78 = -84.2994028713000

x79 = 39.2699083672000

x80 = 93.7241808321000

x81 = 20.4203521504000

x82 = 629.889327280000

x83 = 83.2522056281000

x84 = 53.9306738866000

x85 = -142.942465507000

x86 = 3.66519142919000

x87 = -86.3937977737000

x88 = 60.2138591938000

x89 = -19.3731546971000

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