6*sin(x)^(2)+7*cos(x)-7=0

$$6 sin^{2}{left (x right )} + 7 cos{left (x right )} – 7 = 0$$

Дано уравнение
$$6 sin^{2}{left (x right )} + 7 cos{left (x right )} – 7 = 0$$
преобразуем
$$- 6 cos^{2}{left (x right )} + 7 cos{left (x right )} – 1 = 0$$
$$- 6 cos^{2}{left (x right )} + 7 cos{left (x right )} – 1 = 0$$
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.
$$a = -6$$
$$b = 7$$
$$c = -1$$
, то

D = b^2 – 4 * a * c =

(7)^2 – 4 * (-6) * (-1) = 25

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

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

w2 = (-b – sqrt(D)) / (2*a)

или
$$w_{1} = frac{1}{6}$$
$$w_{2} = 1$$
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
$$x = pi n + {acos}{left (w right )}$$
$$x = pi n + {acos}{left (w right )} – pi$$
Или
$$x = pi n + {acos}{left (w right )}$$
$$x = pi n + {acos}{left (w right )} – pi$$
, где n – любое целое число
подставляем w:
$$x_{1} = pi n + {acos}{left (w_{1} right )}$$
$$x_{1} = pi n + {acos}{left (frac{1}{6} right )}$$
$$x_{1} = pi n + {acos}{left (frac{1}{6} right )}$$
$$x_{2} = pi n + {acos}{left (w_{2} right )}$$
$$x_{2} = pi n + {acos}{left (1 right )}$$
$$x_{2} = pi n$$
$$x_{3} = pi n + {acos}{left (w_{1} right )} – pi$$
$$x_{3} = pi n – pi + {acos}{left (frac{1}{6} right )}$$
$$x_{3} = pi n – pi + {acos}{left (frac{1}{6} right )}$$
$$x_{4} = pi n + {acos}{left (w_{2} right )} – pi$$
$$x_{4} = pi n – pi + {acos}{left (1 right )}$$
$$x_{4} = pi n – pi$$

$$x_{1} = 0$$

/ ____
|/ 35 |
x2 = -2*atan|——|
7 /

$$x_{2} = – 2 {atan}{left (frac{sqrt{35}}{7} right )}$$

/ ____
|/ 35 |
x3 = 2*atan|——|
7 /

$$x_{3} = 2 {atan}{left (frac{sqrt{35}}{7} right )}$$

x1 = 4.87983705960000

x2 = 50.2654824464000

x3 = 92.8444313601000

x4 = 37.6991120003000

x5 = 75.3982237430000

x6 = -6.28318515428000

x7 = 57.9520160122000

x8 = 12.5663704670000

x9 = 95.6511278553000

x10 = -7.68653355475000

x11 = -56.5486676368000

x12 = 83.0847572409000

x13 = 56.5486676220000

x14 = 26.5360894763000

x15 = -32.8192747835000

x16 = -30.0125782883000

x17 = 43.9822971694000

x18 = -57.9520160122000

x19 = 32.8192747835000

x20 = -36.2957635955000

x21 = 100.530964777000

x22 = -92.8444313601000

x23 = 45.3856453978000

x24 = 81.6814091543000

x25 = 18.8495558603000

x26 = 86.5612460529000

x27 = -48.8621342099000

x28 = 48.8621342099000

x29 = -56.5486672244000

x30 = -94.2477794649000

x31 = -95.6511278553000

x32 = 61.4285048242000

x33 = -19633.5507367000

x34 = 99.1276166673000

x35 = 64.2352013194000

x36 = -37.6991118770000

x37 = -39.1024600907000

x38 = -89.3679425481000

x39 = -76.8015719337000

x40 = 1.40334824758000

x41 = 87.9645943355000

x42 = 76.8015719337000

x43 = -20.2529041691000

x44 = 0.0

x45 = 43.9822969538000

x46 = -75.3982238414000

x47 = -81.6814090377000

x48 = 73.9948754386000

x49 = 80.2780607458000

x50 = -51.6688307050000

x51 = 13.9697188619000

x52 = -12.5663705035000

x53 = -25.1327412484000

x54 = 7.68653355475000

x55 = 36.2957635955000

x56 = -69.1150383396000

x57 = -23.7293929811000

x58 = -4.87983705960000

x59 = -87.9645943590000

x60 = -31.4159266875000

x61 = 20.2529041691000

x62 = -50.2654823093000

x63 = 62.8318529807000

x64 = 67.7116901314000

x65 = 42.5789489027000

x66 = 23.7293929811000

x67 = 17.4462076740000

x68 = 94.2477796094000

x69 = 31.4159266239000

x70 = -86.5612460529000

x71 = 30.0125782883000

x72 = -64.2352013194000

x73 = -43.9822971746000

x74 = 11.1630223668000

x75 = -75.3982236624000

x76 = -83.0847572409000

x77 = -13.9697188619000

x78 = 39.1024600907000

x79 = -100.530964774000

x80 = 55.1453195170000

x81 = -99.1276166673000

x82 = -11.1630223668000

x83 = 51.6688307050000

x84 = 89.3679425481000

x85 = -26.5360894763000

x86 = -67.7116901314000

x87 = -80.2780607458000

x88 = 70.5183866266000

x89 = -55.1453195170000

x90 = -1.40334824758000

x91 = -73.9948754386000

x92 = -42.5789489027000

x93 = 6.28318528431000

x94 = -45.3856453978000