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6*cos(x)^(2)+5*sin(x)-7=0

6*cos(x)^(2)+5*sin(x)-7=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
     2                      
6*cos (x) + 5*sin(x) - 7 = 0
(5sin⁡(x)+6cos⁡2(x))−7=0\left(5 \sin{\left(x \right)} + 6 \cos^{2}{\left(x \right)}\right) - 7 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(5sin⁡(x)+6cos⁡2(x))−7=0\left(5 \sin{\left(x \right)} + 6 \cos^{2}{\left(x \right)}\right) - 7 = 0
cambiamos
−6sin⁡2(x)+5sin⁡(x)−1=0- 6 \sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} - 1 = 0
−6sin⁡2(x)+5sin⁡(x)−1=0- 6 \sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} - 1 = 0
Sustituimos
w=sin⁡(x)w = \sin{\left(x \right)}
Es la ecuación de la forma
a*w^2 + b*w + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
w1=D−b2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=−D−b2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=−6a = -6
b=5b = 5
c=−1c = -1
, entonces
D = b^2 - 4 * a * c = 

(5)^2 - 4 * (-6) * (-1) = 1

Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)

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

o
w1=13w_{1} = \frac{1}{3}
w2=12w_{2} = \frac{1}{2}
hacemos cambio inverso
sin⁡(x)=w\sin{\left(x \right)} = w
Tenemos la ecuación
sin⁡(x)=w\sin{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
O
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
, donde n es cualquier número entero
sustituimos w:
x1=2πn+asin⁡(w1)x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}
x1=2πn+asin⁡(13)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{3} \right)}
x1=2πn+asin⁡(13)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{3} \right)}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(12)x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}
x2=2πn+π6x_{2} = 2 \pi n + \frac{\pi}{6}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn−asin⁡(13)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi
x3=2πn−asin⁡(13)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn−asin⁡(12)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi
x4=2πn+5π6x_{4} = 2 \pi n + \frac{5 \pi}{6}
Gráfica
0-80-60-40-2020406080-100100-2010
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
pi   5*pi         /        ___\         /        ___\
-- + ---- + 2*atan\3 - 2*\/ 2 / + 2*atan\3 + 2*\/ 2 /
6     6
2atan⁡(22+3)+(2atan⁡(3−22)+(π6+5π6))2 \operatorname{atan}{\left(2 \sqrt{2} + 3 \right)} + \left(2 \operatorname{atan}{\left(3 - 2 \sqrt{2} \right)} + \left(\frac{\pi}{6} + \frac{5 \pi}{6}\right)\right) 
=
           /        ___\         /        ___\
pi + 2*atan\3 - 2*\/ 2 / + 2*atan\3 + 2*\/ 2 /
2atan⁡(3−22)+2atan⁡(22+3)+π2 \operatorname{atan}{\left(3 - 2 \sqrt{2} \right)} + 2 \operatorname{atan}{\left(2 \sqrt{2} + 3 \right)} + \pi 
producto
pi 5*pi       /        ___\       /        ___\
--*----*2*atan\3 - 2*\/ 2 /*2*atan\3 + 2*\/ 2 /
6   6
π65π6⋅2atan⁡(3−22)2atan⁡(22+3)\frac{\pi}{6} \frac{5 \pi}{6} \cdot 2 \operatorname{atan}{\left(3 - 2 \sqrt{2} \right)} 2 \operatorname{atan}{\left(2 \sqrt{2} + 3 \right)} 
=
    2     /        ___\     /        ___\
5*pi *atan\3 - 2*\/ 2 /*atan\3 + 2*\/ 2 /
-----------------------------------------
                    9
5π2atan⁡(3−22)atan⁡(22+3)9\frac{5 \pi^{2} \operatorname{atan}{\left(3 - 2 \sqrt{2} \right)} \operatorname{atan}{\left(2 \sqrt{2} + 3 \right)}}{9} 
5*pi^2*atan(3 - 2*sqrt(2))*atan(3 + 2*sqrt(2))/9
Respuesta rápida [src] 
     pi
x1 = --
     6
x1=π6x_{1} = \frac{\pi}{6} 
     5*pi
x2 = ----
      6
x2=5π6x_{2} = \frac{5 \pi}{6} 
           /        ___\
x3 = 2*atan\3 - 2*\/ 2 /
x3=2atan⁡(3−22)x_{3} = 2 \operatorname{atan}{\left(3 - 2 \sqrt{2} \right)} 
           /        ___\
x4 = 2*atan\3 + 2*\/ 2 /
x4=2atan⁡(22+3)x_{4} = 2 \operatorname{atan}{\left(2 \sqrt{2} + 3 \right)} 
x4 = 2*atan(2*sqrt(2) + 3)
Respuesta numérica [src] 
x1 = -3.66519142918809
x2 = 38.2227106186758
x3 = 94.5876165171479
x4 = -68.5914396033772
x5 = -418.355421703041
x6 = -56.025068989018
x7 = 53.0672382015724
x8 = 25.6563400043166
x9 = -87.6247573910601
x10 = 82.0212459027887
x11 = 88.3044312099683
x12 = 19.1893928309929
x13 = -66.3132826348398
x14 = -100.007366139275
x15 = 34.2176822800336
x16 = 44.3221340597112
x17 = 27.934496972854
x18 = 90.7663500446499
x19 = 2.80175574413567
x20 = 90.5825881785057
x21 = 0.339836909454122
x22 = 56.8885046740704
x23 = -91.6297857297023
x24 = -41.1805414061214
x25 = 69.6386371545737
x26 = -60.0300973276602
x27 = 2.61799387799149
x28 = -85.3466004225227
x29 = -72.5964679420194
x30 = -53.7469120204806
x31 = -12.0427718387609
x32 = 82.2050077689329
x33 = -47.6474885794452
x34 = -62.4920161623417
x35 = 25.4725781381725
x36 = 75.9218224617533
x37 = -75.0583867767009
x38 = 63.17168998125
x39 = -49.7418836818384
x40 = -81.3415720838805
x41 = 27.7507351067098
x42 = -93.7241808320955
x43 = 40.5008675872132
x44 = 65.6336088159315
x45 = -66.497044500984
x46 = -24.7929043192642
x47 = 15.3681263584948
x48 = -16.0478001774031
x49 = -967067.381613495
x50 = -68.7752014695213
x51 = -12.2265337049051
x52 = 50.6053193668908
x53 = -18.5097190120846
x54 = 122.182276580548
x55 = -123.0457122656
x56 = 97.0495353518295
x57 = 46.6002910282486
x58 = 78.0162175641465
x59 = 100.870801824328
x60 = 71.9167941231111
x61 = -9.7646148702235
x62 = -34.8973560989418
x63 = -31.0760896264438
x64 = -78.879653249199
x65 = 31.9395253114962
x66 = 12.9062075238133
x67 = -22.3309854845827
x68 = -97.7292091707377
x69 = -5.75958653158129
x70 = -28.6141707917623
x71 = 9.08494105131526
x72 = 6.62302221663371
x73 = 69.4548752884296
x74 = -62.3082542961976
x75 = -18.3259571459405
x76 = -41.3643032722656
x77 = -43.642460240803
x78 = 59.350423508752
x79 = -85.1628385563785
x80 = 84.2994028713261
x81 = 46.7840528943928
x82 = 40.317105721069
x83 = 21.6513116656744
x84 = -37.3592749336234
x84 = -37.3592749336234
Gráfico
6*cos(x)^(2)+5*sin(x)-7=0 la ecuación
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