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-5*cos(x)-10*sqrt(3)*sin(x)^2=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
                 ___    2       
-5*cos(x) - 10*\/ 3 *sin (x) = 0
−103sin⁡2(x)−5cos⁡(x)=0- 10 \sqrt{3} \sin^{2}{\left(x \right)} - 5 \cos{\left(x \right)} = 0
Simplificar
Solución detallada
Tenemos la ecuación
−103sin⁡2(x)−5cos⁡(x)=0- 10 \sqrt{3} \sin^{2}{\left(x \right)} - 5 \cos{\left(x \right)} = 0
cambiamos
−103sin⁡2(x)−5cos⁡(x)=0- 10 \sqrt{3} \sin^{2}{\left(x \right)} - 5 \cos{\left(x \right)} = 0
−103(1−cos⁡2(x))−5cos⁡(x)=0- 10 \sqrt{3} \left(1 - \cos^{2}{\left(x \right)}\right) - 5 \cos{\left(x \right)} = 0
Sustituimos
w=cos⁡(x)w = \cos{\left(x \right)}
Abramos la expresión en la ecuación
−5w−103(1−w2)=0- 5 w - 10 \sqrt{3} \left(1 - w^{2}\right) = 0
Obtenemos la ecuación cuadrática
103w2−5w−103=010 \sqrt{3} w^{2} - 5 w - 10 \sqrt{3} = 0
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=103a = 10 \sqrt{3}
b=−5b = -5
c=−103c = - 10 \sqrt{3}
, entonces
D = b^2 - 4 * a * c = 

(-5)^2 - 4 * (10*sqrt(3)) * (-10*sqrt(3)) = 1225

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

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

o
w1=233w_{1} = \frac{2 \sqrt{3}}{3}
w2=−32w_{2} = - \frac{\sqrt{3}}{2}
hacemos cambio inverso
cos⁡(x)=w\cos{\left(x \right)} = w
Tenemos la ecuación
cos⁡(x)=w\cos{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
O
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
, donde n es cualquier número entero
sustituimos w:
x1=πn+acos⁡(w1)x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}
x1=πn+acos⁡(233)x_{1} = \pi n + \operatorname{acos}{\left(\frac{2 \sqrt{3}}{3} \right)}
x1=πn+acos⁡(233)x_{1} = \pi n + \operatorname{acos}{\left(\frac{2 \sqrt{3}}{3} \right)}
x2=πn+acos⁡(w2)x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}
x2=πn+acos⁡(−32)x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}
x2=πn+5π6x_{2} = \pi n + \frac{5 \pi}{6}
x3=πn+acos⁡(w1)−πx_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x3=πn−π+acos⁡(233)x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{2 \sqrt{3}}{3} \right)}
x3=πn−π+acos⁡(233)x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{2 \sqrt{3}}{3} \right)}
x4=πn+acos⁡(w2)−πx_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi
x4=πn−π+acos⁡(−32)x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}
x4=πn−π6x_{4} = \pi n - \frac{\pi}{6}
Gráfica
0-80-60-40-2020406080-100100-2525
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
            /   _____________\
            |  /         ___ |
x1 = -2*atan\\/  7 + 4*\/ 3  /
x1=−2atan⁡(43+7)x_{1} = - 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)} 
           /   _____________\
           |  /         ___ |
x2 = 2*atan\\/  7 + 4*\/ 3  /
x2=2atan⁡(43+7)x_{2} = 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)} 
               /   _____________\
               |  /         ___ |
x3 = -2*I*atanh\\/  7 - 4*\/ 3  /
x3=−2iatanh⁡(7−43)x_{3} = - 2 i \operatorname{atanh}{\left(\sqrt{7 - 4 \sqrt{3}} \right)} 
              /   _____________\
              |  /         ___ |
x4 = 2*I*atanh\\/  7 - 4*\/ 3  /
x4=2iatanh⁡(7−43)x_{4} = 2 i \operatorname{atanh}{\left(\sqrt{7 - 4 \sqrt{3}} \right)} 
x4 = 2*i*atanh(sqrt(7 - 4*sqrt(3)))
Suma y producto de raíces [src] 
suma
        /   _____________\         /   _____________\            /   _____________\            /   _____________\
        |  /         ___ |         |  /         ___ |            |  /         ___ |            |  /         ___ |
- 2*atan\\/  7 + 4*\/ 3  / + 2*atan\\/  7 + 4*\/ 3  / - 2*I*atanh\\/  7 - 4*\/ 3  / + 2*I*atanh\\/  7 - 4*\/ 3  /
((−2atan⁡(43+7)+2atan⁡(43+7))−2iatanh⁡(7−43))+2iatanh⁡(7−43)\left(\left(- 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)} + 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)}\right) - 2 i \operatorname{atanh}{\left(\sqrt{7 - 4 \sqrt{3}} \right)}\right) + 2 i \operatorname{atanh}{\left(\sqrt{7 - 4 \sqrt{3}} \right)} 
=
0
00 
producto
       /   _____________\       /   _____________\           /   _____________\          /   _____________\
       |  /         ___ |       |  /         ___ |           |  /         ___ |          |  /         ___ |
-2*atan\\/  7 + 4*\/ 3  /*2*atan\\/  7 + 4*\/ 3  /*-2*I*atanh\\/  7 - 4*\/ 3  /*2*I*atanh\\/  7 - 4*\/ 3  /
2iatanh⁡(7−43)−2iatanh⁡(7−43)−2atan⁡(43+7)2atan⁡(43+7)2 i \operatorname{atanh}{\left(\sqrt{7 - 4 \sqrt{3}} \right)} - 2 i \operatorname{atanh}{\left(\sqrt{7 - 4 \sqrt{3}} \right)} - 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)} 2 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)} 
=
         /   _____________\       /   _____________\
        2|  /         ___ |      2|  /         ___ |
-16*atan \\/  7 + 4*\/ 3  /*atanh \\/  7 - 4*\/ 3  /
−16atan⁡2(43+7)atanh⁡2(7−43)- 16 \operatorname{atan}^{2}{\left(\sqrt{4 \sqrt{3} + 7} \right)} \operatorname{atanh}^{2}{\left(\sqrt{7 - 4 \sqrt{3}} \right)} 
-16*atan(sqrt(7 + 4*sqrt(3)))^2*atanh(sqrt(7 - 4*sqrt(3)))^2
Respuesta numérica [src] 
x1 = 27.7507351067098
x2 = -91.6297857297023
x3 = 9.94837673636768
x4 = 78.0162175641465
x5 = -60.2138591938044
x6 = 60.2138591938044
x7 = 2.61799387799149
x8 = -3.66519142918809
x9 = 90.5825881785057
x10 = -52.8834763354282
x11 = 85.3466004225227
x12 = -47.6474885794452
x13 = -16.2315620435473
x14 = -84.2994028713261
x15 = 40.317105721069
x16 = 52.8834763354282
x17 = 84.2994028713261
x18 = -90.5825881785057
x19 = 66.497044500984
x20 = 34.0339204138894
x21 = 46.6002910282486
x22 = 22.5147473507269
x23 = -46.6002910282486
x24 = -79.0634151153431
x25 = -78.0162175641465
x26 = -9.94837673636768
x27 = -65.4498469497874
x28 = 71.733032256967
x29 = 140.848070635942
x30 = -34.0339204138894
x31 = 91.6297857297023
x32 = -97.9129710368819
x33 = -2.61799387799149
x34 = -35.081117965086
x35 = -40.317105721069
x36 = 97.9129710368819
x37 = -96.8657734856853
x38 = -53.9306738866248
x39 = 8.90117918517108
x40 = -85.3466004225227
x41 = 53.9306738866248
x42 = 3.66519142918809
x43 = -71.733032256967
x44 = -41.3643032722656
x45 = 21.4675497995303
x46 = 47.6474885794452
x47 = -72.7802298081635
x48 = -27.7507351067098
x49 = 96.8657734856853
x50 = -22.5147473507269
x51 = 41.3643032722656
x52 = -21.4675497995303
x53 = 16.2315620435473
x53 = 16.2315620435473
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