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2sin^2x=3cos la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
     2              
2*sin (x) = 3*cos(x)
2sin⁡2(x)=3cos⁡(x)2 \sin^{2}{\left(x \right)} = 3 \cos{\left(x \right)}
Simplificar
Solución detallada
Tenemos la ecuación
2sin⁡2(x)=3cos⁡(x)2 \sin^{2}{\left(x \right)} = 3 \cos{\left(x \right)}
cambiamos
−3cos⁡(x)−cos⁡(2x)+1=0- 3 \cos{\left(x \right)} - \cos{\left(2 x \right)} + 1 = 0
−2cos⁡2(x)−3cos⁡(x)+2=0- 2 \cos^{2}{\left(x \right)} - 3 \cos{\left(x \right)} + 2 = 0
Sustituimos
w=cos⁡(x)w = \cos{\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=−2a = -2
b=−3b = -3
c=2c = 2
, entonces
D = b^2 - 4 * a * c = 

(-3)^2 - 4 * (-2) * (2) = 25

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

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

o
w1=−2w_{1} = -2
w2=12w_{2} = \frac{1}{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⁡(−2)x_{1} = \pi n + \operatorname{acos}{\left(-2 \right)}
x1=πn+acos⁡(−2)x_{1} = \pi n + \operatorname{acos}{\left(-2 \right)}
x2=πn+acos⁡(w2)x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}
x2=πn+acos⁡(12)x_{2} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}
x2=πn+π3x_{2} = \pi n + \frac{\pi}{3}
x3=πn+acos⁡(w1)−πx_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x3=πn−π+acos⁡(−2)x_{3} = \pi n - \pi + \operatorname{acos}{\left(-2 \right)}
x3=πn−π+acos⁡(−2)x_{3} = \pi n - \pi + \operatorname{acos}{\left(-2 \right)}
x4=πn+acos⁡(w2)−πx_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi
x4=πn−π+acos⁡(12)x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}
x4=πn−2π3x_{4} = \pi n - \frac{2 \pi}{3}
Gráfica
0-80-60-40-2020406080-1001005-5
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
     -pi 
x1 = ----
      3
x1=−π3x_{1} = - \frac{\pi}{3} 
     pi
x2 = --
     3
x2=π3x_{2} = \frac{\pi}{3} 
         /     /  ___\\         /     /  ___\\
x3 = 2*im\atanh\\/ 3 // - 2*I*re\atanh\\/ 3 //
x3=2im⁡(atanh⁡(3))−2ire⁡(atanh⁡(3))x_{3} = 2 \operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)} - 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)} 
           /     /  ___\\         /     /  ___\\
x4 = - 2*im\atanh\\/ 3 // + 2*I*re\atanh\\/ 3 //
x4=−2im⁡(atanh⁡(3))+2ire⁡(atanh⁡(3))x_{4} = - 2 \operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)} + 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)} 
x4 = -2*im(atanh(sqrt(3))) + 2*i*re(atanh(sqrt(3)))
Suma y producto de raíces [src] 
suma
  pi   pi       /     /  ___\\         /     /  ___\\         /     /  ___\\         /     /  ___\\
- -- + -- + 2*im\atanh\\/ 3 // - 2*I*re\atanh\\/ 3 // + - 2*im\atanh\\/ 3 // + 2*I*re\atanh\\/ 3 //
  3    3
((−π3+π3)+(2im⁡(atanh⁡(3))−2ire⁡(atanh⁡(3))))+(−2im⁡(atanh⁡(3))+2ire⁡(atanh⁡(3)))\left(\left(- \frac{\pi}{3} + \frac{\pi}{3}\right) + \left(2 \operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)} - 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)}\right)\right) + \left(- 2 \operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)} + 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)}\right) 
=
0
00 
producto
-pi  pi /    /     /  ___\\         /     /  ___\\\ /      /     /  ___\\         /     /  ___\\\
----*--*\2*im\atanh\\/ 3 // - 2*I*re\atanh\\/ 3 ///*\- 2*im\atanh\\/ 3 // + 2*I*re\atanh\\/ 3 ///
 3   3
−π3π3(2im⁡(atanh⁡(3))−2ire⁡(atanh⁡(3)))(−2im⁡(atanh⁡(3))+2ire⁡(atanh⁡(3)))- \frac{\pi}{3} \frac{\pi}{3} \left(2 \operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)} - 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)}\right) \left(- 2 \operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)} + 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)}\right) 
=
                                               2
    2 /      /     /  ___\\     /     /  ___\\\ 
4*pi *\- I*re\atanh\\/ 3 // + im\atanh\\/ 3 /// 
------------------------------------------------
                       9
4π2(im⁡(atanh⁡(3))−ire⁡(atanh⁡(3)))29\frac{4 \pi^{2} \left(\operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)} - i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{3} \right)}\right)}\right)^{2}}{9} 
4*pi^2*(-i*re(atanh(sqrt(3))) + im(atanh(sqrt(3))))^2/9
Respuesta numérica [src] 
x1 = -82.7286065445312
x2 = -38.7463093942741
x3 = 57.5958653158129
x4 = -38345.2327321658
x5 = -80.634211442138
x6 = 1691.22404518251
x7 = 95.2949771588904
x8 = -61.7846555205993
x9 = -45.0294947014537
x10 = 70.162235930172
x11 = -74.3510261349584
x12 = -55.5014702134197
x13 = -93.2005820564972
x14 = -7.33038285837618
x15 = -13.6135681655558
x16 = -36.6519142918809
x17 = -68.0678408277789
x18 = 218.864288200089
x19 = 36.6519142918809
x20 = 86.9173967493176
x21 = -89.0117918517108
x22 = 32.4631240870945
x23 = 55.5014702134197
x24 = 13.6135681655558
x25 = 76.4454212373516
x26 = -114.144533080429
x27 = -76.4454212373516
x28 = -19.8967534727354
x29 = 80.634211442138
x30 = -63.8790506229925
x31 = 7.33038285837618
x32 = 89.0117918517108
x33 = 24.0855436775217
x34 = 5.23598775598299
x35 = 30.3687289847013
x36 = 82.7286065445312
x37 = 42.9350995990605
x38 = 38.7463093942741
x39 = -24.0855436775217
x40 = -99.4837673636768
x41 = -23317.9478724946
x42 = -1.0471975511966
x43 = 74.3510261349584
x44 = 26.1799387799149
x45 = -935.147413218562
x46 = -95.2949771588904
x47 = 51.3126800086333
x48 = -382.227106186758
x49 = 17.8023583703422
x50 = 63.8790506229925
x51 = 11.5191730631626
x52 = -51.3126800086333
x53 = -11.5191730631626
x54 = -26.1799387799149
x55 = -17.8023583703422
x56 = -32.4631240870945
x57 = -5.23598775598299
x58 = 105.766952670856
x59 = 99.4837673636768
x60 = -57.5958653158129
x61 = 68.0678408277789
x62 = -30.3687289847013
x63 = 61.7846555205993
x64 = 19.8967534727354
x65 = -16783.4351530279
x66 = -49.2182849062401
x67 = -70.162235930172
x67 = -70.162235930172
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