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cos^2x+3cosx=0 la ecuación

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Solución numérica:

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Solución

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

(3)^2 - 4 * (1) * (0) = 9

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

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

o
w1=0w_{1} = 0
w2=−3w_{2} = -3
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⁡(0)x_{1} = \pi n + \operatorname{acos}{\left(0 \right)}
x1=πn+π2x_{1} = \pi n + \frac{\pi}{2}
x2=πn+acos⁡(w2)x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}
x2=πn+acos⁡(−3)x_{2} = \pi n + \operatorname{acos}{\left(-3 \right)}
x2=πn+acos⁡(−3)x_{2} = \pi n + \operatorname{acos}{\left(-3 \right)}
x3=πn+acos⁡(w1)−πx_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x3=πn−π+acos⁡(0)x_{3} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}
x3=πn−π2x_{3} = \pi n - \frac{\pi}{2}
x4=πn+acos⁡(w2)−πx_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi
x4=πn−π+acos⁡(−3)x_{4} = \pi n - \pi + \operatorname{acos}{\left(-3 \right)}
x4=πn−π+acos⁡(−3)x_{4} = \pi n - \pi + \operatorname{acos}{\left(-3 \right)}
Gráfica
0-80-60-40-2020406080-1001005-5
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
pi   3*pi                                                                        
-- + ---- + -re(acos(-3)) + 2*pi - I*im(acos(-3)) + I*im(acos(-3)) + re(acos(-3))
2     2
(re⁡(acos⁡(−3))+iim⁡(acos⁡(−3)))+((π2+3π2)+(−re⁡(acos⁡(−3))+2π−iim⁡(acos⁡(−3))))\left(\operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}\right) + \left(\left(\frac{\pi}{2} + \frac{3 \pi}{2}\right) + \left(- \operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}\right)\right) 
=
4*pi
4π4 \pi 
producto
pi 3*pi                                                                        
--*----*(-re(acos(-3)) + 2*pi - I*im(acos(-3)))*(I*im(acos(-3)) + re(acos(-3)))
2   2
π23π2(−re⁡(acos⁡(−3))+2π−iim⁡(acos⁡(−3)))(re⁡(acos⁡(−3))+iim⁡(acos⁡(−3)))\frac{\pi}{2} \frac{3 \pi}{2} \left(- \operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}\right) 
=
     2                                                                        
-3*pi *(I*im(acos(-3)) + re(acos(-3)))*(-2*pi + I*im(acos(-3)) + re(acos(-3)))
------------------------------------------------------------------------------
                                      4
−3π2(re⁡(acos⁡(−3))+iim⁡(acos⁡(−3)))(−2π+re⁡(acos⁡(−3))+iim⁡(acos⁡(−3)))4- \frac{3 \pi^{2} \left(\operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}\right)}{4} 
-3*pi^2*(i*im(acos(-3)) + re(acos(-3)))*(-2*pi + i*im(acos(-3)) + re(acos(-3)))/4
Respuesta rápida [src] 
     pi
x1 = --
     2
x1=π2x_{1} = \frac{\pi}{2} 
     3*pi
x2 = ----
      2
x2=3π2x_{2} = \frac{3 \pi}{2} 
x3 = -re(acos(-3)) + 2*pi - I*im(acos(-3))
x3=−re⁡(acos⁡(−3))+2π−iim⁡(acos⁡(−3))x_{3} = - \operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)} 
x4 = I*im(acos(-3)) + re(acos(-3))
x4=re⁡(acos⁡(−3))+iim⁡(acos⁡(−3))x_{4} = \operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)} 
x4 = re(acos(-3)) + i*im(acos(-3))
Respuesta numérica [src] 
x1 = 4.71238898038469
x2 = 17.2787595947439
x3 = -89.5353906273091
x4 = 64.4026493985908
x5 = 70.6858347057703
x6 = 36.1283155162826
x7 = -98.9601685880785
x8 = 48.6946861306418
x9 = -58.1194640914112
x10 = 7.85398163397448
x11 = 39.2699081698724
x12 = -95.8185759344887
x13 = -1.5707963267949
x14 = -92.6769832808989
x15 = -23.5619449019235
x16 = 23.5619449019235
x17 = 61.261056745001
x18 = 10224.313291108
x19 = 29.845130209103
x20 = -32.9867228626928
x21 = -51.8362787842316
x22 = -80.1106126665397
x23 = -83.2522053201295
x24 = 67.5442420521806
x25 = 98.9601685880785
x26 = 92.6769832808989
x27 = -39.2699081698724
x28 = 86.3937979737193
x29 = 45.553093477052
x30 = -67.5442420521806
x31 = 51.8362787842316
x32 = 76.9690200129499
x33 = -26.7035375555132
x34 = -4.71238898038469
x35 = 95.8185759344887
x36 = -86.3937979737193
x37 = -10.9955742875643
x38 = 83.2522053201295
x39 = -7.85398163397448
x40 = -36.1283155162826
x41 = -17.2787595947439
x42 = -14.1371669411541
x43 = 20.4203522483337
x44 = 54.9778714378214
x45 = -70.6858347057703
x46 = -48.6946861306418
x47 = -54.9778714378214
x48 = -45.553093477052
x49 = 14.1371669411541
x50 = -73.8274273593601
x51 = 26.7035375555132
x52 = 89.5353906273091
x53 = 10.9955742875643
x54 = 80.1106126665397
x55 = -256.039801267568
x56 = 73.8274273593601
x57 = 58.1194640914112
x58 = -61.261056745001
x59 = 1.5707963267949
x60 = -20.4203522483337
x61 = -42.4115008234622
x62 = 32.9867228626928
x63 = 42.4115008234622
x64 = -76.9690200129499
x65 = -64.4026493985908
x66 = -29.845130209103
x66 = -29.845130209103
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