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cos(2*x)+sqrt(2)*sin(x)+1=0

cos(2*x)+sqrt(2)*sin(x)+1=0 la ecuación

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Solución numérica:

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Solución

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

(sqrt(2))^2 - 4 * (-2) * (2) = 18

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

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

o
w1=−22w_{1} = - \frac{\sqrt{2}}{2}
w2=2w_{2} = \sqrt{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⁡(−22)x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}
x1=2πn−π4x_{1} = 2 \pi n - \frac{\pi}{4}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(2)x_{2} = 2 \pi n + \operatorname{asin}{\left(\sqrt{2} \right)}
x2=2πn+asin⁡(2)x_{2} = 2 \pi n + \operatorname{asin}{\left(\sqrt{2} \right)}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn−asin⁡(−22)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)} + \pi
x3=2πn+5π4x_{3} = 2 \pi n + \frac{5 \pi}{4}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn+π−asin⁡(2)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\sqrt{2} \right)}
x4=2πn+π−asin⁡(2)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\sqrt{2} \right)}
Gráfica
0-80-60-40-2020406080-1001005-5
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
     -3*pi
x1 = -----
       4
x1=−3π4x_{1} = - \frac{3 \pi}{4} 
     -pi 
x2 = ----
      4
x2=−π4x_{2} = - \frac{\pi}{4} 
     pi        /       ___\
x3 = -- - I*log\-1 + \/ 2 /
     2
x3=π2−ilog⁡(−1+2)x_{3} = \frac{\pi}{2} - i \log{\left(-1 + \sqrt{2} \right)} 
     pi        /      ___\
x4 = -- - I*log\1 + \/ 2 /
     2
x4=π2−ilog⁡(1+2)x_{4} = \frac{\pi}{2} - i \log{\left(1 + \sqrt{2} \right)} 
x4 = pi/2 - i*log(1 + sqrt(2))
Suma y producto de raíces [src] 
suma
  3*pi   pi   pi        /       ___\   pi        /      ___\
- ---- - -- + -- - I*log\-1 + \/ 2 / + -- - I*log\1 + \/ 2 /
   4     4    2                        2
(π2−ilog⁡(1+2))+((−3π4−π4)+(π2−ilog⁡(−1+2)))\left(\frac{\pi}{2} - i \log{\left(1 + \sqrt{2} \right)}\right) + \left(\left(- \frac{3 \pi}{4} - \frac{\pi}{4}\right) + \left(\frac{\pi}{2} - i \log{\left(-1 + \sqrt{2} \right)}\right)\right) 
=
       /      ___\        /       ___\
- I*log\1 + \/ 2 / - I*log\-1 + \/ 2 /
−ilog⁡(1+2)−ilog⁡(−1+2)- i \log{\left(1 + \sqrt{2} \right)} - i \log{\left(-1 + \sqrt{2} \right)} 
producto
-3*pi -pi  /pi        /       ___\\ /pi        /      ___\\
-----*----*|-- - I*log\-1 + \/ 2 /|*|-- - I*log\1 + \/ 2 /|
  4    4   \2                     / \2                    /
−3π4(−π4)(π2−ilog⁡(−1+2))(π2−ilog⁡(1+2))- \frac{3 \pi}{4} \left(- \frac{\pi}{4}\right) \left(\frac{\pi}{2} - i \log{\left(-1 + \sqrt{2} \right)}\right) \left(\frac{\pi}{2} - i \log{\left(1 + \sqrt{2} \right)}\right) 
=
    2 /            /      ___\\ /            /       ___\\
3*pi *\pi - 2*I*log\1 + \/ 2 //*\pi - 2*I*log\-1 + \/ 2 //
----------------------------------------------------------
                            64
3π2(π−2ilog⁡(−1+2))(π−2ilog⁡(1+2))64\frac{3 \pi^{2} \left(\pi - 2 i \log{\left(-1 + \sqrt{2} \right)}\right) \left(\pi - 2 i \log{\left(1 + \sqrt{2} \right)}\right)}{64} 
3*pi^2*(pi - 2*i*log(1 + sqrt(2)))*(pi - 2*i*log(-1 + sqrt(2)))/64
Respuesta numérica [src] 
x1 = 49.4800842940392
x2 = 74.6128255227576
x3 = 85.6083998103219
x4 = -574.126057443535
x5 = 66.7588438887831
x6 = -82.4668071567321
x7 = -71.4712328691678
x8 = -13.3517687777566
x9 = -32.2013246992954
x10 = 22.776546738526
x11 = -46.3384916404494
x12 = 99.7455667514759
x13 = 62.0464549083984
x14 = 87.1791961371168
x15 = -57.3340659280137
x16 = -27.4889357189107
x17 = -76.1836218495525
x18 = 93.4623814442964
x19 = 30.6305283725005
x20 = -38.484510006475
x21 = -21.2057504117311
x22 = -84.037603483527
x23 = 98.174770424681
x24 = -69.9004365423729
x25 = 91.8915851175014
x26 = -65.1880475619882
x27 = -63.6172512351933
x28 = 68.329640215578
x29 = 10.2101761241668
x30 = -77.7544181763474
x31 = -25.9181393921158
x32 = 41.6261026600648
x33 = 60.4756585816035
x34 = 5.49778714378214
x35 = 18.0641577581413
x36 = -40.0553063332699
x37 = -19.6349540849362
x38 = -2.35619449019234
x39 = 54.1924732744239
x40 = 3.92699081698724
x41 = 24.3473430653209
x42 = -90.3207887907066
x43 = -33.7721210260903
x44 = 55.7632696012188
x45 = 16.4933614313464
x46 = 11.7809724509617
x47 = 47.9092879672443
x47 = 47.9092879672443
Gráfico
cos(2*x)+sqrt(2)*sin(x)+1=0 la ecuación
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