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cos(2*x)+3*sin(x)-1=0 la ecuación

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Solución numérica:

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Solución

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

(3)^2 - 4 * (-2) * (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=32w_{2} = \frac{3}{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⁡(0)x_{1} = 2 \pi n + \operatorname{asin}{\left(0 \right)}
x1=2πnx_{1} = 2 \pi n
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(32)x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} \right)}
x2=2πn+asin⁡(32)x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} \right)}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn−asin⁡(0)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi
x3=2πn+πx_{3} = 2 \pi n + \pi
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn+π−asin⁡(32)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} \right)}
x4=2πn+π−asin⁡(32)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} \right)}
Gráfica
0-80-60-40-2020406080-1001005-10
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
x1 = 0
x1=0x_{1} = 0 
x2 = pi
x2=πx_{2} = \pi 
               /      ___\
     pi        |3   \/ 5 |
x3 = -- + I*log|- - -----|
     2         \2     2  /
x3=π2+ilog⁡(32−52)x_{3} = \frac{\pi}{2} + i \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)} 
               /      ___\
     pi        |3   \/ 5 |
x4 = -- + I*log|- + -----|
     2         \2     2  /
x4=π2+ilog⁡(52+32)x_{4} = \frac{\pi}{2} + i \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)} 
x4 = pi/2 + i*log(sqrt(5)/2 + 3/2)
Suma y producto de raíces [src] 
suma
               /      ___\             /      ___\
     pi        |3   \/ 5 |   pi        |3   \/ 5 |
pi + -- + I*log|- - -----| + -- + I*log|- + -----|
     2         \2     2  /   2         \2     2  /
(π+(π2+ilog⁡(32−52)))+(π2+ilog⁡(52+32))\left(\pi + \left(\frac{\pi}{2} + i \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}\right)\right) + \left(\frac{\pi}{2} + i \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}\right) 
=
            /      ___\        /      ___\
            |3   \/ 5 |        |3   \/ 5 |
2*pi + I*log|- + -----| + I*log|- - -----|
            \2     2  /        \2     2  /
2π+ilog⁡(32−52)+ilog⁡(52+32)2 \pi + i \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)} + i \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)} 
producto
     /          /      ___\\ /          /      ___\\
     |pi        |3   \/ 5 || |pi        |3   \/ 5 ||
0*pi*|-- + I*log|- - -----||*|-- + I*log|- + -----||
     \2         \2     2  // \2         \2     2  //
0π(π2+ilog⁡(32−52))(π2+ilog⁡(52+32))0 \pi \left(\frac{\pi}{2} + i \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}\right) \left(\frac{\pi}{2} + i \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}\right) 
=
0
00 
0
Respuesta numérica [src] 
x1 = -43.9822971502571
x2 = -31.4159265358979
x3 = 84.8230016469244
x4 = -91.106186954104
x5 = -97.3893722612836
x6 = 91.106186954104
x7 = 210.486707790516
x8 = 6.28318530717959
x9 = -72.2566310325652
x10 = -47.1238898038469
x11 = 94.2477796076938
x12 = 50.2654824574367
x13 = 56.5486677646163
x14 = 43.9822971502571
x15 = 47.1238898038469
x16 = -50.2654824574367
x17 = 37.6991118430775
x18 = -28.2743338823081
x19 = -449.24774946334
x20 = 65.9734457253857
x21 = 15.707963267949
x22 = 28.2743338823081
x23 = -62.8318530717959
x24 = -40.8407044966673
x25 = -6.28318530717959
x26 = -81.6814089933346
x27 = -15.707963267949
x28 = -59.6902604182061
x29 = 72.2566310325652
x30 = 3.14159265358979
x31 = -25.1327412287183
x32 = 21.9911485751286
x33 = -75.398223686155
x34 = -56.5486677646163
x35 = -69.1150383789755
x36 = -84.8230016469244
x37 = 78.5398163397448
x38 = 9.42477796076938
x39 = -304.73448739821
x40 = 207.345115136926
x41 = 62.8318530717959
x42 = -53.4070751110265
x43 = -18.8495559215388
x44 = 25.1327412287183
x45 = 100.530964914873
x46 = -87.9645943005142
x47 = -9.42477796076938
x48 = 81.6814089933346
x49 = 87.9645943005142
x50 = 12.5663706143592
x51 = -34.5575191894877
x52 = 69.1150383789755
x53 = 0.0
x54 = -21.9911485751286
x55 = -37.6991118430775
x56 = -78.5398163397448
x57 = -12.5663706143592
x58 = -109.955742875643
x59 = -94.2477796076938
x60 = 97.3893722612836
x61 = -100.530964914873
x62 = -631.460123371548
x63 = 59.6902604182061
x64 = 53.4070751110265
x65 = 34.5575191894877
x66 = -65.9734457253857
x67 = 18.8495559215388
x67 = 18.8495559215388
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