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cos(2*x)-3*cos(x)=4*cos(x)^(2)/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
                           2   
                      4*cos (x)
cos(2*x) - 3*cos(x) = ---------
                          2
−3cos⁡(x)+cos⁡(2x)=4cos⁡2(x)2- 3 \cos{\left(x \right)} + \cos{\left(2 x \right)} = \frac{4 \cos^{2}{\left(x \right)}}{2}
Simplificar
Solución detallada
Tenemos la ecuación
−3cos⁡(x)+cos⁡(2x)=4cos⁡2(x)2- 3 \cos{\left(x \right)} + \cos{\left(2 x \right)} = \frac{4 \cos^{2}{\left(x \right)}}{2}
cambiamos
−3cos⁡(x)−1=0- 3 \cos{\left(x \right)} - 1 = 0
2cos⁡2(x)−3cos⁡(x)−4cos⁡2(x)2−1=02 \cos^{2}{\left(x \right)} - 3 \cos{\left(x \right)} - \frac{4 \cos^{2}{\left(x \right)}}{2} - 1 = 0
Sustituimos
w=cos⁡(x)w = \cos{\left(x \right)}
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
−3w=1- 3 w = 1
Dividamos ambos miembros de la ecuación en -3
w = 1 / (-3)

Obtenemos la respuesta: w = -1/3
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⁡(−13)x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{3} \right)}
x1=πn+acos⁡(−13)x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{3} \right)}
x2=πn+acos⁡(w1)−πx_{2} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x2=πn−π+acos⁡(−13)x_{2} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{3} \right)}
x2=πn−π+acos⁡(−13)x_{2} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{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
  /     /           ___\         \     /     /           ___\         \
I*\- log\-1 - 2*I*\/ 2 / + log(3)/ + I*\- log\-1 + 2*I*\/ 2 / + log(3)/
i(log⁡(3)−log⁡(−1−22i))+i(log⁡(3)−log⁡(−1+22i))i \left(\log{\left(3 \right)} - \log{\left(-1 - 2 \sqrt{2} i \right)}\right) + i \left(\log{\left(3 \right)} - \log{\left(-1 + 2 \sqrt{2} i \right)}\right) 
=
  /     /           ___\         \     /     /           ___\         \
I*\- log\-1 - 2*I*\/ 2 / + log(3)/ + I*\- log\-1 + 2*I*\/ 2 / + log(3)/
i(log⁡(3)−log⁡(−1−22i))+i(log⁡(3)−log⁡(−1+22i))i \left(\log{\left(3 \right)} - \log{\left(-1 - 2 \sqrt{2} i \right)}\right) + i \left(\log{\left(3 \right)} - \log{\left(-1 + 2 \sqrt{2} i \right)}\right) 
producto
  /     /           ___\         \   /     /           ___\         \
I*\- log\-1 - 2*I*\/ 2 / + log(3)/*I*\- log\-1 + 2*I*\/ 2 / + log(3)/
i(log⁡(3)−log⁡(−1−22i))i(log⁡(3)−log⁡(−1+22i))i \left(\log{\left(3 \right)} - \log{\left(-1 - 2 \sqrt{2} i \right)}\right) i \left(\log{\left(3 \right)} - \log{\left(-1 + 2 \sqrt{2} i \right)}\right) 
=
 /     /           ___\         \ /     /           ___\         \
-\- log\-1 - 2*I*\/ 2 / + log(3)/*\- log\-1 + 2*I*\/ 2 / + log(3)/
−(log⁡(3)−log⁡(−1−22i))(log⁡(3)−log⁡(−1+22i))- \left(\log{\left(3 \right)} - \log{\left(-1 - 2 \sqrt{2} i \right)}\right) \left(\log{\left(3 \right)} - \log{\left(-1 + 2 \sqrt{2} i \right)}\right) 
-(-log(-1 - 2*i*sqrt(2)) + log(3))*(-log(-1 + 2*i*sqrt(2)) + log(3))
Respuesta rápida [src] 
       /     /           ___\         \
x1 = I*\- log\-1 - 2*I*\/ 2 / + log(3)/
x1=i(log⁡(3)−log⁡(−1−22i))x_{1} = i \left(\log{\left(3 \right)} - \log{\left(-1 - 2 \sqrt{2} i \right)}\right) 
       /     /           ___\         \
x2 = I*\- log\-1 + 2*I*\/ 2 / + log(3)/
x2=i(log⁡(3)−log⁡(−1+22i))x_{2} = i \left(\log{\left(3 \right)} - \log{\left(-1 + 2 \sqrt{2} i \right)}\right) 
x2 = i*(log(3) - log(-1 + 2*sqrt(2)*i))
Respuesta numérica [src] 
x1 = -73.487590449906
x2 = -20.7601891577878
x3 = -42.0716639140081
x4 = 155.168999443241
x5 = 35.7884786068285
x6 = -5678.0888844541
x7 = 4.37255207093057
x8 = 10.6557373781102
x9 = 20.7601891577878
x10 = -4.37255207093057
x11 = -39.6097450793265
x12 = -33.326559772147
x13 = 77.3088569224041
x14 = -1.91063323624902
x15 = -14.4770038506082
x16 = 83.5920422295836
x17 = 96.1584128439428
x18 = -77.3088569224041
x19 = -45.8929303865061
x20 = -83.5920422295836
x21 = -92.3371463714448
x22 = 48.3548492211877
x23 = 8.19381854342861
x24 = -16.9389226852897
x25 = -35.7884786068285
x26 = -10.6557373781102
x27 = 71.0256716152245
x28 = -27.0433744649674
x29 = -98.6203316786244
x30 = 52.1761156936857
x31 = 29.5052932996489
x32 = -71.0256716152245
x33 = 86.0539610642652
x34 = 58.4593010008653
x35 = 79.7707757570856
x36 = 16.9389226852897
x37 = 64.7424863080449
x38 = 1.91063323624902
x39 = 67.2044051427264
x40 = -48.3548492211877
x41 = 92.3371463714448
x42 = -52.1761156936857
x43 = -29.5052932996489
x44 = 27.0433744649674
x45 = 23.2221079924693
x46 = -23.2221079924693
x47 = 14.4770038506082
x48 = -971.983089376587
x49 = 60.9212198355468
x50 = -89.8752275367632
x51 = -67.2044051427264
x52 = 54.6380345283673
x53 = 42.0716639140081
x54 = -60.9212198355468
x55 = -64.7424863080449
x56 = -79.7707757570856
x57 = 33.326559772147
x58 = -54.6380345283673
x59 = 3058.00061136021
x60 = 39.6097450793265
x61 = 73.487590449906
x62 = -96.1584128439428
x63 = 89.8752275367632
x64 = 98.6203316786244
x65 = -8.19381854342861
x66 = -86.0539610642652
x67 = -58.4593010008653
x68 = 45.8929303865061
x68 = 45.8929303865061
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