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cos(3*x-pi/4)^2=1 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2/      pi\    
cos |3*x - --| = 1
    \      4 /
cos⁡2(3x−π4)=1\cos^{2}{\left(3 x - \frac{\pi}{4} \right)} = 1
Simplificar
Solución detallada
Tenemos la ecuación
cos⁡2(3x−π4)=1\cos^{2}{\left(3 x - \frac{\pi}{4} \right)} = 1
cambiamos
sin⁡(6x)2−12=0\frac{\sin{\left(6 x \right)}}{2} - \frac{1}{2} = 0
cos⁡2(3x−π4)−1=0\cos^{2}{\left(3 x - \frac{\pi}{4} \right)} - 1 = 0
Sustituimos
w=sin⁡(3x+π4)w = \sin{\left(3 x + \frac{\pi}{4} \right)}
Tenemos la ecuación
cos⁡2(3x−π4)−1=0\cos^{2}{\left(3 x - \frac{\pi}{4} \right)} - 1 = 0
Ya que la potencia en la ecuación es igual a = 2 - contiene un número par 2 en el numerador, entonces
la ecuación tendrá dos raíces reales.
Extraigamos la raíz de potencia 2 de las dos partes de la ecuación:
Obtenemos:
(0w+sin⁡(3x+π4))2=1\sqrt{\left(0 w + \sin{\left(3 x + \frac{\pi}{4} \right)}\right)^{2}} = \sqrt{1}
(0w+sin⁡(3x+π4))2=(−1)1\sqrt{\left(0 w + \sin{\left(3 x + \frac{\pi}{4} \right)}\right)^{2}} = \left(-1\right) \sqrt{1}
o
sin⁡(3x+π4)=1\sin{\left(3 x + \frac{\pi}{4} \right)} = 1
sin⁡(3x+π4)=−1\sin{\left(3 x + \frac{\pi}{4} \right)} = -1
Abrimos los paréntesis en el miembro izquierdo de la ecuación
sin3*x+pi/4 = 1

Esta ecuación no tiene soluciones
Abrimos los paréntesis en el miembro izquierdo de la ecuación
sin3*x+pi/4 = -1

Esta ecuación no tiene soluciones
o

hacemos cambio inverso
sin⁡(3x+π4)=w\sin{\left(3 x + \frac{\pi}{4} \right)} = w
Tenemos la ecuación
sin⁡(3x+π4)=w\sin{\left(3 x + \frac{\pi}{4} \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
3x+π4=2πn+asin⁡(w)3 x + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(w \right)}
3x+π4=2πn−asin⁡(w)+π3 x + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
O
3x+π4=2πn+asin⁡(w)3 x + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(w \right)}
3x+π4=2πn−asin⁡(w)+π3 x + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
, donde n es cualquier número entero
Transportemos
π4\frac{\pi}{4}
al miembro derecho de la ecuación
con el signo opuesto, en total:
3x=2πn+asin⁡(w)−π43 x = 2 \pi n + \operatorname{asin}{\left(w \right)} - \frac{\pi}{4}
3x=2πn−asin⁡(w)+3π43 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \frac{3 \pi}{4}
Dividamos ambos miembros de la ecuación obtenida en
33
sustituimos w:
Gráfica
0-80-60-40-2020406080-10010002
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
  pi   pi   5*pi
- -- + -- + ----
  4    12    12
(−π4+π12)+5π12\left(- \frac{\pi}{4} + \frac{\pi}{12}\right) + \frac{5 \pi}{12} 
=
pi
--
4
π4\frac{\pi}{4} 
producto
-pi  pi 5*pi
----*--*----
 4   12  12
5π12⋅−π4π12\frac{5 \pi}{12} \cdot - \frac{\pi}{4} \frac{\pi}{12} 
=
     3
-5*pi 
------
 576
−5π3576- \frac{5 \pi^{3}}{576} 
-5*pi^3/576
Respuesta rápida [src] 
     -pi 
x1 = ----
      4
x1=−π4x_{1} = - \frac{\pi}{4} 
     pi
x2 = --
     12
x2=π12x_{2} = \frac{\pi}{12} 
     5*pi
x3 = ----
      12
x3=5π12x_{3} = \frac{5 \pi}{12} 
x3 = 5*pi/12
Respuesta numérica [src] 
x1 = 18.0641575904069
x2 = 20.1585529574804
x3 = 47.3856893488309
x4 = -6.02138589730989
x5 = -33.2485222538391
x6 = -50.0036830653347
x7 = -89.797189973996
x8 = -37.4373122668668
x9 = -39.531707620165
x10 = 28.5361333386207
x11 = 22.2529479607571
x12 = 2.35619443223801
x13 = 70.424035246248
x14 = 66.2352451246547
x15 = -12.3045714094311
x16 = -52.0980781057903
x17 = 79.8488133554185
x18 = -67.8060413893714
x19 = 55.7632696509651
x20 = 99.7455667784648
x21 = -85.6083998783531
x22 = 13.8753675471005
x23 = 35.8665162033444
x24 = -92.9387829749902
x25 = 9.68657715106654
x26 = -93.9859802388653
x27 = 77.7544182162793
x28 = 37.9609112670449
x29 = -71.9948316512416
x30 = 0.261799379333194
x31 = 59.9520598481687
x32 = 62.0464548168765
x33 = 24.3473430106795
x34 = -45.814892803953
x35 = 81.9432084286987
x36 = -30.1069295278938
x37 = -61.5228561900619
x38 = 75.6600230667895
x39 = -76.1836217744616
x40 = 15.9697626852522
x41 = 92.41518382174
x42 = 4.45058952366422
x43 = -10.2101760567131
x44 = -32.2013246284748
x45 = 40.05530620807
x46 = 6.5449848141863
x47 = -74.0892266841028
x48 = 57.8576647795972
x49 = 64.1408501103489
x50 = -41.62610272373
x51 = 84.0376034179138
x52 = 33.7721210833439
x53 = -54.1924732011039
x54 = 130.114295757992
x55 = -63.6172513012213
x56 = -17.5405590488288
x57 = -58.3812632463393
x58 = 90.3207887476935
x59 = -98.174770348449
x60 = 44.2440965425203
x61 = -15.4461636998964
x62 = -19.6349541459037
x63 = -23.8237442176309
x64 = -34.2957199248261
x65 = -96.0803752628436
x66 = -8.11578095040642
x67 = 13.8753676266739
x68 = 72.5184304232893
x69 = 11.7809725139856
x70 = -78.2780169853537
x71 = 48.4328866713396
x72 = 88.2263937072014
x73 = -65.7116463631172
x74 = -48.9564853809923
x75 = -83.5140047581305
x76 = -87.7027949433722
x77 = 42.149701543678
x78 = 50.5272818759395
x79 = -100.269165531996
x80 = -56.2868684488565
x81 = -1.8325956303714
x82 = -43.7204977825881
x83 = 46.3384915894085
x84 = -28.012534480777
x85 = -3.92699098017885
x86 = 94.5095789779178
x87 = 68.3296401684154
x88 = -21.7293492017758
x89 = 26.4417380971056
x89 = 26.4417380971056
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