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1-sin^2x=cos^4x la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
       2         4   
1 - sin (x) = cos (x)
1−sin⁡2(x)=cos⁡4(x)1 - \sin^{2}{\left(x \right)} = \cos^{4}{\left(x \right)}
Simplificar
Solución detallada
Tenemos la ecuación
1−sin⁡2(x)=cos⁡4(x)1 - \sin^{2}{\left(x \right)} = \cos^{4}{\left(x \right)}
cambiamos
18−cos⁡(4x)8=0\frac{1}{8} - \frac{\cos{\left(4 x \right)}}{8} = 0
−cos⁡4(x)+cos⁡2(x)=0- \cos^{4}{\left(x \right)} + \cos^{2}{\left(x \right)} = 0
Sustituimos
w=cos⁡(x)w = \cos{\left(x \right)}
Tenemos la ecuación:
−w4+w2=0- w^{4} + w^{2} = 0
Sustituimos
v=w2v = w^{2}
entonces la ecuación será así:
−v2+v=0- v^{2} + v = 0
Es la ecuación de la forma
a*v^2 + b*v + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
v1=D−b2av_{1} = \frac{\sqrt{D} - b}{2 a}
v2=−D−b2av_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=−1a = -1
b=1b = 1
c=0c = 0
, entonces
D = b^2 - 4 * a * c = 

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

Como D > 0 la ecuación tiene dos raíces.
v1 = (-b + sqrt(D)) / (2*a)

v2 = (-b - sqrt(D)) / (2*a)

o
v1=0v_{1} = 0
v2=1v_{2} = 1
Entonces la respuesta definitiva es:
Como
v=w2v = w^{2}
entonces
w1=v1w_{1} = \sqrt{v_{1}}
w2=−v1w_{2} = - \sqrt{v_{1}}
w3=v2w_{3} = \sqrt{v_{2}}
w4=−v2w_{4} = - \sqrt{v_{2}}
entonces:
w1=w_{1} =
0121+01=0\frac{0^{\frac{1}{2}}}{1} + \frac{0}{1} = 0
w2=w_{2} =
01+1121=1\frac{0}{1} + \frac{1^{\frac{1}{2}}}{1} = 1
w3=w_{3} =
(−1)1121+01=−1\frac{\left(-1\right) 1^{\frac{1}{2}}}{1} + \frac{0}{1} = -1
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⁡(1)x_{2} = \pi n + \operatorname{acos}{\left(1 \right)}
x2=πnx_{2} = \pi n
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⁡(1)x_{4} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}
x4=πn−πx_{4} = \pi n - \pi
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
- -- + --
  2    2
−π2+π2- \frac{\pi}{2} + \frac{\pi}{2} 
=
0
00 
producto
  -pi  pi
0*----*--
   2   2
π2⋅0(−π2)\frac{\pi}{2} \cdot 0 \left(- \frac{\pi}{2}\right) 
=
0
00 
0
Respuesta rápida [src] 
x1 = 0
x1=0x_{1} = 0 
     -pi 
x2 = ----
      2
x2=−π2x_{2} = - \frac{\pi}{2} 
     pi
x3 = --
     2
x3=π2x_{3} = \frac{\pi}{2} 
x3 = pi/2
Respuesta numérica [src] 
x1 = -86.3937978789102
x2 = 7.85398173011892
x3 = -65.9734457653935
x4 = -58.1194645366003
x5 = 29.8451303084991
x6 = 51.8362788866811
x7 = 42.4115007365289
x8 = -97.3893723711949
x9 = 117.809724442492
x10 = -42.4115007432387
x11 = 72.2566310277248
x12 = -89.5353907315491
x13 = 92.6769832182628
x14 = 80.1106131511482
x15 = 45.5530935075531
x16 = 65.9734457525462
x17 = 43.9822971692691
x18 = 70.6858346557926
x19 = -64.402649310466
x20 = 26.7035375390573
x21 = 56.5486676469942
x22 = 94.247779609353
x23 = -87.9645943594276
x24 = 21.9911485851564
x25 = 4.71238898608896
x26 = -43.9822971747455
x27 = 28.2743338652921
x28 = -97.3893725907902
x29 = 70.6858338406532
x30 = 87.9645943351391
x31 = 50.2654824463816
x32 = -45.5530935761698
x33 = 86.3937978937855
x34 = 100.530964798296
x35 = 34.5575190717885
x36 = -94.247779486083
x37 = -31.4159266517141
x38 = -15.7079632962205
x39 = -73.8274272808521
x40 = -37.6991118766796
x41 = -75.3982237985682
x42 = -9.42477807759933
x43 = -50.2654823342013
x44 = -95.818575868455
x45 = -100.530965206253
x46 = -29.8451301000724
x47 = -7.85398150696156
x48 = 59.690260541069
x49 = -21.9911485864927
x50 = 73.8274274646672
x51 = 48.6946860958663
x52 = 20.4203521581227
x53 = -51.8362786915081
x54 = -17.2787595621355
x55 = -1.57079626356835
x56 = 72.256630710694
x57 = -28.2743337586152
x58 = 0.0
x59 = 36.1283154718409
x60 = 78.5398162225044
x61 = -14.1371668484631
x62 = -59.6902604569585
x63 = 89.5353906153414
x64 = -72.2566309100272
x65 = 15.7079633917898
x66 = -67.5442421539445
x67 = 64.4026493150839
x68 = -39.2699081045218
x69 = -23.5619449982306
x70 = -6.28318518328035
x71 = -80.1106125854791
x72 = 37.6991119665793
x73 = -20.4203521774723
x74 = 81.6814091152362
x75 = -1.57079642013166
x76 = 12.5663704969137
x77 = 1.57079638652515
x78 = -36.128315427252
x79 = 6.28318528443138
x80 = 23.5619449483644
x81 = 95.8185760424586
x82 = -81.6814090370675
x83 = -61.2610566398387
x84 = 67.5442420634706
x85 = -83.2522051669813
x86 = -53.4070752253874
x87 = -58.1194640062544
x88 = 14.1371670778185
x89 = 95.8185756842062
x89 = 95.8185756842062
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