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cos(2x)+sin(x)^(2)=3/4 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
              2         
cos(2*x) + sin (x) = 3/4
sin⁡2(x)+cos⁡(2x)=34\sin^{2}{\left(x \right)} + \cos{\left(2 x \right)} = \frac{3}{4}
Simplificar
Solución detallada
Tenemos la ecuación
sin⁡2(x)+cos⁡(2x)=34\sin^{2}{\left(x \right)} + \cos{\left(2 x \right)} = \frac{3}{4}
cambiamos
cos⁡2(x)−34=0\cos^{2}{\left(x \right)} - \frac{3}{4} = 0
14−sin⁡2(x)=0\frac{1}{4} - \sin^{2}{\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=−1a = -1
b=0b = 0
c=14c = \frac{1}{4}
, entonces
D = b^2 - 4 * a * c = 

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

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

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

o
w1=−12w_{1} = - \frac{1}{2}
w2=12w_{2} = \frac{1}{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⁡(−12)x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}
x1=2πn−π6x_{1} = 2 \pi n - \frac{\pi}{6}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(12)x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}
x2=2πn+π6x_{2} = 2 \pi n + \frac{\pi}{6}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn−asin⁡(−12)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi
x3=2πn+7π6x_{3} = 2 \pi n + \frac{7 \pi}{6}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn−asin⁡(12)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi
x4=2πn+5π6x_{4} = 2 \pi n + \frac{5 \pi}{6}
Gráfica
0-80-60-40-2020406080-10010002
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
     -5*pi
x1 = -----
       6
x1=−5π6x_{1} = - \frac{5 \pi}{6} 
     -pi 
x2 = ----
      6
x2=−π6x_{2} = - \frac{\pi}{6} 
     pi
x3 = --
     6
x3=π6x_{3} = \frac{\pi}{6} 
     5*pi
x4 = ----
      6
x4=5π6x_{4} = \frac{5 \pi}{6} 
x4 = 5*pi/6
Suma y producto de raíces [src] 
suma
  5*pi   pi   pi   5*pi
- ---- - -- + -- + ----
   6     6    6     6
((−5π6−π6)+π6)+5π6\left(\left(- \frac{5 \pi}{6} - \frac{\pi}{6}\right) + \frac{\pi}{6}\right) + \frac{5 \pi}{6} 
=
0
00 
producto
-5*pi -pi  pi 5*pi
-----*----*--*----
  6    6   6   6
5π6π6⋅−5π6(−π6)\frac{5 \pi}{6} \frac{\pi}{6} \cdot - \frac{5 \pi}{6} \left(- \frac{\pi}{6}\right) 
=
     4
25*pi 
------
 1296
25π41296\frac{25 \pi^{4}}{1296} 
25*pi^4/1296
Respuesta numérica [src] 
x1 = 9.94837673636768
x2 = -31.9395253114962
x3 = -25.6563400043166
x4 = -16.2315620435473
x5 = 84.2994028713261
x6 = -90.5825881785057
x7 = 22.5147473507269
x8 = -46.6002910282486
x9 = -82.2050077689329
x10 = 88.4881930761125
x11 = 44.5058959258554
x12 = 71.733032256967
x13 = 19.3731546971371
x14 = -40.317105721069
x15 = -85.3466004225227
x16 = -93.7241808320955
x17 = -18.3259571459405
x18 = 25.6563400043166
x19 = -41.3643032722656
x20 = -27.7507351067098
x21 = 18.3259571459405
x22 = 16.2315620435473
x23 = 63.3554518473942
x24 = -91.6297857297023
x25 = 24.60914245312
x26 = -47.6474885794452
x27 = 66.497044500984
x28 = 62.3082542961976
x29 = 46.6002910282486
x30 = 5.75958653158129
x31 = -65.4498469497874
x32 = 91.6297857297023
x33 = -35.081117965086
x34 = -5.75958653158129
x35 = 53.9306738866248
x36 = -21.4675497995303
x37 = -100.007366139275
x38 = -69.6386371545737
x39 = 27.7507351067098
x40 = -75.9218224617533
x41 = -56.025068989018
x42 = -60.2138591938044
x43 = 60.2138591938044
x44 = 82.2050077689329
x45 = 90.5825881785057
x46 = -38.2227106186758
x47 = -87.4409955249159
x48 = 52.8834763354282
x49 = 38.2227106186758
x50 = 31.9395253114962
x51 = 34.0339204138894
x52 = -78.0162175641465
x53 = -19.3731546971371
x54 = 69.6386371545737
x55 = -34.0339204138894
x56 = -97.9129710368819
x57 = -933.576616891767
x58 = 97.9129710368819
x59 = -12.0427718387609
x60 = -53.9306738866248
x61 = 8.90117918517108
x62 = 3.66519142918809
x63 = -71.733032256967
x64 = 30.8923277602996
x65 = 47.6474885794452
x66 = 49.7418836818384
x67 = 74.8746249105567
x68 = 78.0162175641465
x69 = 93.7241808320955
x70 = 2.61799387799149
x71 = -3.66519142918809
x72 = -217.293491873294
x73 = 85.3466004225227
x74 = -84.2994028713261
x75 = 40.317105721069
x76 = 131.423292675173
x77 = -24.60914245312
x78 = 12.0427718387609
x79 = 0.523598775598299
x80 = -2685.53812004368
x81 = -9.94837673636768
x82 = -68.5914396033772
x83 = 100.007366139275
x84 = -13.0899693899575
x85 = 56.025068989018
x86 = 125.140107367993
x87 = -63.3554518473942
x88 = -43.4586983746588
x89 = -2.61799387799149
x90 = -81.1578102177363
x91 = -49.7418836818384
x92 = -62.3082542961976
x93 = 68.5914396033772
x94 = 75.9218224617533
x95 = 96.8657734856853
x96 = 41.3643032722656
x96 = 41.3643032722656
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