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sin(x)^2=3/4

sin(x)^2=3/4 la ecuación

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Solución numérica:

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Solución

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

(0)^2 - 4 * (1) * (-3/4) = 3

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

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

o
w1=32w_{1} = \frac{\sqrt{3}}{2}
w2=−32w_{2} = - \frac{\sqrt{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⁡(32)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)}
x1=2πn+π3x_{1} = 2 \pi n + \frac{\pi}{3}
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{\sqrt{3}}{2} \right)}
x2=2πn−π3x_{2} = 2 \pi n - \frac{\pi}{3}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn−asin⁡(32)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)} + \pi
x3=2πn+2π3x_{3} = 2 \pi n + \frac{2 \pi}{3}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn−asin⁡(−32)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi
x4=2πn+4π3x_{4} = 2 \pi n + \frac{4 \pi}{3}
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*pi   4*pi
- -- + -- + ---- + ----
  3    3     3      3
((−π3+π3)+2π3)+4π3\left(\left(- \frac{\pi}{3} + \frac{\pi}{3}\right) + \frac{2 \pi}{3}\right) + \frac{4 \pi}{3} 
=
2*pi
2π2 \pi 
producto
-pi  pi 2*pi 4*pi
----*--*----*----
 3   3   3    3
4π32π3⋅−π3π3\frac{4 \pi}{3} \frac{2 \pi}{3} \cdot - \frac{\pi}{3} \frac{\pi}{3} 
=
     4
-8*pi 
------
  81
−8π481- \frac{8 \pi^{4}}{81} 
-8*pi^4/81
Respuesta rápida [src] 
     -pi 
x1 = ----
      3
x1=−π3x_{1} = - \frac{\pi}{3} 
     pi
x2 = --
     3
x2=π3x_{2} = \frac{\pi}{3} 
     2*pi
x3 = ----
      3
x3=2π3x_{3} = \frac{2 \pi}{3} 
     4*pi
x4 = ----
      3
x4=4π3x_{4} = \frac{4 \pi}{3} 
x4 = 4*pi/3
Respuesta numérica [src] 
x1 = 26.1799387799149
x2 = 11.5191730631626
x3 = -26.1799387799149
x4 = 99.4837673636768
x5 = -35.6047167406843
x6 = -76.4454212373516
x7 = 1515.29485658148
x8 = -13.6135681655558
x9 = -27.2271363311115
x10 = 92.1533845053006
x11 = 90.0589894029074
x12 = -14.6607657167524
x13 = -33.5103216382911
x14 = 4.18879020478639
x15 = 68.0678408277789
x16 = -77.4926187885482
x17 = -5.23598775598299
x18 = 83.7758040957278
x19 = 8.37758040957278
x20 = -378.038315981972
x21 = -99.4837673636768
x22 = -71.2094334813686
x23 = -93.2005820564972
x24 = 2.0943951023932
x25 = 55.5014702134197
x26 = -96.342174710087
x27 = -19.8967534727354
x28 = 54.4542726622231
x29 = -55.5014702134197
x30 = 17.8023583703422
x31 = 61.7846555205993
x32 = -68.0678408277789
x33 = 24.0855436775217
x34 = 30.3687289847013
x35 = 70.162235930172
x36 = 16.7551608191456
x37 = 48.1710873550435
x38 = -11.5191730631626
x39 = 52.3598775598299
x40 = -46.0766922526503
x41 = -54.4542726622231
x42 = 79.5870138909414
x43 = 82.7286065445312
x44 = 63.8790506229925
x45 = 76.4454212373516
x46 = -17.8023583703422
x47 = 41.8879020478639
x48 = 10.471975511966
x49 = -92.1533845053006
x50 = 96.342174710087
x51 = -57.5958653158129
x52 = -48.1710873550435
x53 = -2.0943951023932
x54 = 60.7374579694027
x55 = 98.4365698124802
x56 = -60.7374579694027
x57 = 85.870199198121
x58 = 39.7935069454707
x59 = -32.4631240870945
x60 = -85.870199198121
x61 = 19.8967534727354
x62 = -79.5870138909414
x63 = 32.4631240870945
x64 = -294.262511886244
x65 = -30.3687289847013
x66 = -49.2182849062401
x67 = 20.943951023932
x68 = 77.4926187885482
x69 = -10.471975511966
x70 = 33.5103216382911
x71 = -39.7935069454707
x72 = -4.18879020478639
x73 = -41.8879020478639
x74 = 46.0766922526503
x75 = -24.0855436775217
x76 = -70.162235930172
x77 = -98.4365698124802
x78 = 38.7463093942741
x79 = -90.0589894029074
x80 = 74.3510261349584
x81 = -83.7758040957278
x82 = -61.7846555205993
x83 = -63.8790506229925
x83 = -63.8790506229925
Gráfico
sin(x)^2=3/4 la ecuación
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