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sinx^2-0.5*cosx-1=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2      cos(x)        
sin (x) - ------ - 1 = 0
            2
(sin⁡2(x)−cos⁡(x)2)−1=0\left(\sin^{2}{\left(x \right)} - \frac{\cos{\left(x \right)}}{2}\right) - 1 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(sin⁡2(x)−cos⁡(x)2)−1=0\left(\sin^{2}{\left(x \right)} - \frac{\cos{\left(x \right)}}{2}\right) - 1 = 0
cambiamos
−(cos⁡(x)+12)cos⁡(x)=0- \left(\cos{\left(x \right)} + \frac{1}{2}\right) \cos{\left(x \right)} = 0
−cos⁡2(x)−cos⁡(x)2=0- \cos^{2}{\left(x \right)} - \frac{\cos{\left(x \right)}}{2} = 0
Sustituimos
w=cos⁡(x)w = \cos{\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=−12b = - \frac{1}{2}
c=0c = 0
, entonces
D = b^2 - 4 * a * c = 

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

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=0w_{2} = 0
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⁡(−12)x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}
x1=πn+2π3x_{1} = \pi n + \frac{2 \pi}{3}
x2=πn+acos⁡(w2)x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}
x2=πn+acos⁡(0)x_{2} = \pi n + \operatorname{acos}{\left(0 \right)}
x2=πn+π2x_{2} = \pi n + \frac{\pi}{2}
x3=πn+acos⁡(w1)−πx_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x3=πn−π+acos⁡(−12)x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}
x3=πn−π3x_{3} = \pi n - \frac{\pi}{3}
x4=πn+acos⁡(w2)−πx_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi
x4=πn−π+acos⁡(0)x_{4} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}
x4=πn−π2x_{4} = \pi n - \frac{\pi}{2}
Gráfica
0-80-60-40-2020406080-1001002-2
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
     -2*pi
x1 = -----
       3
x1=−2π3x_{1} = - \frac{2 \pi}{3} 
     -pi 
x2 = ----
      2
x2=−π2x_{2} = - \frac{\pi}{2} 
     pi
x3 = --
     2
x3=π2x_{3} = \frac{\pi}{2} 
     2*pi
x4 = ----
      3
x4=2π3x_{4} = \frac{2 \pi}{3} 
x4 = 2*pi/3
Suma y producto de raíces [src] 
suma
  2*pi   pi   pi   2*pi
- ---- - -- + -- + ----
   3     2    2     3
((−2π3−π2)+π2)+2π3\left(\left(- \frac{2 \pi}{3} - \frac{\pi}{2}\right) + \frac{\pi}{2}\right) + \frac{2 \pi}{3} 
=
0
00 
producto
-2*pi -pi  pi 2*pi
-----*----*--*----
  3    2   2   3
2π3π2⋅−2π3(−π2)\frac{2 \pi}{3} \frac{\pi}{2} \cdot - \frac{2 \pi}{3} \left(- \frac{\pi}{2}\right) 
=
  4
pi 
---
 9
π49\frac{\pi^{4}}{9} 
pi^4/9
Respuesta numérica [src] 
x1 = -64.4026493985908
x2 = -23.5619449019235
x3 = 61.261056745001
x4 = -29.845130209103
x5 = -85.870199198121
x6 = -4.18879020478639
x7 = 4.71238898038469
x8 = -67.0206432765823
x9 = 36.1283155162826
x10 = 10.9955742875643
x11 = 23.5619449019235
x12 = -35.6047167406843
x13 = -17.2787595947439
x14 = 26.7035375555132
x15 = 2.0943951023932
x16 = 33.5103216382911
x17 = 54.9778714378214
x18 = 39.7935069454707
x19 = 39.2699081698724
x20 = 64.4026493985908
x21 = -81877.758534184
x22 = 98.9601685880785
x23 = -39.7935069454707
x24 = -77.4926187885482
x25 = -95.8185759344887
x26 = 85.870199198121
x27 = -86.3937979737193
x28 = 73.8274273593601
x29 = 98.4365698124802
x30 = 67.5442420521806
x31 = -32.9867228626928
x32 = 20.943951023932
x33 = -90.0589894029074
x34 = -41.8879020478639
x35 = 70.6858347057703
x36 = -23.0383461263252
x37 = 60.7374579694027
x38 = 90.0589894029074
x39 = -2.0943951023932
x40 = 41.8879020478639
x41 = -42.4115008234622
x42 = -51.8362787842316
x43 = 58.1194640914112
x44 = -73.8274273593601
x45 = 29.845130209103
x46 = 80.1106126665397
x47 = -58.1194640914112
x48 = 77.4926187885482
x49 = 8.37758040957278
x50 = 52.3598775598299
x51 = -46.0766922526503
x52 = -33.5103216382911
x53 = 32.9867228626928
x54 = 16.7551608191456
x55 = -10.9955742875643
x56 = -27.2271363311115
x57 = 83.2522053201295
x58 = -67.5442420521806
x59 = 96.342174710087
x60 = -83.7758040957278
x61 = -7.85398163397448
x62 = -54.4542726622231
x63 = 89.5353906273091
x64 = -14.1371669411541
x65 = 48.1710873550435
x66 = -10.471975511966
x67 = 76.9690200129499
x68 = 4.18879020478639
x69 = 83.7758040957278
x70 = -4.71238898038469
x71 = -98.9601685880785
x72 = -71.2094334813686
x73 = 14.1371669411541
x74 = 54.4542726622231
x75 = 20.4203522483337
x76 = -70.6858347057703
x77 = -26.7035375555132
x78 = 92.1533845053006
x79 = -111.002940426839
x80 = -48.1710873550435
x81 = 46.0766922526503
x82 = -54.9778714378214
x83 = 10.471975511966
x84 = 17.2787595947439
x85 = -76.9690200129499
x86 = -1072.33029242532
x87 = -92.1533845053006
x88 = -20.4203522483337
x89 = -61.261056745001
x90 = -98.4365698124802
x90 = -98.4365698124802
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