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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           
$$\left(\sin^{2}{\left(x \right)} - \frac{\cos{\left(x \right)}}{2}\right) - 1 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(\sin^{2}{\left(x \right)} - \frac{\cos{\left(x \right)}}{2}\right) - 1 = 0$$
cambiamos
$$- \left(\cos{\left(x \right)} + \frac{1}{2}\right) \cos{\left(x \right)} = 0$$
$$- \cos^{2}{\left(x \right)} - \frac{\cos{\left(x \right)}}{2} = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
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:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -1$$
$$b = - \frac{1}{2}$$
$$c = 0$$
, 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
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = 0$$
hacemos cambio inverso
$$\cos{\left(x \right)} = w$$
Tenemos la ecuación
$$\cos{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
O
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = \pi n + \frac{2 \pi}{3}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$x_{2} = \pi n + \frac{\pi}{2}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{3} = \pi n - \frac{\pi}{3}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
$$x_{4} = \pi n - \frac{\pi}{2}$$
Gráfica
Respuesta rápida [src]
     -2*pi
x1 = -----
       3  
$$x_{1} = - \frac{2 \pi}{3}$$
     -pi 
x2 = ----
      2  
$$x_{2} = - \frac{\pi}{2}$$
     pi
x3 = --
     2 
$$x_{3} = \frac{\pi}{2}$$
     2*pi
x4 = ----
      3  
$$x_{4} = \frac{2 \pi}{3}$$
x4 = 2*pi/3
Suma y producto de raíces [src]
suma
  2*pi   pi   pi   2*pi
- ---- - -- + -- + ----
   3     2    2     3  
$$\left(\left(- \frac{2 \pi}{3} - \frac{\pi}{2}\right) + \frac{\pi}{2}\right) + \frac{2 \pi}{3}$$
=
0
$$0$$
producto
-2*pi -pi  pi 2*pi
-----*----*--*----
  3    2   2   3  
$$\frac{2 \pi}{3} \frac{\pi}{2} \cdot - \frac{2 \pi}{3} \left(- \frac{\pi}{2}\right)$$
=
  4
pi 
---
 9 
$$\frac{\pi^{4}}{9}$$
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