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2*sin(x)^2+5*cos(x)+1=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
     2                      
2*sin (x) + 5*cos(x) + 1 = 0
$$\left(2 \sin^{2}{\left(x \right)} + 5 \cos{\left(x \right)}\right) + 1 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(2 \sin^{2}{\left(x \right)} + 5 \cos{\left(x \right)}\right) + 1 = 0$$
cambiamos
$$5 \cos{\left(x \right)} - \cos{\left(2 x \right)} + 2 = 0$$
$$- 2 \cos^{2}{\left(x \right)} + 5 \cos{\left(x \right)} + 3 = 0$$
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
$$a = -2$$
$$b = 5$$
$$c = 3$$
, entonces
D = b^2 - 4 * a * c = 

(5)^2 - 4 * (-2) * (3) = 49

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

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

o
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = 3$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = \pi n + \frac{2 \pi}{3}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(3 \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(3 \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{3} = \pi n - \frac{\pi}{3}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(3 \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(3 \right)}$$
Gráfica
Suma y producto de raíces [src]
suma
                         /  ___\            /  ___\
  2*pi   2*pi            |\/ 2 |            |\/ 2 |
- ---- + ---- - 2*I*atanh|-----| + 2*I*atanh|-----|
   3      3              \  2  /            \  2  /
$$\left(\left(- \frac{2 \pi}{3} + \frac{2 \pi}{3}\right) - 2 i \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)}\right) + 2 i \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)}$$
=
0
$$0$$
producto
                     /  ___\          /  ___\
-2*pi 2*pi           |\/ 2 |          |\/ 2 |
-----*----*-2*I*atanh|-----|*2*I*atanh|-----|
  3    3             \  2  /          \  2  /
$$2 i \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)} - 2 i \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)} - \frac{2 \pi}{3} \frac{2 \pi}{3}$$
=
              /  ___\
      2      2|\/ 2 |
-16*pi *atanh |-----|
              \  2  /
---------------------
          9          
$$- \frac{16 \pi^{2} \operatorname{atanh}^{2}{\left(\frac{\sqrt{2}}{2} \right)}}{9}$$
-16*pi^2*atanh(sqrt(2)/2)^2/9
Respuesta rápida [src]
     -2*pi
x1 = -----
       3  
$$x_{1} = - \frac{2 \pi}{3}$$
     2*pi
x2 = ----
      3  
$$x_{2} = \frac{2 \pi}{3}$$
               /  ___\
               |\/ 2 |
x3 = -2*I*atanh|-----|
               \  2  /
$$x_{3} = - 2 i \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)}$$
              /  ___\
              |\/ 2 |
x4 = 2*I*atanh|-----|
              \  2  /
$$x_{4} = 2 i \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)}$$
x4 = 2*i*atanh(sqrt(2)/2)
Respuesta numérica [src]
x1 = 64.9262481741891
x2 = -90.0589894029074
x3 = -46.0766922526503
x4 = -41.8879020478639
x5 = -83.7758040957278
x6 = -92.1533845053006
x7 = -58.6430628670095
x8 = -159.174027781883
x9 = -54.4542726622231
x10 = -23.0383461263252
x11 = -85.870199198121
x12 = 23.0383461263252
x13 = -64.9262481741891
x14 = 60.7374579694027
x15 = -29.3215314335047
x16 = -4.18879020478639
x17 = 20.943951023932
x18 = 2.0943951023932
x19 = -178.023583703422
x20 = 77.4926187885482
x21 = 90.0589894029074
x22 = 29.3215314335047
x23 = -16.7551608191456
x24 = 48.1710873550435
x25 = -20.943951023932
x26 = -60.7374579694027
x27 = -77.4926187885482
x28 = 39.7935069454707
x29 = -39.7935069454707
x30 = -2.0943951023932
x31 = -10.471975511966
x32 = -104.71975511966
x33 = -35.6047167406843
x34 = -79.5870138909414
x35 = -586.430628670095
x36 = 14.6607657167524
x37 = 58.6430628670095
x38 = -52.3598775598299
x39 = 79.5870138909414
x40 = 73.3038285837618
x41 = 52.3598775598299
x42 = 8.37758040957278
x43 = 54.4542726622231
x44 = 33.5103216382911
x45 = -98.4365698124802
x46 = 83.7758040957278
x47 = -33.5103216382911
x48 = 71.2094334813686
x49 = 4.18879020478639
x50 = 98.4365698124802
x51 = 96.342174710087
x52 = -14.6607657167524
x53 = -27.2271363311115
x54 = -96.342174710087
x55 = 16.7551608191456
x56 = 10.471975511966
x57 = 92.1533845053006
x58 = -73.3038285837618
x59 = 67.0206432765823
x60 = 35.6047167406843
x61 = -48.1710873550435
x62 = 85.870199198121
x63 = -67.0206432765823
x64 = -8.37758040957278
x65 = 41.8879020478639
x66 = 46.0766922526503
x67 = 27.2271363311115
x68 = -71.2094334813686
x68 = -71.2094334813686