Sr Examen
2sin2(x)+cos(x)-1=0 la ecuación
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Solución
Ha introducido
[src]
2 2*sin (x) + cos(x) - 1 = 0
$$\left(2 \sin^{2}{\left(x \right)} + \cos{\left(x \right)}\right) - 1 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(2 \sin^{2}{\left(x \right)} + \cos{\left(x \right)}\right) - 1 = 0$$
cambiamos
$$\cos{\left(x \right)} - \cos{\left(2 x \right)} = 0$$
$$- 2 \cos^{2}{\left(x \right)} + \cos{\left(x \right)} + 1 = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
Es la ecuación de la forma
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
$$a = -2$$
$$b = 1$$
$$c = 1$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = 1$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = \pi n + \frac{2 \pi}{3}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{2} = \pi n$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{3} = \pi n - \frac{\pi}{3}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{4} = \pi n - \pi$$
$$\left(2 \sin^{2}{\left(x \right)} + \cos{\left(x \right)}\right) - 1 = 0$$
cambiamos
$$\cos{\left(x \right)} - \cos{\left(2 x \right)} = 0$$
$$- 2 \cos^{2}{\left(x \right)} + \cos{\left(x \right)} + 1 = 0$$
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
$$a = -2$$
$$b = 1$$
$$c = 1$$
, entonces
D = b^2 - 4 * a * c =
(1)^2 - 4 * (-2) * (1) = 9
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = 1$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = \pi n + \frac{2 \pi}{3}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{2} = \pi n$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{3} = \pi n - \frac{\pi}{3}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{4} = \pi n - \pi$$
Respuesta rápida
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x1 = 0
$$x_{1} = 0$$
-2*pi
x2 = -----
3
$$x_{2} = - \frac{2 \pi}{3}$$
2*pi
x3 = ----
3
$$x_{3} = \frac{2 \pi}{3}$$
x3 = 2*pi/3
Suma y producto de raíces
[src]
suma
2*pi 2*pi - ---- + ---- 3 3
$$- \frac{2 \pi}{3} + \frac{2 \pi}{3}$$
=
0
$$0$$
producto
-2*pi 2*pi
0*-----*----
3 3
$$\frac{2 \pi}{3} \cdot 0 \left(- \frac{2 \pi}{3}\right)$$
=
0
$$0$$
0
Respuesta numérica
[src]
x1 = -81.6814090379303
x2 = -62.8318529503654
x3 = -46.0766922526503
x4 = -54.4542726622231
x5 = 37.6991120149696
x6 = -6.28318514161788
x7 = 6.28318528426584
x8 = -4.18879020478639
x9 = 90.0589894029074
x10 = 43.9822969706241
x11 = -79.5870138909414
x12 = 100.530964769014
x13 = 12.5663700882745
x14 = 8.37758040957278
x15 = 4.18879020478639
x16 = 98.4365698124802
x17 = -96.342174710087
x18 = -56.5486675394273
x19 = -8.37758040957278
x20 = 46.0766922526503
x21 = -18.8495555012277
x22 = -71.2094334813686
x23 = -41.8879020478639
x24 = -92.1533845053006
x25 = -62.8318537995483
x26 = -100.530964690899
x27 = -29.3215314335047
x28 = -50.2654822985064
x29 = 18.8495557025416
x30 = 77.4926187885482
x31 = 56.5486676119735
x32 = 39.7935069454707
x33 = -39.7935069454707
x34 = 58.6430628670095
x35 = 79.5870138909414
x36 = 0.0
x37 = -33.5103216382911
x38 = 33.5103216382911
x39 = 83.7758040957278
x40 = -43.9822971745925
x41 = -27.2271363311115
x42 = 31.4159267619367
x43 = -73.3038285837618
x44 = -18.8495558006412
x45 = 69.1150383780256
x46 = 25.13274122338
x47 = -48.1710873550435
x48 = 41.8879020478639
x49 = 25.1327417460082
x50 = 94.2477796093525
x51 = 25.1327411125589
x52 = -90.0589894029074
x53 = -85.870199198121
x54 = -77.4926187885482
x55 = -2.0943951023932
x56 = 43.9822971694142
x57 = -25.1327414474833
x58 = -35.6047167406843
x59 = 52.3598775598299
x60 = 54.4542726622231
x61 = -37.6991118771132
x62 = 96.342174710087
x63 = 81.6814091712551
x64 = -18.8495558410301
x65 = 85.870199198121
x66 = -62.8318529623378
x67 = -52.3598775598299
x68 = 69.115037832119
x69 = -12.5663703884691
x70 = -87.9645943588266
x71 = -83.7758040957278
x72 = -69.11503909537
x73 = 60.7374579694027
x74 = 25.1327412731354
x75 = 12.5663704551863
x76 = 62.8318528532238
x77 = 2.0943951023932
x78 = 48.1710873550435
x79 = -10.471975511966
x80 = 69.1150384283402
x81 = 87.9645943357073
x82 = 14.6607657167524
x83 = -94.2477794556977
x84 = -98.4365698124802
x85 = -69.1150385967809
x86 = -75.3982238575994
x87 = 16.7551608191456
x88 = 10.471975511966
x89 = 92.1533845053006
x90 = 50.2654824463501
x91 = -18.8495558711096
x92 = 69.1150383295746
x93 = 35.6047167406843
x94 = -31.4159267013407
x95 = 75.3982239117447
x95 = 75.3982239117447