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Expresiones semejantes
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(cos(2*x)+4)*(cos(2*x)+sqrt(2)/2)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
/ ___\
| \/ 2 |
(cos(2*x) + 4)*|cos(2*x) + -----| = 0
\ 2 /
$$\left(\cos{\left(2 x \right)} + 4\right) \left(\cos{\left(2 x \right)} + \frac{\sqrt{2}}{2}\right) = 0$$
Solución detallada
Tenemos la ecuación
$$\left(\cos{\left(2 x \right)} + 4\right) \left(\cos{\left(2 x \right)} + \frac{\sqrt{2}}{2}\right) = 0$$
cambiamos
$$\frac{\left(\cos{\left(2 x \right)} + 4\right) \left(2 \cos{\left(2 x \right)} + \sqrt{2}\right)}{2} = 0$$
$$\left(\cos{\left(2 x \right)} + 4\right) \left(\cos{\left(2 x \right)} + \frac{\sqrt{2}}{2}\right) = 0$$
Sustituimos
$$w = \cos{\left(2 x \right)}$$
Abramos la expresión en la ecuación
$$\left(w + 4\right) \left(w + \frac{\sqrt{2}}{2}\right) = 0$$
Obtenemos la ecuación cuadrática
$$w^{2} + \frac{\sqrt{2} w}{2} + 4 w + 2 \sqrt{2} = 0$$
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
$$a = 1$$
$$b = \frac{\sqrt{2}}{2} + 4$$
$$c = 2 \sqrt{2}$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = -2 - \frac{\sqrt{2}}{4} + \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2}$$
$$w_{2} = -2 - \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} - \frac{\sqrt{2}}{4}$$
hacemos cambio inverso
$$\cos{\left(2 x \right)} = w$$
Tenemos la ecuación
$$\cos{\left(2 x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$2 x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$2 x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
O
$$2 x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$2 x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, donde n es cualquier número entero
Dividamos ambos miembros de la ecuación obtenida en
$$2$$
sustituimos w:
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(w_{1} \right)}}{2}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{2}}{4} + \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} \right)}}{2}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{2}}{4} + \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} \right)}}{2}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(w_{2} \right)}}{2}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} - \frac{\sqrt{2}}{4} \right)}}{2}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} - \frac{\sqrt{2}}{4} \right)}}{2}$$
$$x_{3} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(w_{1} \right)}}{2} - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi n}{2} - \frac{\pi}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{2}}{4} + \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} \right)}}{2}$$
$$x_{3} = \frac{\pi n}{2} - \frac{\pi}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{2}}{4} + \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} \right)}}{2}$$
$$x_{4} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(w_{2} \right)}}{2} - \frac{\pi}{2}$$
$$x_{4} = \frac{\pi n}{2} - \frac{\pi}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} - \frac{\sqrt{2}}{4} \right)}}{2}$$
$$x_{4} = \frac{\pi n}{2} - \frac{\pi}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} - \frac{\sqrt{2}}{4} \right)}}{2}$$
$$\left(\cos{\left(2 x \right)} + 4\right) \left(\cos{\left(2 x \right)} + \frac{\sqrt{2}}{2}\right) = 0$$
cambiamos
$$\frac{\left(\cos{\left(2 x \right)} + 4\right) \left(2 \cos{\left(2 x \right)} + \sqrt{2}\right)}{2} = 0$$
$$\left(\cos{\left(2 x \right)} + 4\right) \left(\cos{\left(2 x \right)} + \frac{\sqrt{2}}{2}\right) = 0$$
Sustituimos
$$w = \cos{\left(2 x \right)}$$
Abramos la expresión en la ecuación
$$\left(w + 4\right) \left(w + \frac{\sqrt{2}}{2}\right) = 0$$
Obtenemos la ecuación cuadrática
$$w^{2} + \frac{\sqrt{2} w}{2} + 4 w + 2 \sqrt{2} = 0$$
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
$$a = 1$$
$$b = \frac{\sqrt{2}}{2} + 4$$
$$c = 2 \sqrt{2}$$
, entonces
D = b^2 - 4 * a * c =
(4 + sqrt(2)/2)^2 - 4 * (1) * (2*sqrt(2)) = (4 + sqrt(2)/2)^2 - 8*sqrt(2)
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = -2 - \frac{\sqrt{2}}{4} + \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2}$$
$$w_{2} = -2 - \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} - \frac{\sqrt{2}}{4}$$
hacemos cambio inverso
$$\cos{\left(2 x \right)} = w$$
Tenemos la ecuación
$$\cos{\left(2 x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$2 x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$2 x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
O
$$2 x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$2 x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, donde n es cualquier número entero
Dividamos ambos miembros de la ecuación obtenida en
$$2$$
sustituimos w:
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(w_{1} \right)}}{2}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{2}}{4} + \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} \right)}}{2}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{2}}{4} + \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} \right)}}{2}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(w_{2} \right)}}{2}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} - \frac{\sqrt{2}}{4} \right)}}{2}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} - \frac{\sqrt{2}}{4} \right)}}{2}$$
$$x_{3} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(w_{1} \right)}}{2} - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi n}{2} - \frac{\pi}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{2}}{4} + \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} \right)}}{2}$$
$$x_{3} = \frac{\pi n}{2} - \frac{\pi}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{2}}{4} + \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} \right)}}{2}$$
$$x_{4} = \frac{\pi n}{2} + \frac{\operatorname{acos}{\left(w_{2} \right)}}{2} - \frac{\pi}{2}$$
$$x_{4} = \frac{\pi n}{2} - \frac{\pi}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} - \frac{\sqrt{2}}{4} \right)}}{2}$$
$$x_{4} = \frac{\pi n}{2} - \frac{\pi}{2} + \frac{\operatorname{acos}{\left(-2 - \frac{\sqrt{- 8 \sqrt{2} + \left(\frac{\sqrt{2}}{2} + 4\right)^{2}}}{2} - \frac{\sqrt{2}}{4} \right)}}{2}$$
Respuesta rápida
[src]
3*pi
x1 = ----
8
$$x_{1} = \frac{3 \pi}{8}$$
5*pi
x2 = ----
8
$$x_{2} = \frac{5 \pi}{8}$$
re(acos(-4)) I*im(acos(-4))
x3 = pi - ------------ - --------------
2 2
$$x_{3} = - \frac{\operatorname{re}{\left(\operatorname{acos}{\left(-4 \right)}\right)}}{2} + \pi - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(-4 \right)}\right)}}{2}$$
re(acos(-4)) I*im(acos(-4))
x4 = ------------ + --------------
2 2
$$x_{4} = \frac{\operatorname{re}{\left(\operatorname{acos}{\left(-4 \right)}\right)}}{2} + \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(-4 \right)}\right)}}{2}$$
x4 = re(acos(-4))/2 + i*im(acos(-4))/2
Suma y producto de raíces
[src]
suma
3*pi 5*pi re(acos(-4)) I*im(acos(-4)) re(acos(-4)) I*im(acos(-4)) ---- + ---- + pi - ------------ - -------------- + ------------ + -------------- 8 8 2 2 2 2
$$\left(\frac{\operatorname{re}{\left(\operatorname{acos}{\left(-4 \right)}\right)}}{2} + \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(-4 \right)}\right)}}{2}\right) + \left(\left(\frac{3 \pi}{8} + \frac{5 \pi}{8}\right) + \left(- \frac{\operatorname{re}{\left(\operatorname{acos}{\left(-4 \right)}\right)}}{2} + \pi - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(-4 \right)}\right)}}{2}\right)\right)$$
=
2*pi
$$2 \pi$$
producto
3*pi 5*pi / re(acos(-4)) I*im(acos(-4))\ /re(acos(-4)) I*im(acos(-4))\ ----*----*|pi - ------------ - --------------|*|------------ + --------------| 8 8 \ 2 2 / \ 2 2 /
$$\frac{3 \pi}{8} \frac{5 \pi}{8} \left(- \frac{\operatorname{re}{\left(\operatorname{acos}{\left(-4 \right)}\right)}}{2} + \pi - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(-4 \right)}\right)}}{2}\right) \left(\frac{\operatorname{re}{\left(\operatorname{acos}{\left(-4 \right)}\right)}}{2} + \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(-4 \right)}\right)}}{2}\right)$$
=
2
-15*pi *(I*im(acos(-4)) + re(acos(-4)))*(-2*pi + I*im(acos(-4)) + re(acos(-4)))
-------------------------------------------------------------------------------
256
$$- \frac{15 \pi^{2} \left(\operatorname{re}{\left(\operatorname{acos}{\left(-4 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-4 \right)}\right)}\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(-4 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-4 \right)}\right)}\right)}{256}$$
-15*pi^2*(i*im(acos(-4)) + re(acos(-4)))*(-2*pi + i*im(acos(-4)) + re(acos(-4)))/256
Respuesta numérica
[src]
x1 = -27.096236637212
x2 = 70.2931356240716
x3 = -33.3794219443916
x4 = -124.485608898496
x5 = 26.3108384738145
x6 = 17.6714586764426
x7 = 33.3794219443916
x8 = -55.3705705195201
x9 = -92.2842841992002
x10 = -67.9369411338793
x11 = -83.6449044018282
x12 = 86.0010988920206
x13 = 48.3019870489431
x14 = 98.5674695063798
x15 = 111.133840120739
x16 = 8.24668071567321
x17 = 52.2289778659303
x18 = -52.2289778659303
x19 = 42.0188017417635
x20 = -45.9457925587507
x21 = 92.2842841992002
x22 = 74.2201264410589
x23 = 39.6626072515711
x24 = -39.6626072515711
x25 = 16.8860605130451
x26 = 76.5763209312512
x27 = -26.3108384738145
x28 = -99.3528676697772
x29 = -61.6537558266997
x30 = -57.7267650097125
x31 = 54.5851723561227
x32 = -35.7356164345839
x33 = 11.388273369263
x34 = 55.3705705195201
x35 = -736.310778185108
x36 = -49.0873852123405
x37 = 60.8683576633022
x38 = 67.1515429704818
x39 = 5.10508806208341
x40 = 99.3528676697772
x41 = 83.6449044018282
x42 = 1.96349540849362
x43 = -98.5674695063798
x44 = 4.31968989868597
x45 = -64.7953484802895
x46 = -76.5763209312512
x47 = 64.009950316892
x48 = -93.0696823625976
x49 = -96.2112750161874
x50 = -54.5851723561227
x51 = -4.31968989868597
x52 = 20.0276531666349
x53 = 27.096236637212
x54 = 38.8772090881737
x55 = 79.717913584841
x56 = 61.6537558266997
x57 = -71.0785337874691
x58 = -89.9280897090078
x59 = -48.3019870489431
x60 = 23.9546439836222
x61 = -86.0010988920206
x62 = -42.0188017417635
x63 = 45.9457925587507
x64 = -74.2201264410589
x65 = 96.2112750161874
x66 = -10.6028752058656
x67 = 32.5940237809941
x68 = -13.7444678594553
x69 = -5.10508806208341
x70 = 10.6028752058656
x71 = -161.399322578176
x72 = -17.6714586764426
x73 = -23.9546439836222
x74 = 77.3617190946487
x75 = -64.009950316892
x76 = 82.8595062384308
x77 = -77.3617190946487
x78 = -1.96349540849362
x79 = -20.0276531666349
x80 = 89.9280897090078
x81 = -70.2931356240716
x82 = 67.9369411338793
x83 = 30.2378292908018
x84 = -32.5940237809941
x85 = -11.388273369263
x86 = -79.717913584841
x86 = -79.717913584841