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Expresiones idénticas
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- seno de en el grado dos 4𝑥− seno de 4𝑥−2 es igual a cero
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Expresiones con funciones
- Seno sin
- sin(x/3)=1/2
- sin(x-pi/3)=0
- sin(x/2)=1/2
- sin(x)=4
- sin(2*x)=-1/2
sin^24𝑥−sin4𝑥−2=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 sin (4*x) - sin(4*x) - 2 = 0
$$\left(\sin^{2}{\left(4 x \right)} - \sin{\left(4 x \right)}\right) - 2 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(\sin^{2}{\left(4 x \right)} - \sin{\left(4 x \right)}\right) - 2 = 0$$
cambiamos
$$- \sin{\left(4 x \right)} - \frac{\cos{\left(8 x \right)}}{2} - \frac{3}{2} = 0$$
$$\left(\sin^{2}{\left(4 x \right)} - \sin{\left(4 x \right)}\right) - 2 = 0$$
Sustituimos
$$w = \sin{\left(4 x \right)}$$
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
$$a = 1$$
$$b = -1$$
$$c = -2$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = 2$$
$$w_{2} = -1$$
hacemos cambio inverso
$$\sin{\left(4 x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(4 x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$4 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$4 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
Dividamos ambos miembros de la ecuación obtenida en
$$4$$
sustituimos w:
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(w_{1} \right)}}{4}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(w_{2} \right)}}{4}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(-1 \right)}}{4}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\pi}{8}$$
$$x_{3} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(w_{1} \right)}}{4} + \frac{\pi}{4}$$
$$x_{3} = \frac{\pi n}{2} + \frac{\pi}{4} - \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{3} = \frac{\pi n}{2} + \frac{\pi}{4} - \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{4} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(w_{2} \right)}}{4} + \frac{\pi}{4}$$
$$x_{4} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(-1 \right)}}{4} + \frac{\pi}{4}$$
$$x_{4} = \frac{\pi n}{2} + \frac{3 \pi}{8}$$
$$\left(\sin^{2}{\left(4 x \right)} - \sin{\left(4 x \right)}\right) - 2 = 0$$
cambiamos
$$- \sin{\left(4 x \right)} - \frac{\cos{\left(8 x \right)}}{2} - \frac{3}{2} = 0$$
$$\left(\sin^{2}{\left(4 x \right)} - \sin{\left(4 x \right)}\right) - 2 = 0$$
Sustituimos
$$w = \sin{\left(4 x \right)}$$
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
$$a = 1$$
$$b = -1$$
$$c = -2$$
, entonces
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (-2) = 9
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = 2$$
$$w_{2} = -1$$
hacemos cambio inverso
$$\sin{\left(4 x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(4 x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$4 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$4 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
Dividamos ambos miembros de la ecuación obtenida en
$$4$$
sustituimos w:
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(w_{1} \right)}}{4}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(w_{2} \right)}}{4}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(-1 \right)}}{4}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\pi}{8}$$
$$x_{3} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(w_{1} \right)}}{4} + \frac{\pi}{4}$$
$$x_{3} = \frac{\pi n}{2} + \frac{\pi}{4} - \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{3} = \frac{\pi n}{2} + \frac{\pi}{4} - \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{4} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(w_{2} \right)}}{4} + \frac{\pi}{4}$$
$$x_{4} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(-1 \right)}}{4} + \frac{\pi}{4}$$
$$x_{4} = \frac{\pi n}{2} + \frac{3 \pi}{8}$$
Respuesta rápida
[src]
-pi
x1 = ----
8
$$x_{1} = - \frac{\pi}{8}$$
3*pi
x2 = ----
8
$$x_{2} = \frac{3 \pi}{8}$$
re(asin(2)) pi I*im(asin(2))
x3 = - ----------- + -- - -------------
4 4 4
$$x_{3} = - \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4} + \frac{\pi}{4} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4}$$
re(asin(2)) I*im(asin(2))
x4 = ----------- + -------------
4 4
$$x_{4} = \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4}$$
x4 = re(asin(2))/4 + i*im(asin(2))/4
Suma y producto de raíces
[src]
suma
pi 3*pi re(asin(2)) pi I*im(asin(2)) re(asin(2)) I*im(asin(2)) - -- + ---- + - ----------- + -- - ------------- + ----------- + ------------- 8 8 4 4 4 4 4
$$\left(\frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4}\right) + \left(\left(- \frac{\pi}{8} + \frac{3 \pi}{8}\right) + \left(- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4} + \frac{\pi}{4} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4}\right)\right)$$
=
pi -- 2
$$\frac{\pi}{2}$$
producto
-pi 3*pi / re(asin(2)) pi I*im(asin(2))\ /re(asin(2)) I*im(asin(2))\ ----*----*|- ----------- + -- - -------------|*|----------- + -------------| 8 8 \ 4 4 4 / \ 4 4 /
$$- \frac{\pi}{8} \frac{3 \pi}{8} \left(- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4} + \frac{\pi}{4} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4}\right) \left(\frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4}\right)$$
=
2
3*pi *(I*im(asin(2)) + re(asin(2)))*(-pi + I*im(asin(2)) + re(asin(2)))
-----------------------------------------------------------------------
1024
$$\frac{3 \pi^{2} \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)}{1024}$$
3*pi^2*(i*im(asin(2)) + re(asin(2)))*(-pi + i*im(asin(2)) + re(asin(2)))/1024
Respuesta numérica
[src]
x1 = 32.5940236663725
x2 = -82.0741080211412
x3 = -67.9369412926542
x4 = 84.4303025069203
x5 = 56.1559687796345
x6 = -52.2289778464515
x7 = 302.770992033397
x8 = -1.96349547204618
x9 = 5.89048628952707
x10 = 64.0099503038507
x11 = -8.24668073366495
x12 = -55.3705706599853
x13 = 26.3108384365878
x14 = -30.2378292874754
x15 = -45.9457927193195
x16 = -556.45459894507
x17 = -744.950158056443
x18 = 78.1471173590006
x19 = 93.8550805924588
x20 = -1.96349557016575
x21 = 86.0010988906877
x22 = -53.7997741488835
x23 = -89.9280898521286
x24 = -17.6714587216416
x25 = 48.3019870161865
x26 = -75.7909227329734
x27 = -74.2201264093036
x28 = 62.4391539349309
x29 = 20.0276531232969
x30 = -45.9457926797218
x31 = 43.5895979321204
x32 = -97.7820713164945
x33 = -47.5165888974914
x34 = 51.4435795572927
x35 = 13.7444678134058
x36 = -60.0829594427331
x37 = -6.67588458616306
x38 = -91.4988860046927
x39 = -96.2112749750547
x40 = 34.1648201938636
x41 = -3.53429177195145
x42 = 57.7267648828298
x43 = 70.2931355961025
x44 = -39.6626073002566
x45 = 49.8727834419781
x46 = -17.6714586996829
x47 = 4.31968985730964
x48 = 12.1736716073441
x49 = 27.8816348659918
x50 = 40.448005364229
x51 = 101.709061988899
x52 = -25.525440336424
x53 = -31.8086255641596
x54 = -9.81747697874728
x55 = 42.0188017149397
x56 = 35.7356163522263
x57 = -61.6537558785227
x58 = -23.954644072489
x59 = -83.6449044564232
x60 = 18.4568567951268
x61 = 98.567469621515
x62 = -38.091810864829
x63 = 92.2842841763354
x64 = 79.7179134126456
x65 = -85.215700846439
x66 = -16.1006622874121
x67 = -69.5077374540623
x68 = 100.138265889851
x69 = 71.8639320174744
x69 = 71.8639320174744
Gráfico