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- La ecuación:
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Ecuación x^2-11*x+30=0
- Ecuación x^2+y^2=9
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Ecuación -33-y=-19
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Ecuación 11/(x-9)=-10
- Expresar {x} en función de y en la ecuación:
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Expresiones idénticas
- 2sin^2x-11sinx+ nueve = cero
- 2 seno de al cuadrado x menos 11 seno de x más 9 es igual a 0
- 2 seno de al cuadrado x menos 11 seno de x más nueve es igual a cero
- 2sin2x-11sinx+9=0
- 2sin²x-11sinx+9=0
- 2sin en el grado 2x-11sinx+9=0
- 2sin^2x-11sinx+9=O
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Expresiones semejantes
- 2sin^2x-11sinx-9=0
- 2sin^2x+11sinx+9=0
2sin^2x-11sinx+9=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2*sin (x) - 11*sin(x) + 9 = 0
$$\left(2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) + 9 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) + 9 = 0$$
cambiamos
$$2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)} + 9 = 0$$
$$\left(2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) + 9 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
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
$$a = 2$$
$$b = -11$$
$$c = 9$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = \frac{9}{2}$$
$$w_{2} = 1$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{\pi}{2}$$
$$\left(2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) + 9 = 0$$
cambiamos
$$2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)} + 9 = 0$$
$$\left(2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) + 9 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
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
$$a = 2$$
$$b = -11$$
$$c = 9$$
, entonces
D = b^2 - 4 * a * c =
(-11)^2 - 4 * (2) * (9) = 49
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{9}{2}$$
$$w_{2} = 1$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{\pi}{2}$$
Respuesta rápida
[src]
pi
x1 = --
2
$$x_{1} = \frac{\pi}{2}$$
x2 = pi - re(asin(9/2)) - I*im(asin(9/2))
$$x_{2} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}$$
x3 = I*im(asin(9/2)) + re(asin(9/2))
$$x_{3} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}$$
x3 = re(asin(9/2)) + i*im(asin(9/2))
Suma y producto de raíces
[src]
suma
pi -- + pi - re(asin(9/2)) - I*im(asin(9/2)) + I*im(asin(9/2)) + re(asin(9/2)) 2
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}\right) + \left(\frac{\pi}{2} + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}\right)\right)$$
=
3*pi ---- 2
$$\frac{3 \pi}{2}$$
producto
pi --*(pi - re(asin(9/2)) - I*im(asin(9/2)))*(I*im(asin(9/2)) + re(asin(9/2))) 2
$$\frac{\pi}{2} \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}\right)$$
=
-pi*(I*im(asin(9/2)) + re(asin(9/2)))*(-pi + I*im(asin(9/2)) + re(asin(9/2)))
------------------------------------------------------------------------------
2
$$- \frac{\pi \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}\right)}{2}$$
-pi*(i*im(asin(9/2)) + re(asin(9/2)))*(-pi + i*im(asin(9/2)) + re(asin(9/2)))/2
Respuesta numérica
[src]
x1 = -92.6769828953542
x2 = -98.9601680628786
x3 = -61.2610570080406
x4 = 20.4203521467171
x5 = 83.2522048029739
x6 = 89.5353905863307
x7 = 39.2699085820684
x8 = 70.6858346223605
x9 = -626.747734692733
x10 = 45.5530937462752
x11 = 45.5530935153116
x12 = -4.71238857705888
x13 = -10.9955737455676
x14 = -92.6769838313521
x15 = -80.1106125773791
x16 = -54.9778712582553
x17 = -17.2787595215844
x18 = 14.1371671126075
x19 = 1.57079658701878
x20 = 89.5353909055217
x21 = -23.561945012858
x22 = 39.2699076443101
x23 = -54.9778718424277
x24 = -48.6946857362003
x25 = 26.7035375458614
x26 = -54.977870904196
x27 = -86.3937977315325
x28 = -29.845130094999
x29 = -98.9601690014881
x30 = 32.9867224757296
x31 = -86.3937977440352
x32 = 7.85398174444133
x33 = -73.8274272797355
x34 = -10.9955746833544
x35 = 32.986723413003
x36 = 1.57079643579021
x37 = 70.6858344692885
x38 = -4.71238951430128
x39 = -67.5442426460742
x40 = -61.2610565782138
x41 = 51.8362789048843
x42 = 95.8185760651006
x43 = 58.1194643884877
x44 = -48.6946866728566
x45 = -67.5442421723436
x46 = 76.9690196348227
x47 = -42.4115006509881
x48 = 26.7035373101781
x49 = 76.9690205715119
x50 = -42.4115005722496
x51 = 83.2522057411799
x52 = 64.4026493064007
x53 = -36.1283154162869
x54 = -17.2787598489313
x54 = -17.2787598489313