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Ecuación (x-3)*(x+2)=0
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Ecuación (11x+14)-(5x-8)=25
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Ecuación z^2-6*z+10=0
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Ecuación 4/(x-4)=-5
- Expresar {x} en función de y en la ecuación:
- -7*x-13*y=5
- 15*x+1*y=19
- 7*x-1*y=-7
- -11*x-14*y=-3
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Expresiones idénticas
- sin(x)*(dos *sin(x)- uno)+sqrt(tres)*sin(x)+sin(cuatro *pi/ tres)= cero
- seno de (x) multiplicar por (2 multiplicar por seno de (x) menos 1) más raíz cuadrada de (3) multiplicar por seno de (x) más seno de (4 multiplicar por número pi dividir por 3) es igual a 0
- seno de (x) multiplicar por (dos multiplicar por seno de (x) menos uno) más raíz cuadrada de (tres) multiplicar por seno de (x) más seno de (cuatro multiplicar por número pi dividir por tres) es igual a cero
- sin(x)*(2*sin(x)-1)+√(3)*sin(x)+sin(4*pi/3)=0
- sin(x)(2sin(x)-1)+sqrt(3)sin(x)+sin(4pi/3)=0
- sinx2sinx-1+sqrt3sinx+sin4pi/3=0
- sin(x)*(2*sin(x)-1)+sqrt(3)*sin(x)+sin(4*pi/3)=O
- sin(x)*(2*sin(x)-1)+sqrt(3)*sin(x)+sin(4*pi dividir por 3)=0
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Expresiones semejantes
- sin(x)*(2*sin(x)+1)+sqrt(3)*sin(x)+sin(4*pi/3)=0
- sin(x)*(2*sin(x)-1)+sqrt(3)*sin(x)-sin(4*pi/3)=0
- sin(x)*(2*sin(x)-1)-sqrt(3)*sin(x)+sin(4*pi/3)=0
- sinx*(2*sinx-1)+sqrt(3)*sinx+sin(4*pi/3)=0
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Expresiones con funciones
- Seno sin
- sin(x)=(-1)/sqrt(2)
- sin(x)=-5/2
- sin((pi*x)/4)=1
- sin(2*x)-sin(x)=0
- sin(5*x)-cos(4*x)=0
- Seno sin
- sin(x)=(-1)/sqrt(2)
- sin(x)=-5/2
- sin((pi*x)/4)=1
- sin(2*x)-sin(x)=0
- sin(5*x)-cos(4*x)=0
- Raíz cuadrada sqrt
- sqrt(x)=1
- sqrt(x-2)+sqrt(y-2)=2
- sqrt(x^2+9)=2*x-3
- sqrt(x)=-2
- sqrt(2-x)+sqrt(-x-1)=sqrt(-5*x-7)
- Seno sin
- sin(x)=(-1)/sqrt(2)
- sin(x)=-5/2
- sin((pi*x)/4)=1
- sin(2*x)-sin(x)=0
- sin(5*x)-cos(4*x)=0
- Seno sin
- sin(x)=(-1)/sqrt(2)
- sin(x)=-5/2
- sin((pi*x)/4)=1
- sin(2*x)-sin(x)=0
- sin(5*x)-cos(4*x)=0
sin(x)*(2*sin(x)-1)+sqrt(3)*sin(x)+sin(4*pi/3)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
___ /4*pi\
sin(x)*(2*sin(x) - 1) + \/ 3 *sin(x) + sin|----| = 0
\ 3 /
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} + \sqrt{3} \sin{\left(x \right)}\right) + \sin{\left(\frac{4 \pi}{3} \right)} = 0$$
Solución detallada
Tenemos la ecuación
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} + \sqrt{3} \sin{\left(x \right)}\right) + \sin{\left(\frac{4 \pi}{3} \right)} = 0$$
cambiamos
$$2 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + \sqrt{3} \sin{\left(x \right)} - \frac{\sqrt{3}}{2} = 0$$
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} + \sqrt{3} \sin{\left(x \right)}\right) + \sin{\left(\frac{4 \pi}{3} \right)} = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Abramos la expresión en la ecuación
$$w \left(2 w - 1\right) + \sqrt{3} w - \frac{\sqrt{3}}{2} = 0$$
Obtenemos la ecuación cuadrática
$$2 w^{2} - w + \sqrt{3} w - \frac{\sqrt{3}}{2} = 0$$
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
$$a = 2$$
$$b = -1 + \sqrt{3}$$
$$c = - \frac{\sqrt{3}}{2}$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \frac{\sqrt{3}}{4} + \frac{1}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4}$$
$$w_{2} = - \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{1}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{1}{4} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{4} + \frac{\sqrt{3}}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{1}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{4} + \frac{\sqrt{3}}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} \right)} + \pi$$
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} + \sqrt{3} \sin{\left(x \right)}\right) + \sin{\left(\frac{4 \pi}{3} \right)} = 0$$
cambiamos
$$2 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + \sqrt{3} \sin{\left(x \right)} - \frac{\sqrt{3}}{2} = 0$$
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} + \sqrt{3} \sin{\left(x \right)}\right) + \sin{\left(\frac{4 \pi}{3} \right)} = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Abramos la expresión en la ecuación
$$w \left(2 w - 1\right) + \sqrt{3} w - \frac{\sqrt{3}}{2} = 0$$
Obtenemos la ecuación cuadrática
$$2 w^{2} - w + \sqrt{3} w - \frac{\sqrt{3}}{2} = 0$$
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
$$a = 2$$
$$b = -1 + \sqrt{3}$$
$$c = - \frac{\sqrt{3}}{2}$$
, entonces
D = b^2 - 4 * a * c =
(-1 + sqrt(3))^2 - 4 * (2) * (-sqrt(3)/2) = (-1 + sqrt(3))^2 + 4*sqrt(3)
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{\sqrt{3}}{4} + \frac{1}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4}$$
$$w_{2} = - \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{1}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{1}{4} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{4} + \frac{\sqrt{3}}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{1}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{4} + \frac{\sqrt{3}}{4} + \frac{\sqrt{\left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3}}}{4} \right)} + \pi$$
Respuesta rápida
[src]
-pi
x1 = ----
3
$$x_{1} = - \frac{\pi}{3}$$
pi
x2 = --
6
$$x_{2} = \frac{\pi}{6}$$
5*pi
x3 = ----
6
$$x_{3} = \frac{5 \pi}{6}$$
4*pi
x4 = ----
3
$$x_{4} = \frac{4 \pi}{3}$$
x4 = 4*pi/3
Suma y producto de raíces
[src]
suma
pi pi 5*pi 4*pi - -- + -- + ---- + ---- 3 6 6 3
$$\left(\left(- \frac{\pi}{3} + \frac{\pi}{6}\right) + \frac{5 \pi}{6}\right) + \frac{4 \pi}{3}$$
=
2*pi
$$2 \pi$$
producto
-pi pi 5*pi 4*pi ----*--*----*---- 3 6 6 3
$$\frac{4 \pi}{3} \frac{5 \pi}{6} \cdot - \frac{\pi}{3} \frac{\pi}{6}$$
=
4 -5*pi ------ 81
$$- \frac{5 \pi^{4}}{81}$$
-5*pi^4/81
Respuesta numérica
[src]
x1 = -39.7935069454707
x2 = 69.6386371545737
x3 = 25.6563400043166
x4 = -60.2138591938044
x5 = 4.18879020478639
x6 = 84.2994028713261
x7 = 54.4542726622231
x8 = 31.9395253114962
x9 = 24.0855436775217
x10 = 44.5058959258554
x11 = -91.6297857297023
x12 = 17.8023583703422
x13 = 11.5191730631626
x14 = -57.5958653158129
x15 = -49.7418836818384
x16 = 41.8879020478639
x17 = 2.61799387799149
x18 = 8.90117918517108
x19 = 61.7846555205993
x20 = -33.5103216382911
x21 = 27.7507351067098
x22 = -85.3466004225227
x23 = -79.0634151153431
x24 = -97.9129710368819
x25 = -18.3259571459405
x26 = -43.4586983746588
x27 = 86.9173967493176
x28 = -284.837733925475
x29 = 88.4881930761125
x30 = 16.7551608191456
x31 = -62.3082542961976
x32 = -9.94837673636768
x33 = 46.6002910282486
x34 = -5.75958653158129
x35 = -51.3126800086333
x36 = -100.007366139275
x37 = -135.612082879959
x38 = -26.1799387799149
x39 = -32.4631240870945
x40 = 52.8834763354282
x41 = 20140.2269033886
x42 = -87.4409955249159
x43 = 85.870199198121
x44 = -93.7241808320955
x45 = 60.7374579694027
x46 = -77.4926187885482
x47 = -46.0766922526503
x48 = 10.471975511966
x49 = 78.0162175641465
x50 = 82.2050077689329
x51 = -83.7758040957278
x52 = -13.6135681655558
x53 = -53.9306738866248
x54 = -19.8967534727354
x55 = 55.5014702134197
x56 = 63.3554518473942
x57 = 0.523598775598299
x58 = -16.2315620435473
x59 = -56.025068989018
x60 = 1298.52496348378
x61 = -63.8790506229925
x62 = -27.2271363311115
x63 = 30.3687289847013
x64 = 92.1533845053006
x65 = 74.3510261349584
x66 = 48.1710873550435
x67 = 75.9218224617533
x68 = -2.0943951023932
x69 = 99.4837673636768
x70 = -90.0589894029074
x71 = -35.081117965086
x72 = 19.3731546971371
x73 = -68.5914396033772
x74 = 71.733032256967
x75 = -12.0427718387609
x76 = -24.60914245312
x77 = 90.5825881785057
x78 = 98.4365698124802
x79 = -76.4454212373516
x80 = 38.2227106186758
x81 = 40.317105721069
x82 = 13.0899693899575
x83 = -41.3643032722656
x84 = 129.852496348378
x85 = -47.6474885794452
x86 = -3.66519142918809
x87 = -5022.35945553888
x88 = 96.8657734856853
x89 = 34.0339204138894
x90 = 68.0678408277789
x91 = -71.2094334813686
x92 = -70.162235930172
x92 = -70.162235930172