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Expresiones idénticas
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- (log(sin(x))^2+log(sin(x)))/(2cos(x)+√3)=0
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Expresiones semejantes
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Expresiones con funciones
- Logaritmo log
- log(5,x^(2*y))+log(e,(x+y)^5)=17
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- sin(x)=1/2
- sin(x)-sqrt(2)/2=0
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- sin(x)=-0,5
- sin(x)=-1/7
- Logaritmo log
- log(5,x^(2*y))+log(e,(x+y)^5)=17
- log(x^2-x-3)=log(x)
- log(y)-3+4/y=0
- log2(2𝑥−3)=4
- log(15,(x+7)^2)=0
- Seno sin
- sin(x)=1/2
- sin(x)-sqrt(2)/2=0
- sin(x)/x=1
- sin(x)=-0,5
- sin(x)=-1/7
(log(sin(x))^2+log(sin(x)))/(2cos(x)+sqrt3)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2
log (sin(x)) + log(sin(x))
-------------------------- = 0
___
2*cos(x) + \/ 3
$$\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} + \log{\left(\sin{\left(x \right)} \right)}}{2 \cos{\left(x \right)} + \sqrt{3}} = 0$$
Solución detallada
Tenemos la ecuación
$$\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} + \log{\left(\sin{\left(x \right)} \right)}}{2 \cos{\left(x \right)} + \sqrt{3}} = 0$$
cambiamos
$$\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} + \log{\left(\sin{\left(x \right)} \right)} - 2 \cos{\left(x \right)} - \sqrt{3}}{2 \cos{\left(x \right)} + \sqrt{3}} = 0$$
$$\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} + \log{\left(\sin{\left(x \right)} \right)}}{2 \cos{\left(x \right)} + \sqrt{3}} - 1 = 0$$
Sustituimos
$$w = \log{\left(\sin{\left(x \right)} \right)}$$
Abramos la expresión en la ecuación
$$\frac{w^{2} + w}{2 \cos{\left(x \right)} + \sqrt{3}} - 1 = 0$$
Obtenemos la ecuación cuadrática
$$\frac{w^{2}}{2 \cos{\left(x \right)} + \sqrt{3}} + \frac{w}{2 \cos{\left(x \right)} + \sqrt{3}} - 1 = 0$$
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
$$a = \frac{1}{2 \cos{\left(x \right)} + \sqrt{3}}$$
$$b = \frac{1}{2 \cos{\left(x \right)} + \sqrt{3}}$$
$$c = -1$$
, entonces
La ecuación tiene dos raíces.
o
$$w_{1} = \left(\sqrt{\frac{4}{2 \cos{\left(x \right)} + \sqrt{3}} + \frac{1}{\left(2 \cos{\left(x \right)} + \sqrt{3}\right)^{2}}} - \frac{1}{2 \cos{\left(x \right)} + \sqrt{3}}\right) \left(\cos{\left(x \right)} + \frac{\sqrt{3}}{2}\right)$$
$$w_{2} = \left(- \sqrt{\frac{4}{2 \cos{\left(x \right)} + \sqrt{3}} + \frac{1}{\left(2 \cos{\left(x \right)} + \sqrt{3}\right)^{2}}} - \frac{1}{2 \cos{\left(x \right)} + \sqrt{3}}\right) \left(\cos{\left(x \right)} + \frac{\sqrt{3}}{2}\right)$$
hacemos cambio inverso
$$\log{\left(\sin{\left(x \right)} \right)} = w$$
sustituimos w:
$$\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} + \log{\left(\sin{\left(x \right)} \right)}}{2 \cos{\left(x \right)} + \sqrt{3}} = 0$$
cambiamos
$$\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} + \log{\left(\sin{\left(x \right)} \right)} - 2 \cos{\left(x \right)} - \sqrt{3}}{2 \cos{\left(x \right)} + \sqrt{3}} = 0$$
$$\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} + \log{\left(\sin{\left(x \right)} \right)}}{2 \cos{\left(x \right)} + \sqrt{3}} - 1 = 0$$
Sustituimos
$$w = \log{\left(\sin{\left(x \right)} \right)}$$
Abramos la expresión en la ecuación
$$\frac{w^{2} + w}{2 \cos{\left(x \right)} + \sqrt{3}} - 1 = 0$$
Obtenemos la ecuación cuadrática
$$\frac{w^{2}}{2 \cos{\left(x \right)} + \sqrt{3}} + \frac{w}{2 \cos{\left(x \right)} + \sqrt{3}} - 1 = 0$$
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
$$a = \frac{1}{2 \cos{\left(x \right)} + \sqrt{3}}$$
$$b = \frac{1}{2 \cos{\left(x \right)} + \sqrt{3}}$$
$$c = -1$$
, entonces
D = b^2 - 4 * a * c =
(1/(sqrt(3) + 2*cos(x)))^2 - 4 * (1/(sqrt(3) + 2*cos(x))) * (-1) = (sqrt(3) + 2*cos(x))^(-2) + 4/(sqrt(3) + 2*cos(x))
La ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \left(\sqrt{\frac{4}{2 \cos{\left(x \right)} + \sqrt{3}} + \frac{1}{\left(2 \cos{\left(x \right)} + \sqrt{3}\right)^{2}}} - \frac{1}{2 \cos{\left(x \right)} + \sqrt{3}}\right) \left(\cos{\left(x \right)} + \frac{\sqrt{3}}{2}\right)$$
$$w_{2} = \left(- \sqrt{\frac{4}{2 \cos{\left(x \right)} + \sqrt{3}} + \frac{1}{\left(2 \cos{\left(x \right)} + \sqrt{3}\right)^{2}}} - \frac{1}{2 \cos{\left(x \right)} + \sqrt{3}}\right) \left(\cos{\left(x \right)} + \frac{\sqrt{3}}{2}\right)$$
hacemos cambio inverso
$$\log{\left(\sin{\left(x \right)} \right)} = w$$
sustituimos w:
Suma y producto de raíces
[src]
suma
pi / -1\ / -1\ -- + pi - asin\e / + asin\e / 2
$$\operatorname{asin}{\left(e^{-1} \right)} + \left(\frac{\pi}{2} + \left(\pi - \operatorname{asin}{\left(e^{-1} \right)}\right)\right)$$
=
3*pi ---- 2
$$\frac{3 \pi}{2}$$
producto
pi / / -1\\ / -1\ --*\pi - asin\e //*asin\e / 2
$$\frac{\pi}{2} \left(\pi - \operatorname{asin}{\left(e^{-1} \right)}\right) \operatorname{asin}{\left(e^{-1} \right)}$$
=
/ / -1\\ / -1\
pi*\pi - asin\e //*asin\e /
-----------------------------
2
$$\frac{\pi \left(\pi - \operatorname{asin}{\left(e^{-1} \right)}\right) \operatorname{asin}{\left(e^{-1} \right)}}{2}$$
pi*(pi - asin(exp(-1)))*asin(exp(-1))/2
Respuesta rápida
[src]
pi
x1 = --
2
$$x_{1} = \frac{\pi}{2}$$
/ -1\ x2 = pi - asin\e /
$$x_{2} = \pi - \operatorname{asin}{\left(e^{-1} \right)}$$
/ -1\ x3 = asin\e /
$$x_{3} = \operatorname{asin}{\left(e^{-1} \right)}$$
x3 = asin(exp(-1))
Respuesta numérica
[src]
x1 = -29.8451300518943
x2 = -42.411500299771
x3 = -80.110612554579
x4 = 56.9253952726749
x5 = 44.3590246583157
x6 = -4.71238702922319
x7 = 95.8185761186241
x8 = -61.2610568471581
x9 = 7.85398178106424
x10 = 82.0581365013932
x11 = -17.278761192494 + 1.98027421190097e-7*i
x12 = -53.7838026190851
x13 = 78.1630888316863
x14 = 45.5530940785541
x15 = 71.8799035245067
x16 = -73.8274272602741
x17 = -9.80150546882795
x18 = 51.8362789495553
x19 = -61.2610583875251 + 1.45226645040729e-7*i
x20 = -86.3937974853806
x21 = 14.13716719037 - 2.2041050507287e-10*i
x22 = -5.90645779912101
x23 = -36.1283153869449
x24 = 1.57079688812637
x25 = 64.4026493258375
x26 = 70.6858331766713 + 2.97860890603865e-7*i
x27 = 58.1194653651731
x28 = -16.0846907760075
x29 = -23.5619464124793 + 8.19698217858016e-7*i
x30 = -48.6946843694447
x31 = 64.4026478403599 + 8.64539421423235e-7*i
x32 = 38.0758393511361
x33 = -23.5619449852535
x34 = -92.6769816799869
x35 = -97.7660997693422
x36 = 26.7035359951511 + 2.48909677892814e-7*i
x37 = -98.9601681904238 - 3.55787977430897e-7*i
x38 = -67.5442421402291
x39 = 26.703537444496
x40 = -67.544243542434 + 7.9364212507756e-7*i
x41 = 34.1807916814291
x42 = -87.5878667924556
x43 = 89.5353912717917
x44 = -31.0391990278394
x45 = 20.4203507149992 + 8.43314411872054e-7*i
x46 = -49.8887549493781
x47 = -17.2787597021598
x48 = -60.0669879262646
x49 = 39.2699074542285 - 3.08532406117134e-9*i
x50 = 76.96902012591 - 1.62574105359862e-9*i
x51 = 70.685834592425
x52 = -43.6055696421985
x53 = 20.4203521700951
x54 = -93.8710520996352
x55 = 27.8976063742496
x56 = 88.3413218085728
x56 = 88.3413218085728