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Expresiones con funciones
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(log2(sin(x))^2+log2(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))
|-----------| + -----------
\ log(2) / log(2) ___
----------------------------*cos(x) + \/ 3 = 0
2
$$\frac{\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(2 \right)}}\right)^{2} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(2 \right)}}}{2} \cos{\left(x \right)} + \sqrt{3} = 0$$
Solución detallada
Tenemos la ecuación
$$\frac{\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(2 \right)}}\right)^{2} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(2 \right)}}}{2} \cos{\left(x \right)} + \sqrt{3} = 0$$
cambiamos
$$\frac{\log{\left(\sin{\left(x \right)} \right)} \log{\left(2 \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{2 \log{\left(2 \right)}^{2}} - 1 + \sqrt{3} = 0$$
$$\frac{\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(2 \right)}}\right)^{2} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(2 \right)}}}{2} \cos{\left(x \right)} - 1 + \sqrt{3} = 0$$
Sustituimos
$$w = \log{\left(\sin{\left(x \right)} \right)}$$
Abramos la expresión en la ecuación
$$\left(\frac{w^{2}}{2 \log{\left(2 \right)}^{2}} + \frac{w}{2 \log{\left(2 \right)}}\right) \cos{\left(x \right)} - 1 + \sqrt{3} = 0$$
Obtenemos la ecuación cuadrática
$$\frac{w^{2} \cos{\left(x \right)}}{2 \log{\left(2 \right)}^{2}} + \frac{w \cos{\left(x \right)}}{2 \log{\left(2 \right)}} - 1 + \sqrt{3} = 0$$
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
$$a = \frac{\cos{\left(x \right)}}{2 \log{\left(2 \right)}^{2}}$$
$$b = \frac{\cos{\left(x \right)}}{2 \log{\left(2 \right)}}$$
$$c = -1 + \sqrt{3}$$
, entonces
La ecuación tiene dos raíces.
o
$$w_{1} = \frac{\left(\sqrt{\frac{\cos^{2}{\left(x \right)}}{4 \log{\left(2 \right)}^{2}} - \frac{2 \left(-1 + \sqrt{3}\right) \cos{\left(x \right)}}{\log{\left(2 \right)}^{2}}} - \frac{\cos{\left(x \right)}}{2 \log{\left(2 \right)}}\right) \log{\left(2 \right)}^{2}}{\cos{\left(x \right)}}$$
$$w_{2} = \frac{\left(- \sqrt{\frac{\cos^{2}{\left(x \right)}}{4 \log{\left(2 \right)}^{2}} - \frac{2 \left(-1 + \sqrt{3}\right) \cos{\left(x \right)}}{\log{\left(2 \right)}^{2}}} - \frac{\cos{\left(x \right)}}{2 \log{\left(2 \right)}}\right) \log{\left(2 \right)}^{2}}{\cos{\left(x \right)}}$$
hacemos cambio inverso
$$\log{\left(\sin{\left(x \right)} \right)} = w$$
sustituimos w:
$$\frac{\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(2 \right)}}\right)^{2} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(2 \right)}}}{2} \cos{\left(x \right)} + \sqrt{3} = 0$$
cambiamos
$$\frac{\log{\left(\sin{\left(x \right)} \right)} \log{\left(2 \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{2 \log{\left(2 \right)}^{2}} - 1 + \sqrt{3} = 0$$
$$\frac{\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(2 \right)}}\right)^{2} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(2 \right)}}}{2} \cos{\left(x \right)} - 1 + \sqrt{3} = 0$$
Sustituimos
$$w = \log{\left(\sin{\left(x \right)} \right)}$$
Abramos la expresión en la ecuación
$$\left(\frac{w^{2}}{2 \log{\left(2 \right)}^{2}} + \frac{w}{2 \log{\left(2 \right)}}\right) \cos{\left(x \right)} - 1 + \sqrt{3} = 0$$
Obtenemos la ecuación cuadrática
$$\frac{w^{2} \cos{\left(x \right)}}{2 \log{\left(2 \right)}^{2}} + \frac{w \cos{\left(x \right)}}{2 \log{\left(2 \right)}} - 1 + \sqrt{3} = 0$$
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
$$a = \frac{\cos{\left(x \right)}}{2 \log{\left(2 \right)}^{2}}$$
$$b = \frac{\cos{\left(x \right)}}{2 \log{\left(2 \right)}}$$
$$c = -1 + \sqrt{3}$$
, entonces
D = b^2 - 4 * a * c =
(cos(x)/(2*log(2)))^2 - 4 * (cos(x)/(2*log(2)^2)) * (-1 + sqrt(3)) = cos(x)^2/(4*log(2)^2) - 2*(-1 + sqrt(3))*cos(x)/log(2)^2
La ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{\left(\sqrt{\frac{\cos^{2}{\left(x \right)}}{4 \log{\left(2 \right)}^{2}} - \frac{2 \left(-1 + \sqrt{3}\right) \cos{\left(x \right)}}{\log{\left(2 \right)}^{2}}} - \frac{\cos{\left(x \right)}}{2 \log{\left(2 \right)}}\right) \log{\left(2 \right)}^{2}}{\cos{\left(x \right)}}$$
$$w_{2} = \frac{\left(- \sqrt{\frac{\cos^{2}{\left(x \right)}}{4 \log{\left(2 \right)}^{2}} - \frac{2 \left(-1 + \sqrt{3}\right) \cos{\left(x \right)}}{\log{\left(2 \right)}^{2}}} - \frac{\cos{\left(x \right)}}{2 \log{\left(2 \right)}}\right) \log{\left(2 \right)}^{2}}{\cos{\left(x \right)}}$$
hacemos cambio inverso
$$\log{\left(\sin{\left(x \right)} \right)} = w$$
sustituimos w:
Respuesta numérica
[src]
x1 = -472.647306056146 + 0.0348779365518103*i
x2 = 58.6341270642077 + 1.2387993371625*i
x3 = 12.7967349327576 - 0.580681494672827*i
x4 = -1.40840801767671 + 0.0348779365518103*i
x5 = 64.9173123713873 - 1.2387993371625*i
x6 = -70.5234463966522 - 0.0348779365518103*i
x7 = 99.1225568971967 - 0.0348779365518103*i
x8 = -81.4510446749362 + 0.580681494672827*i
x9 = -29.3304672363065 - 1.2387993371625*i
x10 = 50.4958467758351 + 0.580681494672827*i
x11 = -89.3730023181909 - 0.0348779365518103*i
x12 = 28.0893463579863
x13 = 75.6285880045534 + 0.580681494672827*i
x14 = -83.0898170110113 - 0.0348779365518103*i
x15 = 78.354828815423
x16 = -70.5234463966522 + 0.0348779365518103*i
x17 = 72.0716435082434
x18 = -32.8243345535746 + 0.0348779365518103*i
x19 = 100.761329233272 - 0.580681494672827*i
x20 = -45.3907051679338 - 0.0348779365518103*i
x21 = 67.7066303612987 + 0.0348779365518103*i
x22 = 61.4234450541192 - 0.0348779365518103*i
x23 = 30.0075185182212 + 0.0348779365518103*i
x24 = 42.5738891325804 - 0.0348779365518103*i
x25 = 55.1402597469396 + 0.0348779365518103*i
x26 = -57.957075782293 + 0.0348779365518103*i
x27 = 81.911773311733 - 0.580681494672827*i
x28 = -75.1678593677566 - 0.580681494672827*i
x29 = 42.5738891325804 + 0.0348779365518103*i
x30 = -85.8791350009228 - 1.2387993371625*i
x31 = 86.5561862828375 + 0.0348779365518103*i
x32 = -24.9023769103199 - 0.580681494672827*i
x33 = -20.2579639392155 + 0.0348779365518103*i
x34 = 69.3454026973739 + 0.580681494672827*i
x35 = 17.441147903862 - 0.0348779365518103*i
x36 = 4.87477728950287 - 0.0348779365518103*i
x37 = 92.8393715900171 - 0.0348779365518103*i
x38 = 23.7243332110416 + 0.0348779365518103*i
x39 = -98.4455056152819 + 1.2387993371625*i
x40 = -60.7463937722044 + 1.2387993371625*i
x41 = -13.9747786320359 + 0.0348779365518103*i
x42 = -67.029579079384 + 1.2387993371625*i
x43 = -75.1678593677566 + 0.580681494672827*i
x44 = -26.5411492463951 + 0.0348779365518103*i
x45 = -64.2402610894726 + 0.0348779365518103*i
x46 = 1450.00739794081 + 0.0348779365518103*i
x47 = 80.2730009756579 + 0.0348779365518103*i
x48 = -95.6561876253705 + 0.0348779365518103*i
x49 = 37.9294761614759 + 0.580681494672827*i
x50 = -37.4687475246791 + 0.580681494672827*i
x51 = -10.4809113147677 - 1.2387993371625*i
x52 = -76.8066317038318 + 0.0348779365518103*i
x53 = -1.40840801767671 - 0.0348779365518103*i
x54 = 36.2907038254008 + 0.0348779365518103*i
x55 = 34.3725316651659
x56 = 61.4234450541192 + 0.0348779365518103*i
x57 = 55.1402597469396 - 0.0348779365518103*i
x58 = 94.4781439260922 + 0.580681494672827*i
x59 = 11.1579625966825 + 0.0348779365518103*i
x60 = 73.9898156684783 + 0.0348779365518103*i
x61 = 17.441147903862 + 0.0348779365518103*i
x62 = -7.6915933248563 + 0.0348779365518103*i
x63 = 0.230364318398399 - 0.580681494672827*i
x64 = -39.1075198607542 - 0.0348779365518103*i
x65 = -51.6738904751134 + 0.0348779365518103*i
x66 = -296.718117455117 + 0.0348779365518103*i
x67 = 48.85707443976 - 0.0348779365518103*i
x68 = 6.51354962557799 + 0.580681494672827*i
x69 = -100.300600596475 - 0.580681494672827*i
x70 = -32.8243345535746 - 0.0348779365518103*i
x71 = -50.0351181390383 - 0.580681494672827*i
x72 = 14.6518299139506 + 1.2387993371625*i
x73 = -68.8846740605771 - 0.580681494672827*i
x74 = -59.8752479425279
x75 = -15.8929507922708
x75 = -15.8929507922708