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log(cos(x))*x*(sqrt(3)/2)=1 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
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log(cos(x))*x*----- = 1
                2      
$$\frac{\sqrt{3}}{2} x \log{\left(\cos{\left(x \right)} \right)} = 1$$
Solución detallada
Tenemos la ecuación
$$\frac{\sqrt{3}}{2} x \log{\left(\cos{\left(x \right)} \right)} = 1$$
cambiamos
$$\frac{\sqrt{3} x \log{\left(\cos{\left(x \right)} \right)}}{2} - 1 = 0$$
$$\frac{\sqrt{3}}{2} x \log{\left(\cos{\left(x \right)} \right)} - 1 = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
Tenemos la ecuación
$$\frac{\sqrt{3} x \log{\left(w \right)}}{2} - 1 = 0$$
$$\frac{\sqrt{3} x \log{\left(w \right)}}{2} = 1$$
Devidimos ambás partes de la ecuación por el multiplicador de log =x*sqrt(3)/2
$$\log{\left(w \right)} = \frac{2 \sqrt{3}}{3 x}$$
Es la ecuación de la forma:
log(v)=p

Por definición log
v=e^p

entonces
$$w = e^{\frac{1}{\frac{1}{2} \sqrt{3} x}}$$
simplificamos
$$w = e^{\frac{2 \sqrt{3}}{3 x}}$$
hacemos cambio inverso
$$\cos{\left(x \right)} = w$$
Tenemos la ecuación
$$\cos{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
O
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, donde n es cualquier número entero
sustituimos w:
Gráfica
Respuesta numérica [src]
x1 = -31.6842673910973
x2 = 81.6815836979795 - 0.168542590001074*i
x3 = -12.1370457313664
x4 = -75.5725893690224
x5 = -12.9819211022713
x6 = 62.8321491518962 - 0.192303225544944*i
x7 = 100.531080044771 + 0.151855385274509*i
x8 = -81.5134864068444
x9 = 87.964744837959 + 0.162384485455842*i
x10 = 25.134625216394 + 0.305427949612451*i
x11 = -94.4038674527582
x12 = -18.4999023228677
x13 = -94.0914338892939
x14 = -87.8027701812223
x15 = 43.9829045351223 + 0.230145130842218*i
x16 = -50.0515040422965
x17 = 94.2479106664787 - 0.156855520190678*i
x18 = 50.2659464854494 + 0.215164782143379*i
x19 = -50.478560745876
x20 = -63.0226951795667
x21 = 62.8321491518962 + 0.192303225544944*i
x22 = 25.134625216394 - 0.305427949612451*i
x23 = 69.1152828033185 + 0.183303215728435*i
x24 = -5.66621694198888
x25 = -6.84774240096047
x26 = -81.848988689
x27 = -37.9445657794717
x28 = -56.7497131342722
x29 = -88.1261225646145
x30 = 12.5741192507654 - 0.434940646110801*i
x31 = -69.2970861724669
x32 = 75.3984288814445 - 0.175458973368231*i
x33 = -31.1453023535666
x34 = -56.3469099729739
x35 = 43.9829045351223 - 0.230145130842218*i
x36 = -19.1929678913582
x37 = -62.640433115955
x38 = 31.4171251850265 - 0.272778977146702*i
x39 = -37.4520658111118
x40 = 18.8529376632546 - 0.353528564269807*i
x41 = -43.7535629699458
x42 = -68.932512492568
x43 = 37.6999409306527 - 0.248763577048458*i
x44 = 31.4171251850265 + 0.272778977146702*i
x45 = 6.31544342745461 - 0.62078397347028*i
x46 = -100.379576010306
x47 = 0.783987613426113 + 1.15338565815753*i
x48 = 69.1152828033185 - 0.183303215728435*i
x49 = 37.6999409306527 + 0.248763577048458*i
x50 = 50.2659464854494 - 0.215164782143379*i
x51 = 81.6815836979795 + 0.168542590001074*i
x52 = 0.783987613426113 - 1.15338565815753*i
x53 = -75.2234558990625
x54 = 94.2479106664787 + 0.156855520190678*i
x55 = -44.2098582207138
x56 = 87.964744837959 - 0.162384485455842*i
x57 = 18.8529376632546 + 0.353528564269807*i
x58 = 56.5490337883634 + 0.20277385518947*i
x59 = -24.8301272099416
x60 = 56.5490337883634 - 0.20277385518947*i
x61 = -1.18410247987361
x62 = -25.4318091003211
x62 = -25.4318091003211