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log2((sin2x+2sinx+cosx+1)/(2sinx+1))=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   /sin(2*x) + 2*sin(x) + cos(x) + 1\    
log|--------------------------------|    
   \          2*sin(x) + 1          /    
------------------------------------- = 0
                log(2)
log⁡(((2sin⁡(x)+sin⁡(2x))+cos⁡(x))+12sin⁡(x)+1)log⁡(2)=0\frac{\log{\left(\frac{\left(\left(2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right) + \cos{\left(x \right)}\right) + 1}{2 \sin{\left(x \right)} + 1} \right)}}{\log{\left(2 \right)}} = 0
Simplificar
Solución detallada
Tenemos la ecuación
log⁡(((2sin⁡(x)+sin⁡(2x))+cos⁡(x))+12sin⁡(x)+1)log⁡(2)=0\frac{\log{\left(\frac{\left(\left(2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right) + \cos{\left(x \right)}\right) + 1}{2 \sin{\left(x \right)} + 1} \right)}}{\log{\left(2 \right)}} = 0
cambiamos
log⁡(cos⁡(x)+1)log⁡(2)−1=0\frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{\log{\left(2 \right)}} - 1 = 0
log⁡(((2sin⁡(x)+sin⁡(2x))+cos⁡(x))+12sin⁡(x)+1)log⁡(2)−1=0\frac{\log{\left(\frac{\left(\left(2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right) + \cos{\left(x \right)}\right) + 1}{2 \sin{\left(x \right)} + 1} \right)}}{\log{\left(2 \right)}} - 1 = 0
Sustituimos
w=log⁡(2sin⁡(x)2sin⁡(x)+1+sin⁡(2x)2sin⁡(x)+1+cos⁡(x)2sin⁡(x)+1+12sin⁡(x)+1)w = \log{\left(\frac{2 \sin{\left(x \right)}}{2 \sin{\left(x \right)} + 1} + \frac{\sin{\left(2 x \right)}}{2 \sin{\left(x \right)} + 1} + \frac{\cos{\left(x \right)}}{2 \sin{\left(x \right)} + 1} + \frac{1}{2 \sin{\left(x \right)} + 1} \right)}
Tenemos la ecuación:
log⁡(((2sin⁡(x)+sin⁡(2x))+cos⁡(x))+12sin⁡(x)+1)log⁡(2)−1=0\frac{\log{\left(\frac{\left(\left(2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right) + \cos{\left(x \right)}\right) + 1}{2 \sin{\left(x \right)} + 1} \right)}}{\log{\left(2 \right)}} - 1 = 0
Usamos la regla de proporciones:
De a1/b1 = a2/b2 se deduce a1*b2 = a2*b1,
En nuestro caso
a1 = log((1 + 2*sin(x) + cos(x) + sin(2*x))/(1 + 2*sin(x)))

b1 = log(2)

a2 = 1

b2 = 1

signo obtendremos la ecuación
log⁡(2sin⁡(x)+sin⁡(2x)+cos⁡(x)+12sin⁡(x)+1)=log⁡(2)\log{\left(\frac{2 \sin{\left(x \right)} + \sin{\left(2 x \right)} + \cos{\left(x \right)} + 1}{2 \sin{\left(x \right)} + 1} \right)} = \log{\left(2 \right)}
log⁡(2sin⁡(x)+sin⁡(2x)+cos⁡(x)+12sin⁡(x)+1)=log⁡(2)\log{\left(\frac{2 \sin{\left(x \right)} + \sin{\left(2 x \right)} + \cos{\left(x \right)} + 1}{2 \sin{\left(x \right)} + 1} \right)} = \log{\left(2 \right)}
Abrimos los paréntesis en el miembro izquierdo de la ecuación
log1+2*sin+x + cosx + sin2*x)/1+/2*sin+/x)) = log(2)

Abrimos los paréntesis en el miembro derecho de la ecuación
log1+2*sin+x + cosx + sin2*x)/1+/2*sin+/x)) = log2

Esta ecuación no tiene soluciones
hacemos cambio inverso
log⁡(2sin⁡(x)2sin⁡(x)+1+sin⁡(2x)2sin⁡(x)+1+cos⁡(x)2sin⁡(x)+1+12sin⁡(x)+1)=w\log{\left(\frac{2 \sin{\left(x \right)}}{2 \sin{\left(x \right)} + 1} + \frac{\sin{\left(2 x \right)}}{2 \sin{\left(x \right)} + 1} + \frac{\cos{\left(x \right)}}{2 \sin{\left(x \right)} + 1} + \frac{1}{2 \sin{\left(x \right)} + 1} \right)} = w
sustituimos w:
Gráfica
0-80-60-40-2020406080-100100-2525
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
0
00 
=
0
00 
producto
1
11 
=
1
11 
1
Respuesta numérica [src] 
x1 = 1.5707963267949
x2 = 39.2699081698724
x3 = -39.2699081698724
x4 = 14.1371669411541
x5 = -70.6858347057703
x6 = 32.9867228626928
x7 = 61.261056745001
x8 = 36.1283155162826
x9 = -61.261056745001
x10 = -105.243353895258
x11 = -36.1283155162826
x12 = -48.6946861306418
x13 = 7.85398163397448
x14 = 70.6858347057703
x15 = -80.1106126665397
x16 = 42.4115008234622
x17 = 51.8362787842316
x18 = -98.9601685880785
x19 = 80.1106126665397
x20 = -64.4026493985908
x21 = -29.845130209103
x22 = 98.9601685880785
x23 = 83.2522053201295
x24 = -89.5353906273091
x25 = -86.3937979737193
x26 = -10.9955742875643
x27 = 17.2787595947439
x28 = -26.7035375555132
x29 = -54.9778714378214
x30 = -92.6769832808989
x31 = 89.5353906273091
x32 = 58.1194640914112
x33 = 4.71238898038469
x34 = 20.4203522483337
x35 = 73.8274273593601
x36 = -67.5442420521806
x37 = 45.553093477052
x38 = -58.1194640914112
x39 = -51.8362787842316
x40 = -83.2522053201295
x41 = -76.9690200129499
x42 = -14.1371669411541
x43 = -73.8274273593601
x44 = 92.6769832808989
x45 = -20.4203522483337
x46 = 64.4026493985908
x47 = 95.8185759344887
x48 = 67.5442420521806
x49 = 54.9778714378214
x50 = 48.6946861306418
x51 = -4.71238898038469
x52 = 76.9690200129499
x53 = -7.85398163397448
x54 = -45.553093477052
x55 = -95.8185759344887
x56 = 29.845130209103
x57 = 26.7035375555132
x58 = -17.2787595947439
x59 = -32.9867228626928
x60 = -1.5707963267949
x61 = 23.5619449019235
x62 = 10.9955742875643
x63 = 86.3937979737193
x64 = -23.5619449019235
x65 = -42.4115008234622
x65 = -42.4115008234622
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