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Trazado el gráfico de la función/ log(sin(x))/cot(x)

Gráfico de la función y = log(sin(x))/cot(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       log(sin(x))
f(x) = -----------
          cot(x)
f(x)=log⁡(sin⁡(x))cot⁡(x)f{\left(x \right)} = \frac{\log{\left(\sin{\left(x \right)} \right)}}{\cot{\left(x \right)}} 
f = log(sin(x))/cot(x)
Gráfico de la función
02468-8-6-4-2-10101.0-1.0
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=1.5707963267949x_{1} = 1.5707963267949
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
log⁡(sin⁡(x))cot⁡(x)=0\frac{\log{\left(\sin{\left(x \right)} \right)}}{\cot{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−105.243353895258x_{1} = -105.243353895258
x2=−23.5619449019235x_{2} = -23.5619449019235
x3=−91.106186954104x_{3} = -91.106186954104
x4=−29.845130209103x_{4} = -29.845130209103
x5=−28.2743338823081x_{5} = -28.2743338823081
x6=1.5707963267949x_{6} = 1.5707963267949
x7=−12.5663706143592x_{7} = -12.5663706143592
x8=−65.9734457253857x_{8} = -65.9734457253857
x9=26.7035375555132x_{9} = 26.7035375555132
x10=−48.6946861306418x_{10} = -48.6946861306418
x11=−17.2787595947439x_{11} = -17.2787595947439
x12=−80.1106126665397x_{12} = -80.1106126665397
x13=−10.9955742875643x_{13} = -10.9955742875643
x14=39.2699081698724x_{14} = 39.2699081698724
x15=64.4026493985908x_{15} = 64.4026493985908
x16=15.707963267949x_{16} = 15.707963267949
x17=−10.9955742875643x_{17} = -10.9955742875643
x18=56.5486677646163x_{18} = 56.5486677646163
x19=−47.1238898038469x_{19} = -47.1238898038469
x20=−34.5575191894877x_{20} = -34.5575191894877
x21=−86.3937979737193x_{21} = -86.3937979737193
x22=−81.6814089933346x_{22} = -81.6814089933346
x23=−37.6991118430775x_{23} = -37.6991118430775
x24=−92.6769832808989x_{24} = -92.6769832808989
x25=72.2566310325652x_{25} = 72.2566310325652
x26=70.6858347057703x_{26} = 70.6858347057703
x27=43.9822971502571x_{27} = 43.9822971502571
x28=−62.8318530717959x_{28} = -62.8318530717959
x29=−84.8230016469244x_{29} = -84.8230016469244
x30=−42.4115008234622x_{30} = -42.4115008234622
x31=−31.4159265358979x_{31} = -31.4159265358979
x32=58.1194640914112x_{32} = 58.1194640914112
x33=51.8362787842316x_{33} = 51.8362787842316
x34=−73.8274273593601x_{34} = -73.8274273593601
x35=−59.6902604182061x_{35} = -59.6902604182061
x36=83.2522053201295x_{36} = 83.2522053201295
x37=47.1238898038469x_{37} = 47.1238898038469
x38=−75.398223686155x_{38} = -75.398223686155
x39=94.2477796076938x_{39} = 94.2477796076938
x40=62.8318530717959x_{40} = 62.8318530717959
x41=−36.1283155162826x_{41} = -36.1283155162826
x42=28.2743338823081x_{42} = 28.2743338823081
x43=100.530964914873x_{43} = 100.530964914873
x44=32.9867228626928x_{44} = 32.9867228626928
x45=−54.9778714378214x_{45} = -54.9778714378214
x46=−15.707963267949x_{46} = -15.707963267949
x47=−111.526539202438x_{47} = -111.526539202438
x48=−67.5442420521806x_{48} = -67.5442420521806
x49=−43.9822971502571x_{49} = -43.9822971502571
x50=−72.2566310325652x_{50} = -72.2566310325652
x51=−3.14159265358979x_{51} = -3.14159265358979
x52=1.57079632679489x_{52} = 1.57079632679489
x53=−69.1150383789755x_{53} = -69.1150383789755
x54=−21.9911485751286x_{54} = -21.9911485751286
x55=89.5353906273091x_{55} = 89.5353906273091
x56=9.42477796076938x_{56} = 9.42477796076938
x57=−97.3893722612836x_{57} = -97.3893722612836
x58=7.85398163397448x_{58} = 7.85398163397448
x59=50.2654824574367x_{59} = 50.2654824574367
x60=76.9690200129499x_{60} = 76.9690200129499
x61=6.28318530717959x_{61} = 6.28318530717959
x62=25.1327412287183x_{62} = 25.1327412287183
x63=−4.71238898038469x_{63} = -4.71238898038469
x64=−98.9601685880785x_{64} = -98.9601685880785
x65=14.1371669411541x_{65} = 14.1371669411541
x66=65.9734457253857x_{66} = 65.9734457253857
x67=−25.1327412287183x_{67} = -25.1327412287183
x68=20.4203522483337x_{68} = 20.4203522483337
x69=45.553093477052x_{69} = 45.553093477052
x70=95.8185759344887x_{70} = 95.8185759344887
x71=−56.5486677646163x_{71} = -56.5486677646163
x72=3.1415926535898x_{72} = 3.1415926535898
x73=12.5663706143592x_{73} = 12.5663706143592
x74=−87.9645943005142x_{74} = -87.9645943005142
x75=59.6902604182061x_{75} = 59.6902604182061
x76=−50.2654824574367x_{76} = -50.2654824574367
x77=21.9911485751286x_{77} = 21.9911485751286
x78=−48.6946861306418x_{78} = -48.6946861306418
x79=−61.261056745001x_{79} = -61.261056745001
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en log(sin(x))/cot(x).
log⁡(sin⁡(0))cot⁡(0)\frac{\log{\left(\sin{\left(0 \right)} \right)}}{\cot{\left(0 \right)}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Asíntotas verticales
Hay:
x1=1.5707963267949x_{1} = 1.5707963267949
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(log⁡(sin⁡(x))cot⁡(x))y = \lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\cot{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(log⁡(sin⁡(x))cot⁡(x))y = \lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\cot{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función log(sin(x))/cot(x), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(log⁡(sin⁡(x))xcot⁡(x))y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{x \cot{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(log⁡(sin⁡(x))xcot⁡(x))y = x \lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{x \cot{\left(x \right)}}\right)
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
log⁡(sin⁡(x))cot⁡(x)=−log⁡(−sin⁡(x))cot⁡(x)\frac{\log{\left(\sin{\left(x \right)} \right)}}{\cot{\left(x \right)}} = - \frac{\log{\left(- \sin{\left(x \right)} \right)}}{\cot{\left(x \right)}}
- No
log⁡(sin⁡(x))cot⁡(x)=log⁡(−sin⁡(x))cot⁡(x)\frac{\log{\left(\sin{\left(x \right)} \right)}}{\cot{\left(x \right)}} = \frac{\log{\left(- \sin{\left(x \right)} \right)}}{\cot{\left(x \right)}}
- No
es decir, función
no es
par ni impar
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