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sin(x)/sin(x+pi/4)
Trazado el gráfico de la función/ sin(x)/sin(x+pi/4)

Gráfico de la función y = sin(x)/sin(x+pi/4)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          sin(x)  
f(x) = -----------
          /    pi\
       sin|x + --|
          \    4 /
f(x)=sin⁡(x)sin⁡(x+π4)f{\left(x \right)} = \frac{\sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}} 
f = sin(x)/sin(x + pi/4)
Gráfico de la función
02468-8-6-4-2-1010-200200
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−0.785398163397448x_{1} = -0.785398163397448
x2=2.35619449019234x_{2} = 2.35619449019234
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:
sin⁡(x)sin⁡(x+π4)=0\frac{\sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=12.5663706143592x_{1} = 12.5663706143592
x2=53.4070751110265x_{2} = 53.4070751110265
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=37.6991118430775x_{4} = 37.6991118430775
x5=97.3893722612836x_{5} = 97.3893722612836
x6=78.5398163397448x_{6} = 78.5398163397448
x7=−59.6902604182061x_{7} = -59.6902604182061
x8=−65.9734457253857x_{8} = -65.9734457253857
x9=0x_{9} = 0
x10=−31.4159265358979x_{10} = -31.4159265358979
x11=−50.2654824574367x_{11} = -50.2654824574367
x12=−21.9911485751286x_{12} = -21.9911485751286
x13=6.28318530717959x_{13} = 6.28318530717959
x14=−34.5575191894877x_{14} = -34.5575191894877
x15=−94.2477796076938x_{15} = -94.2477796076938
x16=−69.1150383789755x_{16} = -69.1150383789755
x17=−15.707963267949x_{17} = -15.707963267949
x18=21.9911485751286x_{18} = 21.9911485751286
x19=69.1150383789755x_{19} = 69.1150383789755
x20=62.8318530717959x_{20} = 62.8318530717959
x21=50.2654824574367x_{21} = 50.2654824574367
x22=81.6814089933346x_{22} = 81.6814089933346
x23=100.530964914873x_{23} = 100.530964914873
x24=−40.8407044966673x_{24} = -40.8407044966673
x25=9.42477796076938x_{25} = 9.42477796076938
x26=−87.9645943005142x_{26} = -87.9645943005142
x27=34.5575191894877x_{27} = 34.5575191894877
x28=65.9734457253857x_{28} = 65.9734457253857
x29=−62.8318530717959x_{29} = -62.8318530717959
x30=−18.8495559215388x_{30} = -18.8495559215388
x31=−28.2743338823081x_{31} = -28.2743338823081
x32=−56.5486677646163x_{32} = -56.5486677646163
x33=−53.4070751110265x_{33} = -53.4070751110265
x34=−37.6991118430775x_{34} = -37.6991118430775
x35=−25.1327412287183x_{35} = -25.1327412287183
x36=−100.530964914873x_{36} = -100.530964914873
x37=−9.42477796076938x_{37} = -9.42477796076938
x38=40.8407044966673x_{38} = 40.8407044966673
x39=−91.106186954104x_{39} = -91.106186954104
x40=−75.398223686155x_{40} = -75.398223686155
x41=18.8495559215388x_{41} = 18.8495559215388
x42=87.9645943005142x_{42} = 87.9645943005142
x43=59.6902604182061x_{43} = 59.6902604182061
x44=−6.28318530717959x_{44} = -6.28318530717959
x45=25.1327412287183x_{45} = 25.1327412287183
x46=47.1238898038469x_{46} = 47.1238898038469
x47=91.106186954104x_{47} = 91.106186954104
x48=28.2743338823081x_{48} = 28.2743338823081
x49=56.5486677646163x_{49} = 56.5486677646163
x50=−43.9822971502571x_{50} = -43.9822971502571
x51=−47.1238898038469x_{51} = -47.1238898038469
x52=−3.14159265358979x_{52} = -3.14159265358979
x53=31.4159265358979x_{53} = 31.4159265358979
x54=94.2477796076938x_{54} = 94.2477796076938
x55=−12.5663706143592x_{55} = -12.5663706143592
x56=75.398223686155x_{56} = 75.398223686155
x57=−72.2566310325652x_{57} = -72.2566310325652
x58=−84.8230016469244x_{58} = -84.8230016469244
x59=84.8230016469244x_{59} = 84.8230016469244
x60=72.2566310325652x_{60} = 72.2566310325652
x61=−81.6814089933346x_{61} = -81.6814089933346
x62=43.9822971502571x_{62} = 43.9822971502571
x63=−78.5398163397448x_{63} = -78.5398163397448
x64=15.707963267949x_{64} = 15.707963267949
x65=3.14159265358979x_{65} = 3.14159265358979
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 sin(x)/sin(x + pi/4).
sin⁡(0)sin⁡(π4)\frac{\sin{\left(0 \right)}}{\sin{\left(\frac{\pi}{4} \right)}}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
−sin⁡(x)cos⁡(x+π4)sin⁡2(x+π4)+cos⁡(x)sin⁡(x+π4)=0- \frac{\sin{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}} + \frac{\cos{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
(1+2cos⁡2(x+π4)sin⁡2(x+π4))sin⁡(x)−sin⁡(x)−2cos⁡(x)cos⁡(x+π4)sin⁡(x+π4)sin⁡(x+π4)=0\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−99.7455667514759x_{1} = -99.7455667514759
x2=73.0420291959627x_{2} = 73.0420291959627
x3=−96.6039740978861x_{3} = -96.6039740978861
x4=−8.63937979737193x_{4} = -8.63937979737193
x5=−24.3473430653209x_{5} = -24.3473430653209
x6=−71.4712328691678x_{6} = -71.4712328691678
x7=76.1836218495525x_{7} = 76.1836218495525
x8=−30.6305283725005x_{8} = -30.6305283725005
x9=−5.49778714378214x_{9} = -5.49778714378214
x10=3.92699081698724x_{10} = 3.92699081698724
x11=−62.0464549083984x_{11} = -62.0464549083984
x12=−33.7721210260903x_{12} = -33.7721210260903
x13=−36.9137136796801x_{13} = -36.9137136796801
x14=−87.1791961371168x_{14} = -87.1791961371168
x15=95.0331777710912x_{15} = 95.0331777710912
x16=−40.0553063332699x_{16} = -40.0553063332699
x17=−58.9048622548086x_{17} = -58.9048622548086
x18=38.484510006475x_{18} = 38.484510006475
x19=−84.037603483527x_{19} = -84.037603483527
x20=66.7588438887831x_{20} = 66.7588438887831
x21=−43.1968989868597x_{21} = -43.1968989868597
x22=51.0508806208341x_{22} = 51.0508806208341
x23=0.785398163397448x_{23} = 0.785398163397448
x24=69.9004365423729x_{24} = 69.9004365423729
x25=−27.4889357189107x_{25} = -27.4889357189107
x26=101.316363078271x_{26} = 101.316363078271
x27=85.6083998103219x_{27} = 85.6083998103219
x28=54.1924732744239x_{28} = 54.1924732744239
x29=13.3517687777566x_{29} = 13.3517687777566
x30=88.7499924639117x_{30} = 88.7499924639117
x31=98.174770424681x_{31} = 98.174770424681
x32=−21.2057504117311x_{32} = -21.2057504117311
x33=−68.329640215578x_{33} = -68.329640215578
x34=10.2101761241668x_{34} = 10.2101761241668
x35=−11.7809724509617x_{35} = -11.7809724509617
x36=19.6349540849362x_{36} = 19.6349540849362
x37=41.6261026600648x_{37} = 41.6261026600648
x38=−55.7632696012188x_{38} = -55.7632696012188
x39=35.3429173528852x_{39} = 35.3429173528852
x40=82.4668071567321x_{40} = 82.4668071567321
x41=−93.4623814442964x_{41} = -93.4623814442964
x42=−90.3207887907066x_{42} = -90.3207887907066
x43=44.7676953136546x_{43} = 44.7676953136546
x44=−49.4800842940392x_{44} = -49.4800842940392
x45=−74.6128255227576x_{45} = -74.6128255227576
x46=32.2013246992954x_{46} = 32.2013246992954
x47=−2.35619449019234x_{47} = -2.35619449019234
x48=−65.1880475619882x_{48} = -65.1880475619882
x49=57.3340659280137x_{49} = 57.3340659280137
x50=22.776546738526x_{50} = 22.776546738526
x51=79.3252145031423x_{51} = 79.3252145031423
x52=63.6172512351933x_{52} = 63.6172512351933
x53=−46.3384916404494x_{53} = -46.3384916404494
x54=29.0597320457056x_{54} = 29.0597320457056
x55=25.9181393921158x_{55} = 25.9181393921158
x56=−52.621676947629x_{56} = -52.621676947629
x57=−14.9225651045515x_{57} = -14.9225651045515
x58=47.9092879672443x_{58} = 47.9092879672443
x59=60.4756585816035x_{59} = 60.4756585816035
x60=16.4933614313464x_{60} = 16.4933614313464
x61=−77.7544181763474x_{61} = -77.7544181763474
x62=91.8915851175014x_{62} = 91.8915851175014
x63=−80.8960108299372x_{63} = -80.8960108299372
x64=7.06858347057703x_{64} = 7.06858347057703
x65=−18.0641577581413x_{65} = -18.0641577581413
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=−0.785398163397448x_{1} = -0.785398163397448
x2=2.35619449019234x_{2} = 2.35619449019234

lim⁡x→−0.785398163397448−((1+2cos⁡2(x+π4)sin⁡2(x+π4))sin⁡(x)−sin⁡(x)−2cos⁡(x)cos⁡(x+π4)sin⁡(x+π4)sin⁡(x+π4))=−1(1.4142135623731sin⁡(0.785398163397448+0.25π)cos⁡(0.785398163397448+0.25π)+1.41421356237309sin⁡2(0.785398163397448+0.25π))cos⁡3(0.785398163397448+0.25π)\lim_{x \to -0.785398163397448^-}\left(\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = - \frac{1 \left(1.4142135623731 \sin{\left(0.785398163397448 + 0.25 \pi \right)} \cos{\left(0.785398163397448 + 0.25 \pi \right)} + 1.41421356237309 \sin^{2}{\left(0.785398163397448 + 0.25 \pi \right)}\right)}{\cos^{3}{\left(0.785398163397448 + 0.25 \pi \right)}}
lim⁡x→−0.785398163397448+((1+2cos⁡2(x+π4)sin⁡2(x+π4))sin⁡(x)−sin⁡(x)−2cos⁡(x)cos⁡(x+π4)sin⁡(x+π4)sin⁡(x+π4))=1(−1.4142135623731sin⁡(0.785398163397448−0.25π)cos⁡(0.785398163397448−0.25π)+1.41421356237309cos⁡2(0.785398163397448−0.25π))sin⁡3(0.785398163397448−0.25π)\lim_{x \to -0.785398163397448^+}\left(\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = \frac{1 \left(- 1.4142135623731 \sin{\left(0.785398163397448 - 0.25 \pi \right)} \cos{\left(0.785398163397448 - 0.25 \pi \right)} + 1.41421356237309 \cos^{2}{\left(0.785398163397448 - 0.25 \pi \right)}\right)}{\sin^{3}{\left(0.785398163397448 - 0.25 \pi \right)}}
- los límites no son iguales, signo
x1=−0.785398163397448x_{1} = -0.785398163397448
- es el punto de flexión
lim⁡x→2.35619449019234−((1+2cos⁡2(x+π4)sin⁡2(x+π4))sin⁡(x)−sin⁡(x)−2cos⁡(x)cos⁡(x+π4)sin⁡(x+π4)sin⁡(x+π4))=1(1.41421356237309sin⁡(0.25π+2.35619449019234)cos⁡(0.25π+2.35619449019234)+1.4142135623731cos⁡2(0.25π+2.35619449019234))sin⁡3(0.25π+2.35619449019234)\lim_{x \to 2.35619449019234^-}\left(\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = \frac{1 \left(1.41421356237309 \sin{\left(0.25 \pi + 2.35619449019234 \right)} \cos{\left(0.25 \pi + 2.35619449019234 \right)} + 1.4142135623731 \cos^{2}{\left(0.25 \pi + 2.35619449019234 \right)}\right)}{\sin^{3}{\left(0.25 \pi + 2.35619449019234 \right)}}
lim⁡x→2.35619449019234+((1+2cos⁡2(x+π4)sin⁡2(x+π4))sin⁡(x)−sin⁡(x)−2cos⁡(x)cos⁡(x+π4)sin⁡(x+π4)sin⁡(x+π4))=1(1.41421356237309sin⁡(0.25π+2.35619449019234)cos⁡(0.25π+2.35619449019234)+1.4142135623731cos⁡2(0.25π+2.35619449019234))sin⁡3(0.25π+2.35619449019234)\lim_{x \to 2.35619449019234^+}\left(\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = \frac{1 \left(1.41421356237309 \sin{\left(0.25 \pi + 2.35619449019234 \right)} \cos{\left(0.25 \pi + 2.35619449019234 \right)} + 1.4142135623731 \cos^{2}{\left(0.25 \pi + 2.35619449019234 \right)}\right)}{\sin^{3}{\left(0.25 \pi + 2.35619449019234 \right)}}
- los límites son iguales, es decir omitimos el punto correspondiente

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[101.316363078271,∞)\left[101.316363078271, \infty\right)
Convexa en los intervalos
(−∞,−99.7455667514759]\left(-\infty, -99.7455667514759\right]
Asíntotas verticales
Hay:
x1=−0.785398163397448x_{1} = -0.785398163397448
x2=2.35619449019234x_{2} = 2.35619449019234
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→−∞(sin⁡(x)sin⁡(x+π4))y = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(sin⁡(x)sin⁡(x+π4))y = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)/sin(x + pi/4), 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→−∞(sin⁡(x)xsin⁡(x+π4))y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sin{\left(x + \frac{\pi}{4} \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(sin⁡(x)xsin⁡(x+π4))y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sin{\left(x + \frac{\pi}{4} \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:
sin⁡(x)sin⁡(x+π4)=sin⁡(x)sin⁡(x−π4)\frac{\sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}} = \frac{\sin{\left(x \right)}}{\sin{\left(x - \frac{\pi}{4} \right)}}
- No
sin⁡(x)sin⁡(x+π4)=−sin⁡(x)sin⁡(x−π4)\frac{\sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}} = - \frac{\sin{\left(x \right)}}{\sin{\left(x - \frac{\pi}{4} \right)}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = sin(x)/sin(x+pi/4)
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