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Plotting a function graph/ ctg(4x)*cos(4x)

Graphing y = ctg(4x)*cos(4x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = cot(4*x)*cos(4*x)
f(x)=cos⁡(4x)cot⁡(4x)f{\left(x \right)} = \cos{\left(4 x \right)} \cot{\left(4 x \right)} 
f = cos(4*x)*cot(4*x)
The graph of the function
02468-8-6-4-2-1010-500500
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
cos⁡(4x)cot⁡(4x)=0\cos{\left(4 x \right)} \cot{\left(4 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π8x_{1} = - \frac{\pi}{8}
x2=π8x_{2} = \frac{\pi}{8}
Numerical solution
x1=12.1736716122318x_{1} = 12.1736716122318
x2=−21.5984496198436x_{2} = -21.5984496198436
x3=−45.9457926721836x_{3} = -45.9457926721836
x4=−3.5342919993209x_{4} = -3.5342919993209
x5=−75.7909227335983x_{5} = -75.7909227335983
x6=−23.9546440649227x_{6} = -23.9546440649227
x7=92.2842841747879x_{7} = 92.2842841747879
x8=−82.0741079971979x_{8} = -82.0741079971979
x9=−60.0829594119627x_{9} = -60.0829594119627
x10=−61.653755896837x_{10} = -61.653755896837
x11=−79.7179136077217x_{11} = -79.7179136077217
x12=20.0276531264367x_{12} = 20.0276531264367
x13=−1.96349546669253x_{13} = -1.96349546669253
x14=66.3661447194599x_{14} = 66.3661447194599
x15=−38.0918108260377x_{15} = -38.0918108260377
x16=−39.6626073145636x_{16} = -39.6626073145636
x17=8.24668071815482x_{17} = 8.24668071815482
x18=5.89048634868838x_{18} = 5.89048634868838
x19=−13.7444678635245x_{19} = -13.7444678635245
x20=−78.1471170599108x_{20} = -78.1471170599108
x21=44.3749961357343x_{21} = 44.3749961357343
x22=−97.7820713170403x_{22} = -97.7820713170403
x23=48.3019870121468x_{23} = 48.3019870121468
x24=67.9369411998163x_{24} = 67.9369411998163
x25=−20.0276531365334x_{25} = -20.0276531365334
x26=62.4391538011772x_{26} = 62.4391538011772
x27=−16.1006622392305x_{27} = -16.1006622392305
x28=−87.57189538582x_{28} = -87.57189538582
x29=30.2378293000822x_{29} = 30.2378293000822
x30=78.1471174281992x_{30} = 78.1471174281992
x31=−100.138265656808x_{31} = -100.138265656808
x32=−65.5807467955783x_{32} = -65.5807467955783
x33=−57.7267650264066x_{33} = -57.7267650264066
x34=34.1648202052439x_{34} = 34.1648202052439
x35=18.4568566055691x_{35} = 18.4568566055691
x36=89.928089784421x_{36} = 89.928089784421
x37=−17.6714587325842x_{37} = -17.6714587325842
x38=−93.8550803321541x_{38} = -93.8550803321541
x39=88.3572933026447x_{39} = 88.3572933026447
x40=70.2931355934875x_{40} = 70.2931355934875
x41=−5.89048609822403x_{41} = -5.89048609822403
x42=−34.1648198530086x_{42} = -34.1648198530086
x43=−31.808625563961x_{43} = -31.808625563961
x44=71.8639321223223x_{44} = 71.8639321223223
x45=22.3838475513636x_{45} = 22.3838475513636
x46=84.430302394159x_{46} = 84.430302394159
x47=−42.0188017207944x_{47} = -42.0188017207944
x48=49.8727835290591x_{48} = 49.8727835290591
x49=−86.0010988907069x_{49} = -86.0010988907069
x50=−56.1559684590926x_{50} = -56.1559684590926
x51=96.2112750483112x_{51} = 96.2112750483112
x52=56.1559688055632x_{52} = 56.1559688055632
x53=−64.009950305446x_{53} = -64.009950305446
x54=−43.5895982070279x_{54} = -43.5895982070279
x55=−35.7356164450251x_{55} = -35.7356164450251
x56=74.2201264650387x_{56} = 74.2201264650387
x57=1.96349544751322x_{57} = 1.96349544751322
x58=93.8550807184008x_{58} = 93.8550807184008
x59=40.4480052052875x_{59} = 40.4480052052875
x60=52.2289778823416x_{60} = 52.2289778823416
x61=16.8860606369495x_{61} = 16.8860606369495
x62=−83.6449044794693x_{62} = -83.6449044794693
x63=−27.8816347272427x_{63} = -27.8816347272427
x64=27.8816349380003x_{64} = 27.8816349380003
x65=45.9457926155531x_{65} = 45.9457926155531
x66=−67.9369412987172x_{66} = -67.9369412987172
x67=−53.7997741493363x_{67} = -53.7997741493363
x68=42.0188017163919x_{68} = 42.0188017163919
x69=26.3108384307197x_{69} = 26.3108384307197
x70=23.9546440314963x_{70} = 23.9546440314963
x71=4.31968984915916x_{71} = 4.31968984915916
x72=−9.81747697702242x_{72} = -9.81747697702242
x73=86.0010988906926x_{73} = 86.0010988906926
x74=64.0099503042551x_{74} = 64.0099503042551
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cot(4*x)*cos(4*x).
cos⁡(0⋅4)cot⁡(0⋅4)\cos{\left(0 \cdot 4 \right)} \cot{\left(0 \cdot 4 \right)}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(cos⁡(4x)cot⁡(4x))y = \lim_{x \to -\infty}\left(\cos{\left(4 x \right)} \cot{\left(4 x \right)}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(cos⁡(4x)cot⁡(4x))y = \lim_{x \to \infty}\left(\cos{\left(4 x \right)} \cot{\left(4 x \right)}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(4*x)*cos(4*x), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(cos⁡(4x)cot⁡(4x)x)y = x \lim_{x \to -\infty}\left(\frac{\cos{\left(4 x \right)} \cot{\left(4 x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cos⁡(4x)cot⁡(4x)x)y = x \lim_{x \to \infty}\left(\frac{\cos{\left(4 x \right)} \cot{\left(4 x \right)}}{x}\right)
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
cos⁡(4x)cot⁡(4x)=−cos⁡(4x)cot⁡(4x)\cos{\left(4 x \right)} \cot{\left(4 x \right)} = - \cos{\left(4 x \right)} \cot{\left(4 x \right)}
- No
cos⁡(4x)cot⁡(4x)=cos⁡(4x)cot⁡(4x)\cos{\left(4 x \right)} \cot{\left(4 x \right)} = \cos{\left(4 x \right)} \cot{\left(4 x \right)}
- No
so, the function
not is
neither even, nor odd
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