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Plotting a function graph/ asin(x)*cot(x)

Graphing y = asin(x)*cot(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = asin(x)*cot(x)
f(x)=cot⁡(x)asin⁡(x)f{\left(x \right)} = \cot{\left(x \right)} \operatorname{asin}{\left(x \right)} 
f = cot(x)*asin(x)
The graph of the function
02468-8-6-4-2-10100.81.1
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:
cot⁡(x)asin⁡(x)=0\cot{\left(x \right)} \operatorname{asin}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
Numerical solution
x1=29.845130209103x_{1} = 29.845130209103
x2=−61.261056745001x_{2} = -61.261056745001
x3=7.85398163397448x_{3} = 7.85398163397448
x4=−83.2522053201295x_{4} = -83.2522053201295
x5=−20.4203522483337x_{5} = -20.4203522483337
x6=−70.6858347057703x_{6} = -70.6858347057703
x7=17.2787595947439x_{7} = 17.2787595947439
x8=67.5442420521806x_{8} = 67.5442420521806
x9=−42.4115008234622x_{9} = -42.4115008234622
x10=98.9601685880785x_{10} = 98.9601685880785
x11=4.71238898038469x_{11} = 4.71238898038469
x12=−95.8185759344887x_{12} = -95.8185759344887
x13=−32.9867228626928x_{13} = -32.9867228626928
x14=89.5353906273091x_{14} = 89.5353906273091
x15=39.2699081698724x_{15} = 39.2699081698724
x16=−17.2787595947439x_{16} = -17.2787595947439
x17=92.6769832808989x_{17} = 92.6769832808989
x18=−64.4026493985908x_{18} = -64.4026493985908
x19=14.1371669411541x_{19} = 14.1371669411541
x20=−48.6946861306418x_{20} = -48.6946861306418
x21=−17.2787595947439x_{21} = -17.2787595947439
x22=−76.9690200129499x_{22} = -76.9690200129499
x23=54.9778714378214x_{23} = 54.9778714378214
x24=−89.5353906273091x_{24} = -89.5353906273091
x25=45.553093477052x_{25} = 45.553093477052
x26=−23.5619449019235x_{26} = -23.5619449019235
x27=−1.5707963267949x_{27} = -1.5707963267949
x28=−67.5442420521806x_{28} = -67.5442420521806
x29=−10.9955742875643x_{29} = -10.9955742875643
x30=61.261056745001x_{30} = 61.261056745001
x31=−29.845130209103x_{31} = -29.845130209103
x32=76.9690200129499x_{32} = 76.9690200129499
x33=95.8185759344887x_{33} = 95.8185759344887
x34=70.6858347057703x_{34} = 70.6858347057703
x35=−26.7035375555132x_{35} = -26.7035375555132
x36=−7.85398163397448x_{36} = -7.85398163397448
x37=73.8274273593601x_{37} = 73.8274273593601
x38=64.4026493985908x_{38} = 64.4026493985908
x39=83.2522053201295x_{39} = 83.2522053201295
x40=−98.9601685880785x_{40} = -98.9601685880785
x41=−39.2699081698724x_{41} = -39.2699081698724
x42=32.9867228626928x_{42} = 32.9867228626928
x43=−73.8274273593601x_{43} = -73.8274273593601
x44=−36.1283155162826x_{44} = -36.1283155162826
x45=−45.553093477052x_{45} = -45.553093477052
x46=48.6946861306418x_{46} = 48.6946861306418
x47=−14.1371669411541x_{47} = -14.1371669411541
x48=51.8362787842316x_{48} = 51.8362787842316
x49=−54.9778714378214x_{49} = -54.9778714378214
x50=58.1194640914112x_{50} = 58.1194640914112
x51=10.9955742875643x_{51} = 10.9955742875643
x52=23.5619449019235x_{52} = 23.5619449019235
x53=−39.2699081698724x_{53} = -39.2699081698724
x54=−4.71238898038469x_{54} = -4.71238898038469
x55=36.1283155162826x_{55} = 36.1283155162826
x56=−51.8362787842316x_{56} = -51.8362787842316
x57=−86.3937979737193x_{57} = -86.3937979737193
x58=−58.1194640914112x_{58} = -58.1194640914112
x59=42.4115008234622x_{59} = 42.4115008234622
x60=−92.6769832808989x_{60} = -92.6769832808989
x61=20.4203522483337x_{61} = 20.4203522483337
x62=−80.1106126665397x_{62} = -80.1106126665397
x63=1.5707963267949x_{63} = 1.5707963267949
x64=26.7035375555132x_{64} = 26.7035375555132
x65=86.3937979737193x_{65} = 86.3937979737193
x66=80.1106126665397x_{66} = 80.1106126665397
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to asin(x)*cot(x).
cot⁡(0)asin⁡(0)\cot{\left(0 \right)} \operatorname{asin}{\left(0 \right)}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
(−cot⁡2(x)−1)asin⁡(x)+cot⁡(x)1−x2=0\left(- \cot^{2}{\left(x \right)} - 1\right) \operatorname{asin}{\left(x \right)} + \frac{\cot{\left(x \right)}}{\sqrt{1 - x^{2}}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
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→−∞(cot⁡(x)asin⁡(x))y = \lim_{x \to -\infty}\left(\cot{\left(x \right)} \operatorname{asin}{\left(x \right)}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(cot⁡(x)asin⁡(x))y = \lim_{x \to \infty}\left(\cot{\left(x \right)} \operatorname{asin}{\left(x \right)}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of asin(x)*cot(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→−∞(cot⁡(x)asin⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(x \right)} \operatorname{asin}{\left(x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(x)asin⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{\cot{\left(x \right)} \operatorname{asin}{\left(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:
cot⁡(x)asin⁡(x)=cot⁡(x)asin⁡(x)\cot{\left(x \right)} \operatorname{asin}{\left(x \right)} = \cot{\left(x \right)} \operatorname{asin}{\left(x \right)}
- Yes
cot⁡(x)asin⁡(x)=−cot⁡(x)asin⁡(x)\cot{\left(x \right)} \operatorname{asin}{\left(x \right)} = - \cot{\left(x \right)} \operatorname{asin}{\left(x \right)}
- No
so, the function
is
even
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