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Plotting a function graph/ (x*cos(x))/sin(x)

Graphing y = (x*cos(x))/sin(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       x*cos(x)
f(x) = --------
        sin(x)
f(x)=xcos⁡(x)sin⁡(x)f{\left(x \right)} = \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} 
f = (x*cos(x))/sin(x)
The graph of the function
02468-8-6-4-2-1010-50005000
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
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:
xcos⁡(x)sin⁡(x)=0\frac{x \cos{\left(x \right)}}{\sin{\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=7.85398163397448x_{1} = 7.85398163397448
x2=−73.8274273593601x_{2} = -73.8274273593601
x3=−54.9778714378214x_{3} = -54.9778714378214
x4=73.8274273593601x_{4} = 73.8274273593601
x5=−26.7035375555132x_{5} = -26.7035375555132
x6=−1.5707963267949x_{6} = -1.5707963267949
x7=−95.8185759344887x_{7} = -95.8185759344887
x8=−39.2699081698724x_{8} = -39.2699081698724
x9=−4.71238898038469x_{9} = -4.71238898038469
x10=14.1371669411541x_{10} = 14.1371669411541
x11=10.9955742875643x_{11} = 10.9955742875643
x12=58.1194640914112x_{12} = 58.1194640914112
x13=70.6858347057703x_{13} = 70.6858347057703
x14=−36.1283155162826x_{14} = -36.1283155162826
x15=54.9778714378214x_{15} = 54.9778714378214
x16=23.5619449019235x_{16} = 23.5619449019235
x17=−92.6769832808989x_{17} = -92.6769832808989
x18=−86.3937979737193x_{18} = -86.3937979737193
x19=−10.9955742875643x_{19} = -10.9955742875643
x20=92.6769832808989x_{20} = 92.6769832808989
x21=39.2699081698724x_{21} = 39.2699081698724
x22=−32.9867228626928x_{22} = -32.9867228626928
x23=98.9601685880785x_{23} = 98.9601685880785
x24=36.1283155162826x_{24} = 36.1283155162826
x25=−7.85398163397448x_{25} = -7.85398163397448
x26=−58.1194640914112x_{26} = -58.1194640914112
x27=−67.5442420521806x_{27} = -67.5442420521806
x28=−61.261056745001x_{28} = -61.261056745001
x29=26.7035375555132x_{29} = 26.7035375555132
x30=86.3937979737193x_{30} = 86.3937979737193
x31=−48.6946861306418x_{31} = -48.6946861306418
x32=51.8362787842316x_{32} = 51.8362787842316
x33=−42.4115008234622x_{33} = -42.4115008234622
x34=−89.5353906273091x_{34} = -89.5353906273091
x35=−98.9601685880785x_{35} = -98.9601685880785
x36=−14.1371669411541x_{36} = -14.1371669411541
x37=80.1106126665397x_{37} = 80.1106126665397
x38=−64.4026493985908x_{38} = -64.4026493985908
x39=95.8185759344887x_{39} = 95.8185759344887
x40=1.5707963267949x_{40} = 1.5707963267949
x41=45.553093477052x_{41} = 45.553093477052
x42=−17.2787595947439x_{42} = -17.2787595947439
x43=4.71238898038469x_{43} = 4.71238898038469
x44=48.6946861306418x_{44} = 48.6946861306418
x45=76.9690200129499x_{45} = 76.9690200129499
x46=−45.553093477052x_{46} = -45.553093477052
x47=20.4203522483337x_{47} = 20.4203522483337
x48=17.2787595947439x_{48} = 17.2787595947439
x49=−83.2522053201295x_{49} = -83.2522053201295
x50=−20.4203522483337x_{50} = -20.4203522483337
x51=−80.1106126665397x_{51} = -80.1106126665397
x52=61.261056745001x_{52} = 61.261056745001
x53=32.9867228626928x_{53} = 32.9867228626928
x54=64.4026493985908x_{54} = 64.4026493985908
x55=−23.5619449019235x_{55} = -23.5619449019235
x56=29.845130209103x_{56} = 29.845130209103
x57=42.4115008234622x_{57} = 42.4115008234622
x58=89.5353906273091x_{58} = 89.5353906273091
x59=−51.8362787842316x_{59} = -51.8362787842316
x60=−70.6858347057703x_{60} = -70.6858347057703
x61=83.2522053201295x_{61} = 83.2522053201295
x62=67.5442420521806x_{62} = 67.5442420521806
x63=−29.845130209103x_{63} = -29.845130209103
x64=−76.9690200129499x_{64} = -76.9690200129499
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x*cos(x))/sin(x).
0cos⁡(0)sin⁡(0)\frac{0 \cos{\left(0 \right)}}{\sin{\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
−xcos⁡2(x)sin⁡2(x)+−xsin⁡(x)+cos⁡(x)sin⁡(x)=0- \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{- x \sin{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=−4.40913187710091⋅10−17x_{1} = -4.40913187710091 \cdot 10^{-17}
x2=9.39240573061262⋅10−15x_{2} = 9.39240573061262 \cdot 10^{-15}
x3=−4.21743821735313⋅10−16x_{3} = -4.21743821735313 \cdot 10^{-16}
x4=−3.61900755242328⋅10−17x_{4} = -3.61900755242328 \cdot 10^{-17}
x5=4.48304559831337⋅10−17x_{5} = 4.48304559831337 \cdot 10^{-17}
x6=3.14012048853875⋅10−14x_{6} = 3.14012048853875 \cdot 10^{-14}
x7=5.91573285178007⋅10−15x_{7} = 5.91573285178007 \cdot 10^{-15}
x8=−1.13179325061883⋅10−14x_{8} = -1.13179325061883 \cdot 10^{-14}
x9=9.74762070728167⋅10−18x_{9} = 9.74762070728167 \cdot 10^{-18}
x10=−1.1002546748705⋅10−15x_{10} = -1.1002546748705 \cdot 10^{-15}
x11=−7.5836731643092⋅10−16x_{11} = -7.5836731643092 \cdot 10^{-16}
x12=−5.75417603992226⋅10−15x_{12} = -5.75417603992226 \cdot 10^{-15}
x13=1.64531346120711⋅10−15x_{13} = 1.64531346120711 \cdot 10^{-15}
x14=−3.51091717186253⋅10−15x_{14} = -3.51091717186253 \cdot 10^{-15}
x15=−1.59933636933227⋅10−17x_{15} = -1.59933636933227 \cdot 10^{-17}
x16=−6.42168935285657⋅10−16x_{16} = -6.42168935285657 \cdot 10^{-16}
x17=−7.90161711738172⋅10−15x_{17} = -7.90161711738172 \cdot 10^{-15}
x18=1.05753608890005⋅10−18x_{18} = 1.05753608890005 \cdot 10^{-18}
x19=6.85337988706529⋅10−19x_{19} = 6.85337988706529 \cdot 10^{-19}
x20=1.2698036406442⋅10−14x_{20} = 1.2698036406442 \cdot 10^{-14}
x21=−1.51633573535514⋅10−16x_{21} = -1.51633573535514 \cdot 10^{-16}
x22=1.06627540746173⋅10−14x_{22} = 1.06627540746173 \cdot 10^{-14}
x23=−4.18830956653573⋅10−16x_{23} = -4.18830956653573 \cdot 10^{-16}
x24=3.51403758286318⋅10−18x_{24} = 3.51403758286318 \cdot 10^{-18}
x25=−2.79377951045871⋅10−14x_{25} = -2.79377951045871 \cdot 10^{-14}
x26=1.64286535305518⋅10−16x_{26} = 1.64286535305518 \cdot 10^{-16}
x27=1.24026702406092⋅10−18x_{27} = 1.24026702406092 \cdot 10^{-18}
x28=3.28842331082471⋅10−18x_{28} = 3.28842331082471 \cdot 10^{-18}
x29=−1.40086013599021⋅10−16x_{29} = -1.40086013599021 \cdot 10^{-16}
x30=4.38463740569535⋅10−16x_{30} = 4.38463740569535 \cdot 10^{-16}
x31=−2.43657728705196⋅10−18x_{31} = -2.43657728705196 \cdot 10^{-18}
x32=−1.2559595098069⋅10−18x_{32} = -1.2559595098069 \cdot 10^{-18}
x33=2.36949247901661⋅10−17x_{33} = 2.36949247901661 \cdot 10^{-17}
x34=−7.18633372811494⋅10−17x_{34} = -7.18633372811494 \cdot 10^{-17}
x35=−4.92334219541076⋅10−15x_{35} = -4.92334219541076 \cdot 10^{-15}
x36=4.34273978333441⋅10−14x_{36} = 4.34273978333441 \cdot 10^{-14}
x37=−2.33425664095118⋅10−16x_{37} = -2.33425664095118 \cdot 10^{-16}
x38=2.99361568894398⋅10−15x_{38} = 2.99361568894398 \cdot 10^{-15}
x39=−2.40864592765222⋅10−15x_{39} = -2.40864592765222 \cdot 10^{-15}
x40=−2.10507703847062⋅10−19x_{40} = -2.10507703847062 \cdot 10^{-19}
x41=−3.84704756316661⋅10−17x_{41} = -3.84704756316661 \cdot 10^{-17}
x42=−4.19289120984576⋅10−18x_{42} = -4.19289120984576 \cdot 10^{-18}
x43=1.76954434151675⋅10−14x_{43} = 1.76954434151675 \cdot 10^{-14}
x44=−8.00122769514424⋅10−15x_{44} = -8.00122769514424 \cdot 10^{-15}
x45=−6.18947246968182⋅10−19x_{45} = -6.18947246968182 \cdot 10^{-19}
x46=−2.39625603482602⋅10−19x_{46} = -2.39625603482602 \cdot 10^{-19}
x47=1.81525026069804⋅10−18x_{47} = 1.81525026069804 \cdot 10^{-18}
x48=−7.24952705142948⋅10−15x_{48} = -7.24952705142948 \cdot 10^{-15}
x49=1.61579342741466⋅10−15x_{49} = 1.61579342741466 \cdot 10^{-15}
x50=−4.06303754303798⋅10−17x_{50} = -4.06303754303798 \cdot 10^{-17}
x51=−9.70142387854361⋅10−15x_{51} = -9.70142387854361 \cdot 10^{-15}
x52=−2.71634397358289⋅10−15x_{52} = -2.71634397358289 \cdot 10^{-15}
x53=−1.13133098959488⋅10−16x_{53} = -1.13133098959488 \cdot 10^{-16}
x54=−2.05466024628649⋅10−17x_{54} = -2.05466024628649 \cdot 10^{-17}
x55=−9.70783817428321⋅10−18x_{55} = -9.70783817428321 \cdot 10^{-18}
x56=1.14239635295548⋅10−17x_{56} = 1.14239635295548 \cdot 10^{-17}
x57=7.84600979716324⋅10−17x_{57} = 7.84600979716324 \cdot 10^{-17}
x58=−5.9968655068791⋅10−16x_{58} = -5.9968655068791 \cdot 10^{-16}
x59=4.90143809392486⋅10−16x_{59} = 4.90143809392486 \cdot 10^{-16}
x60=−8.92994949606852⋅10−19x_{60} = -8.92994949606852 \cdot 10^{-19}
x61=4.94984072088645⋅10−19x_{61} = 4.94984072088645 \cdot 10^{-19}
x62=3.28789431740595⋅10−14x_{62} = 3.28789431740595 \cdot 10^{-14}
x63=−1.09687894517034⋅10−14x_{63} = -1.09687894517034 \cdot 10^{-14}
x64=−1.16385961067265⋅10−14x_{64} = -1.16385961067265 \cdot 10^{-14}
The values of the extrema at the points:
(-4.409131877100905e-17, 1)

(9.392405730612624e-15, 1)

(-4.217438217353127e-16, 1)

(-3.619007552423282e-17, 1)

(4.4830455983133666e-17, 1)

(3.140120488538752e-14, 1)

(5.915732851780069e-15, 1)

(-1.1317932506188329e-14, 1)

(9.747620707281665e-18, 1)

(-1.1002546748704967e-15, 1)

(-7.583673164309204e-16, 1)

(-5.75417603992226e-15, 1)

(1.6453134612071067e-15, 1)

(-3.5109171718625317e-15, 1)

(-1.5993363693322655e-17, 1)

(-6.421689352856568e-16, 1)

(-7.901617117381719e-15, 1)

(1.0575360889000531e-18, 1)

(6.85337988706529e-19, 1)

(1.2698036406441962e-14, 1)

(-1.5163357353551354e-16, 1)

(1.0662754074617275e-14, 1)

(-4.188309566535729e-16, 1)

(3.514037582863182e-18, 1)

(-2.7937795104587108e-14, 1)

(1.6428653530551778e-16, 1)

(1.2402670240609207e-18, 1)

(3.288423310824711e-18, 1)

(-1.4008601359902102e-16, 1)

(4.3846374056953494e-16, 1)

(-2.4365772870519606e-18, 1)

(-1.2559595098068954e-18, 1)

(2.369492479016613e-17, 1)

(-7.186333728114936e-17, 1)

(-4.923342195410756e-15, 1)

(4.3427397833344134e-14, 1)

(-2.3342566409511843e-16, 1)

(2.9936156889439767e-15, 1)

(-2.40864592765222e-15, 1)

(-2.1050770384706248e-19, 1)

(-3.84704756316661e-17, 1)

(-4.192891209845756e-18, 1)

(1.7695443415167507e-14, 1)

(-8.001227695144237e-15, 1)

(-6.189472469681823e-19, 1)

(-2.396256034826019e-19, 1)

(1.81525026069804e-18, 1)

(-7.249527051429481e-15, 1)

(1.6157934274146616e-15, 1)

(-4.0630375430379806e-17, 1)

(-9.701423878543611e-15, 1)

(-2.7163439735828863e-15, 1)

(-1.1313309895948811e-16, 1)

(-2.0546602462864907e-17, 1)

(-9.707838174283208e-18, 1)

(1.1423963529554787e-17, 1)

(7.846009797163235e-17, 1)

(-5.996865506879105e-16, 1)

(4.901438093924855e-16, 1)

(-8.929949496068521e-19, 1)

(4.949840720886448e-19, 1)

(3.287894317405953e-14, 1)

(-1.0968789451703356e-14, 1)

(-1.163859610672651e-14, 1)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
Maxima of the function at points:
x64=−4.40913187710091⋅10−17x_{64} = -4.40913187710091 \cdot 10^{-17}
x64=9.39240573061262⋅10−15x_{64} = 9.39240573061262 \cdot 10^{-15}
x64=−4.21743821735313⋅10−16x_{64} = -4.21743821735313 \cdot 10^{-16}
x64=−3.61900755242328⋅10−17x_{64} = -3.61900755242328 \cdot 10^{-17}
x64=4.48304559831337⋅10−17x_{64} = 4.48304559831337 \cdot 10^{-17}
x64=3.14012048853875⋅10−14x_{64} = 3.14012048853875 \cdot 10^{-14}
x64=5.91573285178007⋅10−15x_{64} = 5.91573285178007 \cdot 10^{-15}
x64=−1.13179325061883⋅10−14x_{64} = -1.13179325061883 \cdot 10^{-14}
x64=9.74762070728167⋅10−18x_{64} = 9.74762070728167 \cdot 10^{-18}
x64=−1.1002546748705⋅10−15x_{64} = -1.1002546748705 \cdot 10^{-15}
x64=−7.5836731643092⋅10−16x_{64} = -7.5836731643092 \cdot 10^{-16}
x64=−5.75417603992226⋅10−15x_{64} = -5.75417603992226 \cdot 10^{-15}
x64=1.64531346120711⋅10−15x_{64} = 1.64531346120711 \cdot 10^{-15}
x64=−3.51091717186253⋅10−15x_{64} = -3.51091717186253 \cdot 10^{-15}
x64=−1.59933636933227⋅10−17x_{64} = -1.59933636933227 \cdot 10^{-17}
x64=−6.42168935285657⋅10−16x_{64} = -6.42168935285657 \cdot 10^{-16}
x64=−7.90161711738172⋅10−15x_{64} = -7.90161711738172 \cdot 10^{-15}
x64=1.05753608890005⋅10−18x_{64} = 1.05753608890005 \cdot 10^{-18}
x64=6.85337988706529⋅10−19x_{64} = 6.85337988706529 \cdot 10^{-19}
x64=1.2698036406442⋅10−14x_{64} = 1.2698036406442 \cdot 10^{-14}
x64=−1.51633573535514⋅10−16x_{64} = -1.51633573535514 \cdot 10^{-16}
x64=1.06627540746173⋅10−14x_{64} = 1.06627540746173 \cdot 10^{-14}
x64=−4.18830956653573⋅10−16x_{64} = -4.18830956653573 \cdot 10^{-16}
x64=3.51403758286318⋅10−18x_{64} = 3.51403758286318 \cdot 10^{-18}
x64=−2.79377951045871⋅10−14x_{64} = -2.79377951045871 \cdot 10^{-14}
x64=1.64286535305518⋅10−16x_{64} = 1.64286535305518 \cdot 10^{-16}
x64=1.24026702406092⋅10−18x_{64} = 1.24026702406092 \cdot 10^{-18}
x64=3.28842331082471⋅10−18x_{64} = 3.28842331082471 \cdot 10^{-18}
x64=−1.40086013599021⋅10−16x_{64} = -1.40086013599021 \cdot 10^{-16}
x64=4.38463740569535⋅10−16x_{64} = 4.38463740569535 \cdot 10^{-16}
x64=−2.43657728705196⋅10−18x_{64} = -2.43657728705196 \cdot 10^{-18}
x64=−1.2559595098069⋅10−18x_{64} = -1.2559595098069 \cdot 10^{-18}
x64=2.36949247901661⋅10−17x_{64} = 2.36949247901661 \cdot 10^{-17}
x64=−7.18633372811494⋅10−17x_{64} = -7.18633372811494 \cdot 10^{-17}
x64=−4.92334219541076⋅10−15x_{64} = -4.92334219541076 \cdot 10^{-15}
x64=4.34273978333441⋅10−14x_{64} = 4.34273978333441 \cdot 10^{-14}
x64=−2.33425664095118⋅10−16x_{64} = -2.33425664095118 \cdot 10^{-16}
x64=2.99361568894398⋅10−15x_{64} = 2.99361568894398 \cdot 10^{-15}
x64=−2.40864592765222⋅10−15x_{64} = -2.40864592765222 \cdot 10^{-15}
x64=−2.10507703847062⋅10−19x_{64} = -2.10507703847062 \cdot 10^{-19}
x64=−3.84704756316661⋅10−17x_{64} = -3.84704756316661 \cdot 10^{-17}
x64=−4.19289120984576⋅10−18x_{64} = -4.19289120984576 \cdot 10^{-18}
x64=1.76954434151675⋅10−14x_{64} = 1.76954434151675 \cdot 10^{-14}
x64=−8.00122769514424⋅10−15x_{64} = -8.00122769514424 \cdot 10^{-15}
x64=−6.18947246968182⋅10−19x_{64} = -6.18947246968182 \cdot 10^{-19}
x64=−2.39625603482602⋅10−19x_{64} = -2.39625603482602 \cdot 10^{-19}
x64=1.81525026069804⋅10−18x_{64} = 1.81525026069804 \cdot 10^{-18}
x64=−7.24952705142948⋅10−15x_{64} = -7.24952705142948 \cdot 10^{-15}
x64=1.61579342741466⋅10−15x_{64} = 1.61579342741466 \cdot 10^{-15}
x64=−4.06303754303798⋅10−17x_{64} = -4.06303754303798 \cdot 10^{-17}
x64=−9.70142387854361⋅10−15x_{64} = -9.70142387854361 \cdot 10^{-15}
x64=−2.71634397358289⋅10−15x_{64} = -2.71634397358289 \cdot 10^{-15}
x64=−1.13133098959488⋅10−16x_{64} = -1.13133098959488 \cdot 10^{-16}
x64=−2.05466024628649⋅10−17x_{64} = -2.05466024628649 \cdot 10^{-17}
x64=−9.70783817428321⋅10−18x_{64} = -9.70783817428321 \cdot 10^{-18}
x64=1.14239635295548⋅10−17x_{64} = 1.14239635295548 \cdot 10^{-17}
x64=7.84600979716324⋅10−17x_{64} = 7.84600979716324 \cdot 10^{-17}
x64=−5.9968655068791⋅10−16x_{64} = -5.9968655068791 \cdot 10^{-16}
x64=4.90143809392486⋅10−16x_{64} = 4.90143809392486 \cdot 10^{-16}
x64=−8.92994949606852⋅10−19x_{64} = -8.92994949606852 \cdot 10^{-19}
x64=4.94984072088645⋅10−19x_{64} = 4.94984072088645 \cdot 10^{-19}
x64=3.28789431740595⋅10−14x_{64} = 3.28789431740595 \cdot 10^{-14}
x64=−1.09687894517034⋅10−14x_{64} = -1.09687894517034 \cdot 10^{-14}
x64=−1.16385961067265⋅10−14x_{64} = -1.16385961067265 \cdot 10^{-14}
Decreasing at intervals
(−∞,−2.79377951045871⋅10−14]\left(-\infty, -2.79377951045871 \cdot 10^{-14}\right]
Increasing at intervals
[4.34273978333441⋅10−14,∞)\left[4.34273978333441 \cdot 10^{-14}, \infty\right)
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
x(1+2cos⁡2(x)sin⁡2(x))cos⁡(x)−xcos⁡(x)+2(xsin⁡(x)−cos⁡(x))cos⁡(x)sin⁡(x)−2sin⁡(x)sin⁡(x)=0\frac{x \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)} - x \cos{\left(x \right)} + \frac{2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 \sin{\left(x \right)}}{\sin{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=70.6716857116195x_{1} = 70.6716857116195
x2=−48.6741442319544x_{2} = -48.6741442319544
x3=51.8169824872797x_{3} = 51.8169824872797
x4=48.6741442319544x_{4} = 48.6741442319544
x5=17.2207552719308x_{5} = 17.2207552719308
x6=−36.1006222443756x_{6} = -36.1006222443756
x7=45.5311340139913x_{7} = 45.5311340139913
x8=98.9500628243319x_{8} = 98.9500628243319
x9=−89.5242209304172x_{9} = -89.5242209304172
x10=−4.49340945790906x_{10} = -4.49340945790906
x11=−64.3871195905574x_{11} = -64.3871195905574
x12=29.811598790893x_{12} = 29.811598790893
x13=61.2447302603744x_{13} = 61.2447302603744
x14=39.2444323611642x_{14} = 39.2444323611642
x15=80.0981286289451x_{15} = 80.0981286289451
x16=−7.72525183693771x_{16} = -7.72525183693771
x17=−20.3713029592876x_{17} = -20.3713029592876
x18=−86.3822220347287x_{18} = -86.3822220347287
x19=−32.9563890398225x_{19} = -32.9563890398225
x20=−10.9041216594289x_{20} = -10.9041216594289
x21=−70.6716857116195x_{21} = -70.6716857116195
x22=−61.2447302603744x_{22} = -61.2447302603744
x23=92.6661922776228x_{23} = 92.6661922776228
x24=−92.6661922776228x_{24} = -92.6661922776228
x25=67.5294347771441x_{25} = 67.5294347771441
x26=−14.0661939128315x_{26} = -14.0661939128315
x27=54.9596782878889x_{27} = 54.9596782878889
x28=−45.5311340139913x_{28} = -45.5311340139913
x29=−39.2444323611642x_{29} = -39.2444323611642
x30=89.5242209304172x_{30} = 89.5242209304172
x31=86.3822220347287x_{31} = 86.3822220347287
x32=4.49340945790906x_{32} = 4.49340945790906
x33=−42.3879135681319x_{33} = -42.3879135681319
x34=58.1022547544956x_{34} = 58.1022547544956
x35=−23.519452498689x_{35} = -23.519452498689
x36=36.1006222443756x_{36} = 36.1006222443756
x37=−80.0981286289451x_{37} = -80.0981286289451
x38=7.72525183693771x_{38} = 7.72525183693771
x39=−76.9560263103312x_{39} = -76.9560263103312
x40=73.8138806006806x_{40} = 73.8138806006806
x41=−98.9500628243319x_{41} = -98.9500628243319
x42=95.8081387868617x_{42} = 95.8081387868617
x43=26.6660542588127x_{43} = 26.6660542588127
x44=−17.2207552719308x_{44} = -17.2207552719308
x45=−26.6660542588127x_{45} = -26.6660542588127
x46=20.3713029592876x_{46} = 20.3713029592876
x47=−54.9596782878889x_{47} = -54.9596782878889
x48=−73.8138806006806x_{48} = -73.8138806006806
x49=−58.1022547544956x_{49} = -58.1022547544956
x50=23.519452498689x_{50} = 23.519452498689
x51=83.2401924707234x_{51} = 83.2401924707234
x52=−95.8081387868617x_{52} = -95.8081387868617
x53=−29.811598790893x_{53} = -29.811598790893
x54=−51.8169824872797x_{54} = -51.8169824872797
x55=10.9041216594289x_{55} = 10.9041216594289
x56=14.0661939128315x_{56} = 14.0661939128315
x57=32.9563890398225x_{57} = 32.9563890398225
x58=64.3871195905574x_{58} = 64.3871195905574
x59=−67.5294347771441x_{59} = -67.5294347771441
x60=76.9560263103312x_{60} = 76.9560263103312
x61=−83.2401924707234x_{61} = -83.2401924707234
x62=42.3879135681319x_{62} = 42.3879135681319
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979

lim⁡x→0−(x(1+2cos⁡2(x)sin⁡2(x))cos⁡(x)−xcos⁡(x)+2(xsin⁡(x)−cos⁡(x))cos⁡(x)sin⁡(x)−2sin⁡(x)sin⁡(x))=−23\lim_{x \to 0^-}\left(\frac{x \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)} - x \cos{\left(x \right)} + \frac{2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 \sin{\left(x \right)}}{\sin{\left(x \right)}}\right) = - \frac{2}{3}
lim⁡x→0+(x(1+2cos⁡2(x)sin⁡2(x))cos⁡(x)−xcos⁡(x)+2(xsin⁡(x)−cos⁡(x))cos⁡(x)sin⁡(x)−2sin⁡(x)sin⁡(x))=−23\lim_{x \to 0^+}\left(\frac{x \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)} - x \cos{\left(x \right)} + \frac{2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 \sin{\left(x \right)}}{\sin{\left(x \right)}}\right) = - \frac{2}{3}
- limits are equal, then skip the corresponding point
lim⁡x→3.14159265358979−(x(1+2cos⁡2(x)sin⁡2(x))cos⁡(x)−xcos⁡(x)+2(xsin⁡(x)−cos⁡(x))cos⁡(x)sin⁡(x)−2sin⁡(x)sin⁡(x))=−3.42095462014284⋅1048\lim_{x \to 3.14159265358979^-}\left(\frac{x \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)} - x \cos{\left(x \right)} + \frac{2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 \sin{\left(x \right)}}{\sin{\left(x \right)}}\right) = -3.42095462014284 \cdot 10^{48}
lim⁡x→3.14159265358979+(x(1+2cos⁡2(x)sin⁡2(x))cos⁡(x)−xcos⁡(x)+2(xsin⁡(x)−cos⁡(x))cos⁡(x)sin⁡(x)−2sin⁡(x)sin⁡(x))=−3.42095462014284⋅1048\lim_{x \to 3.14159265358979^+}\left(\frac{x \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)} - x \cos{\left(x \right)} + \frac{2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 \sin{\left(x \right)}}{\sin{\left(x \right)}}\right) = -3.42095462014284 \cdot 10^{48}
- limits are equal, then skip the corresponding point

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[−4.49340945790906,4.49340945790906]\left[-4.49340945790906, 4.49340945790906\right]
Convex at the intervals
(−∞,−98.9500628243319]\left(-\infty, -98.9500628243319\right]
Vertical asymptotes
Have:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
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→−∞(xcos⁡(x)sin⁡(x))y = \lim_{x \to -\infty}\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}\right)
True

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

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cos⁡(x)sin⁡(x))y = x \lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\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:
xcos⁡(x)sin⁡(x)=xcos⁡(x)sin⁡(x)\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} = \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}
- No
xcos⁡(x)sin⁡(x)=−xcos⁡(x)sin⁡(x)\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} = - \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}
- No
so, the function
not is
neither even, nor odd
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