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

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

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       cos(x)*cos(x)
f(x) = -------------
           sin(x)
f(x)=cos⁡(x)cos⁡(x)sin⁡(x)f{\left(x \right)} = \frac{\cos{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} 
f = (cos(x)*cos(x))/sin(x)
The graph of the function
02468-8-6-4-2-1010-10001000
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:
cos⁡(x)cos⁡(x)sin⁡(x)=0\frac{\cos{\left(x \right)} \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=3π2x_{2} = \frac{3 \pi}{2}
Numerical solution
x1=−76.969019142094x_{1} = -76.969019142094
x2=42.4115007257482x_{2} = 42.4115007257482
x3=−48.6946856519848x_{3} = -48.6946856519848
x4=−26.7035384718653x_{4} = -26.7035384718653
x5=29.8451303260505x_{5} = 29.8451303260505
x6=23.5619451851288x_{6} = 23.5619451851288
x7=67.5442423466955x_{7} = 67.5442423466955
x8=45.553093765886x_{8} = 45.553093765886
x9=10.9955752430018x_{9} = 10.9955752430018
x10=−39.2699084457792x_{10} = -39.2699084457792
x11=−51.8362786889097x_{11} = -51.8362786889097
x12=−7.85398149471446x_{12} = -7.85398149471446
x13=70.6858344565908x_{13} = 70.6858344565908
x14=83.2522045003433x_{14} = 83.2522045003433
x15=−45.5530935939383x_{15} = -45.5530935939383
x16=4.71238871530383x_{16} = 4.71238871530383
x17=51.836278906418x_{17} = 51.836278906418
x18=−29.8451300946099x_{18} = -29.8451300946099
x19=20.4203521458051x_{19} = 20.4203521458051
x20=80.1106131287292x_{20} = 80.1106131287292
x21=48.6946858762043x_{21} = 48.6946858762043
x22=−20.4203519762382x_{22} = -20.4203519762382
x23=36.1283159260006x_{23} = 36.1283159260006
x24=−36.1283154153854x_{24} = -36.1283154153854
x25=−10.9955733545245x_{25} = -10.9955733545245
x26=−58.1194639960073x_{26} = -58.1194639960073
x27=−48.6946870685899x_{27} = -48.6946870685899
x28=−67.5442421738335x_{28} = -67.5442421738335
x29=76.9690195630622x_{29} = 76.9690195630622
x30=−54.9778705470218x_{30} = -54.9778705470218
x31=−64.4026491374242x_{31} = -64.4026491374242
x32=58.1194643607763x_{32} = 58.1194643607763
x33=−4.71238848455677x_{33} = -4.71238848455677
x34=−86.3937977179549x_{34} = -86.3937977179549
x35=−32.9867219511814x_{35} = -32.9867219511814
x36=−14.1371668348422x_{36} = -14.1371668348422
x37=−98.9601677364429x_{37} = -98.9601677364429
x38=−1.57079643412171x_{38} = -1.57079643412171
x39=89.5353909275596x_{39} = 89.5353909275596
x40=17.2787587191641x_{40} = 17.2787587191641
x41=92.6769830369374x_{41} = 92.6769830369374
x42=1.57079660442153x_{42} = 1.57079660442153
x43=−10.9955747699649x_{43} = -10.9955747699649
x44=−70.6858342354489x_{44} = -70.6858342354489
x45=−95.8185758679732x_{45} = -95.8185758679732
x46=61.2610572679436x_{46} = 61.2610572679436
x47=86.3937978856968x_{47} = 86.3937978856968
x48=−54.9778719374446x_{48} = -54.9778719374446
x49=−26.7035370683564x_{49} = -26.7035370683564
x50=7.85398174563321x_{50} = 7.85398174563321
x51=−73.8274272796526x_{51} = -73.8274272796526
x52=73.8274274867432x_{52} = 73.8274274867432
x53=32.9867223968539x_{53} = 32.9867223968539
x54=39.2699086835855x_{54} = 39.2699086835855
x55=−76.9690205214496x_{55} = -76.9690205214496
x56=14.1371671153205x_{56} = 14.1371671153205
x57=61.2610559072772x_{57} = 61.2610559072772
x58=98.960168145952x_{58} = 98.960168145952
x59=54.9778709800313x_{59} = 54.9778709800313
x60=−98.9601691056411x_{60} = -98.9601691056411
x61=−4.71238987596373x_{61} = -4.71238987596373
x62=10.9955738135239x_{62} = 10.9955738135239
x63=−32.9867233536188x_{63} = -32.9867233536188
x64=−89.5353907537234x_{64} = -89.5353907537234
x65=76.9690210413954x_{65} = 76.9690210413954
x66=64.4026493057109x_{66} = 64.4026493057109
x67=−42.4115005568527x_{67} = -42.4115005568527
x68=26.7035372957759x_{68} = 26.7035372957759
x69=−92.676982818755x_{69} = -92.676982818755
x70=−61.2610570263942x_{70} = -61.2610570263942
x71=17.2787600994214x_{71} = 17.2787600994214
x72=39.2699073135637x_{72} = 39.2699073135637
x73=−17.2787598652139x_{73} = -17.2787598652139
x74=32.9867238414546x_{74} = 32.9867238414546
x75=98.9601696430262x_{75} = 98.9601696430262
x76=83.2522058525039x_{76} = 83.2522058525039
x77=−92.6769842647274x_{77} = -92.6769842647274
x78=95.8185760670326x_{78} = 95.8185760670326
x79=−83.2522056070609x_{79} = -83.2522056070609
x80=−80.1106125767207x_{80} = -80.1106125767207
x81=54.9778724408964x_{81} = 54.9778724408964
x82=−70.6858356661912x_{82} = -70.6858356661912
x83=−23.5619450140352x_{83} = -23.5619450140352
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (cos(x)*cos(x))/sin(x).
cos⁡(0)cos⁡(0)sin⁡(0)\frac{\cos{\left(0 \right)} \cos{\left(0 \right)}}{\sin{\left(0 \right)}}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
−2cos⁡(x)−cos⁡3(x)sin⁡2(x)=0- 2 \cos{\left(x \right)} - \frac{\cos^{3}{\left(x \right)}}{\sin^{2}{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
The values of the extrema at the points:
 -pi     
(----, 0)
  2      

 pi    
(--, 0)
 2


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:
Minima of the function at points:
x1=π2x_{1} = \frac{\pi}{2}
Maxima of the function at points:
x1=−π2x_{1} = - \frac{\pi}{2}
Decreasing at intervals
(−∞,−π2]∪[π2,∞)\left(-\infty, - \frac{\pi}{2}\right] \cup \left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
[−π2,π2]\left[- \frac{\pi}{2}, \frac{\pi}{2}\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→−∞(cos⁡(x)cos⁡(x)sin⁡(x))y = \lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} \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→∞(cos⁡(x)cos⁡(x)sin⁡(x))y = \lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (cos(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⁡2(x)xsin⁡(x))y = x \lim_{x \to -\infty}\left(\frac{\cos^{2}{\left(x \right)}}{x \sin{\left(x \right)}}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cos⁡2(x)xsin⁡(x))y = x \lim_{x \to \infty}\left(\frac{\cos^{2}{\left(x \right)}}{x \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:
cos⁡(x)cos⁡(x)sin⁡(x)=−cos⁡2(x)sin⁡(x)\frac{\cos{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} = - \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)}}
- No
cos⁡(x)cos⁡(x)sin⁡(x)=cos⁡2(x)sin⁡(x)\frac{\cos{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} = \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)}}
- No
so, the function
not is
neither even, nor odd
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