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Plotting a function graph/ cos(2x)/sin(x)

Graphing y = cos(2x)/sin(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       cos(2*x)
f(x) = --------
        sin(x)
f(x)=cos⁡(2x)sin⁡(x)f{\left(x \right)} = \frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}} 
f = cos(2*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⁡(2x)sin⁡(x)=0\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π4x_{1} = \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}
Numerical solution
x1=−5.49778714378214x_{1} = -5.49778714378214
x2=−38.484510006475x_{2} = -38.484510006475
x3=−40.0553063332699x_{3} = -40.0553063332699
x4=−43.1968989868597x_{4} = -43.1968989868597
x5=30.6305283725005x_{5} = 30.6305283725005
x6=84.037603483527x_{6} = 84.037603483527
x7=13.3517687777566x_{7} = 13.3517687777566
x8=85.6083998103219x_{8} = 85.6083998103219
x9=54.1924732744239x_{9} = 54.1924732744239
x10=−68.329640215578x_{10} = -68.329640215578
x11=19.6349540849362x_{11} = 19.6349540849362
x12=11.7809724509617x_{12} = 11.7809724509617
x13=68.329640215578x_{13} = 68.329640215578
x14=44.7676953136546x_{14} = 44.7676953136546
x15=−54.1924732744239x_{15} = -54.1924732744239
x16=−47.9092879672443x_{16} = -47.9092879672443
x17=−18.0641577581413x_{17} = -18.0641577581413
x18=−16.4933614313464x_{18} = -16.4933614313464
x19=90.3207887907066x_{19} = 90.3207887907066
x20=8.63937979737193x_{20} = 8.63937979737193
x21=−84.037603483527x_{21} = -84.037603483527
x22=−69.9004365423729x_{22} = -69.9004365423729
x23=−19.6349540849362x_{23} = -19.6349540849362
x24=49.4800842940392x_{24} = 49.4800842940392
x25=−27.4889357189107x_{25} = -27.4889357189107
x26=−3.92699081698724x_{26} = -3.92699081698724
x27=41.6261026600648x_{27} = 41.6261026600648
x28=−55.7632696012188x_{28} = -55.7632696012188
x29=−76.1836218495525x_{29} = -76.1836218495525
x30=62.0464549083984x_{30} = 62.0464549083984
x31=−32.2013246992954x_{31} = -32.2013246992954
x32=71.4712328691678x_{32} = 71.4712328691678
x33=−65.1880475619882x_{33} = -65.1880475619882
x34=−46.3384916404494x_{34} = -46.3384916404494
x35=25.9181393921158x_{35} = 25.9181393921158
x36=−77.7544181763474x_{36} = -77.7544181763474
x37=47.9092879672443x_{37} = 47.9092879672443
x38=93.4623814442964x_{38} = 93.4623814442964
x39=91.8915851175014x_{39} = 91.8915851175014
x40=24.3473430653209x_{40} = 24.3473430653209
x41=38.484510006475x_{41} = 38.484510006475
x42=−22.776546738526x_{42} = -22.776546738526
x43=99.7455667514759x_{43} = 99.7455667514759
x44=40.0553063332699x_{44} = 40.0553063332699
x45=66.7588438887831x_{45} = 66.7588438887831
x46=−13.3517687777566x_{46} = -13.3517687777566
x47=88.7499924639117x_{47} = 88.7499924639117
x48=−21.2057504117311x_{48} = -21.2057504117311
x49=98.174770424681x_{49} = 98.174770424681
x50=10.2101761241668x_{50} = 10.2101761241668
x51=−90.3207887907066x_{51} = -90.3207887907066
x52=55.7632696012188x_{52} = 55.7632696012188
x53=−49.4800842940392x_{53} = -49.4800842940392
x54=22.776546738526x_{54} = 22.776546738526
x55=−79.3252145031423x_{55} = -79.3252145031423
x56=60.4756585816035x_{56} = 60.4756585816035
x57=74.6128255227576x_{57} = 74.6128255227576
x58=−99.7455667514759x_{58} = -99.7455667514759
x59=−24.3473430653209x_{59} = -24.3473430653209
x60=−71.4712328691678x_{60} = -71.4712328691678
x61=76.1836218495525x_{61} = 76.1836218495525
x62=3.92699081698724x_{62} = 3.92699081698724
x63=−62.0464549083984x_{63} = -62.0464549083984
x64=−33.7721210260903x_{64} = -33.7721210260903
x65=18.0641577581413x_{65} = 18.0641577581413
x66=−36.9137136796801x_{66} = -36.9137136796801
x67=−87.1791961371168x_{67} = -87.1791961371168
x68=−41.6261026600648x_{68} = -41.6261026600648
x69=−85.6083998103219x_{69} = -85.6083998103219
x70=−35.3429173528852x_{70} = -35.3429173528852
x71=52.621676947629x_{71} = 52.621676947629
x72=69.9004365423729x_{72} = 69.9004365423729
x73=96.6039740978861x_{73} = 96.6039740978861
x74=146.869456555323x_{74} = 146.869456555323
x75=−82.4668071567321x_{75} = -82.4668071567321
x76=−63.6172512351933x_{76} = -63.6172512351933
x77=−11.7809724509617x_{77} = -11.7809724509617
x78=27.4889357189107x_{78} = 27.4889357189107
x79=−10.2101761241668x_{79} = -10.2101761241668
x80=82.4668071567321x_{80} = 82.4668071567321
x81=46.3384916404494x_{81} = 46.3384916404494
x82=−93.4623814442964x_{82} = -93.4623814442964
x83=−60.4756585816035x_{83} = -60.4756585816035
x84=−91.8915851175014x_{84} = -91.8915851175014
x85=32.2013246992954x_{85} = 32.2013246992954
x86=−98.174770424681x_{86} = -98.174770424681
x87=−2.35619449019234x_{87} = -2.35619449019234
x88=−107.59954838545x_{88} = -107.59954838545
x89=63.6172512351933x_{89} = 63.6172512351933
x90=−9156.95718705085x_{90} = -9156.95718705085
x91=5.49778714378214x_{91} = 5.49778714378214
x92=−57.3340659280137x_{92} = -57.3340659280137
x93=77.7544181763474x_{93} = 77.7544181763474
x94=−25.9181393921158x_{94} = -25.9181393921158
x95=16.4933614313464x_{95} = 16.4933614313464
x96=2.35619449019234x_{96} = 2.35619449019234
x97=33.7721210260903x_{97} = 33.7721210260903
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(2*x)/sin(x).
cos⁡(0⋅2)sin⁡(0)\frac{\cos{\left(0 \cdot 2 \right)}}{\sin{\left(0 \right)}}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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⁡(2x)sin⁡(x))y = \lim_{x \to -\infty}\left(\frac{\cos{\left(2 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⁡(2x)sin⁡(x))y = \lim_{x \to \infty}\left(\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(2*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⁡(2x)xsin⁡(x))y = x \lim_{x \to -\infty}\left(\frac{\cos{\left(2 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⁡(2x)xsin⁡(x))y = x \lim_{x \to \infty}\left(\frac{\cos{\left(2 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⁡(2x)sin⁡(x)=−cos⁡(2x)sin⁡(x)\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}} = - \frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}}
- No
cos⁡(2x)sin⁡(x)=cos⁡(2x)sin⁡(x)\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}} = \frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}}
- No
so, the function
not is
neither even, nor odd
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