Mister Exam
Graphing y = cos(2x)/sin(x)
The solution
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cos(2*x)
f(x) = --------
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
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
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:
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{4}$$
$$x_{2} = \frac{3 \pi}{4}$$
Numerical solution
$$x_{1} = -5.49778714378214$$
$$x_{2} = -38.484510006475$$
$$x_{3} = -40.0553063332699$$
$$x_{4} = -43.1968989868597$$
$$x_{5} = 30.6305283725005$$
$$x_{6} = 84.037603483527$$
$$x_{7} = 13.3517687777566$$
$$x_{8} = 85.6083998103219$$
$$x_{9} = 54.1924732744239$$
$$x_{10} = -68.329640215578$$
$$x_{11} = 19.6349540849362$$
$$x_{12} = 11.7809724509617$$
$$x_{13} = 68.329640215578$$
$$x_{14} = 44.7676953136546$$
$$x_{15} = -54.1924732744239$$
$$x_{16} = -47.9092879672443$$
$$x_{17} = -18.0641577581413$$
$$x_{18} = -16.4933614313464$$
$$x_{19} = 90.3207887907066$$
$$x_{20} = 8.63937979737193$$
$$x_{21} = -84.037603483527$$
$$x_{22} = -69.9004365423729$$
$$x_{23} = -19.6349540849362$$
$$x_{24} = 49.4800842940392$$
$$x_{25} = -27.4889357189107$$
$$x_{26} = -3.92699081698724$$
$$x_{27} = 41.6261026600648$$
$$x_{28} = -55.7632696012188$$
$$x_{29} = -76.1836218495525$$
$$x_{30} = 62.0464549083984$$
$$x_{31} = -32.2013246992954$$
$$x_{32} = 71.4712328691678$$
$$x_{33} = -65.1880475619882$$
$$x_{34} = -46.3384916404494$$
$$x_{35} = 25.9181393921158$$
$$x_{36} = -77.7544181763474$$
$$x_{37} = 47.9092879672443$$
$$x_{38} = 93.4623814442964$$
$$x_{39} = 91.8915851175014$$
$$x_{40} = 24.3473430653209$$
$$x_{41} = 38.484510006475$$
$$x_{42} = -22.776546738526$$
$$x_{43} = 99.7455667514759$$
$$x_{44} = 40.0553063332699$$
$$x_{45} = 66.7588438887831$$
$$x_{46} = -13.3517687777566$$
$$x_{47} = 88.7499924639117$$
$$x_{48} = -21.2057504117311$$
$$x_{49} = 98.174770424681$$
$$x_{50} = 10.2101761241668$$
$$x_{51} = -90.3207887907066$$
$$x_{52} = 55.7632696012188$$
$$x_{53} = -49.4800842940392$$
$$x_{54} = 22.776546738526$$
$$x_{55} = -79.3252145031423$$
$$x_{56} = 60.4756585816035$$
$$x_{57} = 74.6128255227576$$
$$x_{58} = -99.7455667514759$$
$$x_{59} = -24.3473430653209$$
$$x_{60} = -71.4712328691678$$
$$x_{61} = 76.1836218495525$$
$$x_{62} = 3.92699081698724$$
$$x_{63} = -62.0464549083984$$
$$x_{64} = -33.7721210260903$$
$$x_{65} = 18.0641577581413$$
$$x_{66} = -36.9137136796801$$
$$x_{67} = -87.1791961371168$$
$$x_{68} = -41.6261026600648$$
$$x_{69} = -85.6083998103219$$
$$x_{70} = -35.3429173528852$$
$$x_{71} = 52.621676947629$$
$$x_{72} = 69.9004365423729$$
$$x_{73} = 96.6039740978861$$
$$x_{74} = 146.869456555323$$
$$x_{75} = -82.4668071567321$$
$$x_{76} = -63.6172512351933$$
$$x_{77} = -11.7809724509617$$
$$x_{78} = 27.4889357189107$$
$$x_{79} = -10.2101761241668$$
$$x_{80} = 82.4668071567321$$
$$x_{81} = 46.3384916404494$$
$$x_{82} = -93.4623814442964$$
$$x_{83} = -60.4756585816035$$
$$x_{84} = -91.8915851175014$$
$$x_{85} = 32.2013246992954$$
$$x_{86} = -98.174770424681$$
$$x_{87} = -2.35619449019234$$
$$x_{88} = -107.59954838545$$
$$x_{89} = 63.6172512351933$$
$$x_{90} = -9156.95718705085$$
$$x_{91} = 5.49778714378214$$
$$x_{92} = -57.3340659280137$$
$$x_{93} = 77.7544181763474$$
$$x_{94} = -25.9181393921158$$
$$x_{95} = 16.4933614313464$$
$$x_{96} = 2.35619449019234$$
$$x_{97} = 33.7721210260903$$
so we need to solve the equation:
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{4}$$
$$x_{2} = \frac{3 \pi}{4}$$
Numerical solution
$$x_{1} = -5.49778714378214$$
$$x_{2} = -38.484510006475$$
$$x_{3} = -40.0553063332699$$
$$x_{4} = -43.1968989868597$$
$$x_{5} = 30.6305283725005$$
$$x_{6} = 84.037603483527$$
$$x_{7} = 13.3517687777566$$
$$x_{8} = 85.6083998103219$$
$$x_{9} = 54.1924732744239$$
$$x_{10} = -68.329640215578$$
$$x_{11} = 19.6349540849362$$
$$x_{12} = 11.7809724509617$$
$$x_{13} = 68.329640215578$$
$$x_{14} = 44.7676953136546$$
$$x_{15} = -54.1924732744239$$
$$x_{16} = -47.9092879672443$$
$$x_{17} = -18.0641577581413$$
$$x_{18} = -16.4933614313464$$
$$x_{19} = 90.3207887907066$$
$$x_{20} = 8.63937979737193$$
$$x_{21} = -84.037603483527$$
$$x_{22} = -69.9004365423729$$
$$x_{23} = -19.6349540849362$$
$$x_{24} = 49.4800842940392$$
$$x_{25} = -27.4889357189107$$
$$x_{26} = -3.92699081698724$$
$$x_{27} = 41.6261026600648$$
$$x_{28} = -55.7632696012188$$
$$x_{29} = -76.1836218495525$$
$$x_{30} = 62.0464549083984$$
$$x_{31} = -32.2013246992954$$
$$x_{32} = 71.4712328691678$$
$$x_{33} = -65.1880475619882$$
$$x_{34} = -46.3384916404494$$
$$x_{35} = 25.9181393921158$$
$$x_{36} = -77.7544181763474$$
$$x_{37} = 47.9092879672443$$
$$x_{38} = 93.4623814442964$$
$$x_{39} = 91.8915851175014$$
$$x_{40} = 24.3473430653209$$
$$x_{41} = 38.484510006475$$
$$x_{42} = -22.776546738526$$
$$x_{43} = 99.7455667514759$$
$$x_{44} = 40.0553063332699$$
$$x_{45} = 66.7588438887831$$
$$x_{46} = -13.3517687777566$$
$$x_{47} = 88.7499924639117$$
$$x_{48} = -21.2057504117311$$
$$x_{49} = 98.174770424681$$
$$x_{50} = 10.2101761241668$$
$$x_{51} = -90.3207887907066$$
$$x_{52} = 55.7632696012188$$
$$x_{53} = -49.4800842940392$$
$$x_{54} = 22.776546738526$$
$$x_{55} = -79.3252145031423$$
$$x_{56} = 60.4756585816035$$
$$x_{57} = 74.6128255227576$$
$$x_{58} = -99.7455667514759$$
$$x_{59} = -24.3473430653209$$
$$x_{60} = -71.4712328691678$$
$$x_{61} = 76.1836218495525$$
$$x_{62} = 3.92699081698724$$
$$x_{63} = -62.0464549083984$$
$$x_{64} = -33.7721210260903$$
$$x_{65} = 18.0641577581413$$
$$x_{66} = -36.9137136796801$$
$$x_{67} = -87.1791961371168$$
$$x_{68} = -41.6261026600648$$
$$x_{69} = -85.6083998103219$$
$$x_{70} = -35.3429173528852$$
$$x_{71} = 52.621676947629$$
$$x_{72} = 69.9004365423729$$
$$x_{73} = 96.6039740978861$$
$$x_{74} = 146.869456555323$$
$$x_{75} = -82.4668071567321$$
$$x_{76} = -63.6172512351933$$
$$x_{77} = -11.7809724509617$$
$$x_{78} = 27.4889357189107$$
$$x_{79} = -10.2101761241668$$
$$x_{80} = 82.4668071567321$$
$$x_{81} = 46.3384916404494$$
$$x_{82} = -93.4623814442964$$
$$x_{83} = -60.4756585816035$$
$$x_{84} = -91.8915851175014$$
$$x_{85} = 32.2013246992954$$
$$x_{86} = -98.174770424681$$
$$x_{87} = -2.35619449019234$$
$$x_{88} = -107.59954838545$$
$$x_{89} = 63.6172512351933$$
$$x_{90} = -9156.95718705085$$
$$x_{91} = 5.49778714378214$$
$$x_{92} = -57.3340659280137$$
$$x_{93} = 77.7544181763474$$
$$x_{94} = -25.9181393921158$$
$$x_{95} = 16.4933614313464$$
$$x_{96} = 2.35619449019234$$
$$x_{97} = 33.7721210260903$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}}\right)$$
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
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cos{\left(2 x \right)}}{x \sin{\left(x \right)}}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cos{\left(2 x \right)}}{x \sin{\left(x \right)}}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cos{\left(2 x \right)}}{x \sin{\left(x \right)}}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cos{\left(2 x \right)}}{x \sin{\left(x \right)}}\right)$$
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:
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}} = - \frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}}$$
- No
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}} = \frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}} = - \frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}}$$
- No
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}} = \frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd