Mister Exam
Graphing y = sinxcos2xsin3x
The solution
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f(x) = sin(x)*cos(2*x)*sin(3*x)
$$f{\left(x \right)} = \sin{\left(x \right)} \cos{\left(2 x \right)} \sin{\left(3 x \right)}$$
f = (sin(x)*cos(2*x))*sin(3*x)
The graph of the function
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:
$$\sin{\left(x \right)} \cos{\left(2 x \right)} \sin{\left(3 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{4}$$
$$x_{3} = \frac{\pi}{4}$$
Numerical solution
$$x_{1} = -75.3982236173947$$
$$x_{2} = -13.6135681655558$$
$$x_{3} = 96.342174710087$$
$$x_{4} = -40.0553063332699$$
$$x_{5} = -83.7758040957278$$
$$x_{6} = -57.5958653158129$$
$$x_{7} = 84.037603483527$$
$$x_{8} = 84.8230017843543$$
$$x_{9} = 54.1924732744239$$
$$x_{10} = -87.9645943611421$$
$$x_{11} = 36.9137136796801$$
$$x_{12} = 63.8790506229925$$
$$x_{13} = -24.0855436775217$$
$$x_{14} = 8.37758040957278$$
$$x_{15} = -2.0943951023932$$
$$x_{16} = 75.3982238975926$$
$$x_{17} = 68.329640215578$$
$$x_{18} = 41.8879020478639$$
$$x_{19} = -97.3893723252802$$
$$x_{20} = 70.162235930172$$
$$x_{21} = 1.0471975511966$$
$$x_{22} = -19.8967534727354$$
$$x_{23} = -39.7935069454707$$
$$x_{24} = -47.9092879672443$$
$$x_{25} = 90.3207887907066$$
$$x_{26} = 8.63937979737193$$
$$x_{27} = -35.6047167406843$$
$$x_{28} = -41.8879020478639$$
$$x_{29} = 95.0331777710912$$
$$x_{30} = -97.3893720452059$$
$$x_{31} = 26.1799387799149$$
$$x_{32} = -69.9004365423729$$
$$x_{33} = 48.1710873550435$$
$$x_{34} = 50.2654824464746$$
$$x_{35} = -3.92699081698724$$
$$x_{36} = -77.4926187885482$$
$$x_{37} = 72.2566310277404$$
$$x_{38} = -55.7632696012188$$
$$x_{39} = 62.0464549083984$$
$$x_{40} = -32.2013246992954$$
$$x_{41} = -68.0678408277789$$
$$x_{42} = -15.7079632953739$$
$$x_{43} = -81.6814090344003$$
$$x_{44} = -77.7544181763474$$
$$x_{45} = 52.3598775598299$$
$$x_{46} = 85.870199198121$$
$$x_{47} = -37.6991118753633$$
$$x_{48} = 92.1533845053006$$
$$x_{49} = 40.0553063332699$$
$$x_{50} = -59.6902604550405$$
$$x_{51} = 6.28318528492648$$
$$x_{52} = -72.2566312054435$$
$$x_{53} = -63.8790506229925$$
$$x_{54} = -94.2477797173708$$
$$x_{55} = 28.2743338655404$$
$$x_{56} = -631.460123530361$$
$$x_{57} = 65.9734457515969$$
$$x_{58} = 98.174770424681$$
$$x_{59} = 10.2101761241668$$
$$x_{60} = -65.9734457663932$$
$$x_{61} = -50.2654824968965$$
$$x_{62} = 19.8967534727354$$
$$x_{63} = -90.0589894029074$$
$$x_{64} = -21.9911485866033$$
$$x_{65} = 74.3510261349584$$
$$x_{66} = 77.4926187885482$$
$$x_{67} = -49.4800842940392$$
$$x_{68} = -61.7846555205993$$
$$x_{69} = 4.18879020478639$$
$$x_{70} = 94.2477796093546$$
$$x_{71} = -43.9822971751915$$
$$x_{72} = -99.7455667514759$$
$$x_{73} = -46.0766922526503$$
$$x_{74} = -17.8023583703422$$
$$x_{75} = 43.9822971688428$$
$$x_{76} = -33.7721210260903$$
$$x_{77} = 76.1836218495525$$
$$x_{78} = 18.0641577581413$$
$$x_{79} = 21.9911485850485$$
$$x_{80} = 0$$
$$x_{81} = 30.3687289847013$$
$$x_{82} = 1276.27201552085$$
$$x_{83} = -79.5870138909414$$
$$x_{84} = -85.870199198121$$
$$x_{85} = -11.7809724509617$$
$$x_{86} = -28.2743339229441$$
$$x_{87} = 87.9645943334654$$
$$x_{88} = -91.8915851175014$$
$$x_{89} = 32.2013246992954$$
$$x_{90} = 63.6172512351933$$
$$x_{91} = -25.9181393921158$$
so we need to solve the equation:
$$\sin{\left(x \right)} \cos{\left(2 x \right)} \sin{\left(3 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{4}$$
$$x_{3} = \frac{\pi}{4}$$
Numerical solution
$$x_{1} = -75.3982236173947$$
$$x_{2} = -13.6135681655558$$
$$x_{3} = 96.342174710087$$
$$x_{4} = -40.0553063332699$$
$$x_{5} = -83.7758040957278$$
$$x_{6} = -57.5958653158129$$
$$x_{7} = 84.037603483527$$
$$x_{8} = 84.8230017843543$$
$$x_{9} = 54.1924732744239$$
$$x_{10} = -87.9645943611421$$
$$x_{11} = 36.9137136796801$$
$$x_{12} = 63.8790506229925$$
$$x_{13} = -24.0855436775217$$
$$x_{14} = 8.37758040957278$$
$$x_{15} = -2.0943951023932$$
$$x_{16} = 75.3982238975926$$
$$x_{17} = 68.329640215578$$
$$x_{18} = 41.8879020478639$$
$$x_{19} = -97.3893723252802$$
$$x_{20} = 70.162235930172$$
$$x_{21} = 1.0471975511966$$
$$x_{22} = -19.8967534727354$$
$$x_{23} = -39.7935069454707$$
$$x_{24} = -47.9092879672443$$
$$x_{25} = 90.3207887907066$$
$$x_{26} = 8.63937979737193$$
$$x_{27} = -35.6047167406843$$
$$x_{28} = -41.8879020478639$$
$$x_{29} = 95.0331777710912$$
$$x_{30} = -97.3893720452059$$
$$x_{31} = 26.1799387799149$$
$$x_{32} = -69.9004365423729$$
$$x_{33} = 48.1710873550435$$
$$x_{34} = 50.2654824464746$$
$$x_{35} = -3.92699081698724$$
$$x_{36} = -77.4926187885482$$
$$x_{37} = 72.2566310277404$$
$$x_{38} = -55.7632696012188$$
$$x_{39} = 62.0464549083984$$
$$x_{40} = -32.2013246992954$$
$$x_{41} = -68.0678408277789$$
$$x_{42} = -15.7079632953739$$
$$x_{43} = -81.6814090344003$$
$$x_{44} = -77.7544181763474$$
$$x_{45} = 52.3598775598299$$
$$x_{46} = 85.870199198121$$
$$x_{47} = -37.6991118753633$$
$$x_{48} = 92.1533845053006$$
$$x_{49} = 40.0553063332699$$
$$x_{50} = -59.6902604550405$$
$$x_{51} = 6.28318528492648$$
$$x_{52} = -72.2566312054435$$
$$x_{53} = -63.8790506229925$$
$$x_{54} = -94.2477797173708$$
$$x_{55} = 28.2743338655404$$
$$x_{56} = -631.460123530361$$
$$x_{57} = 65.9734457515969$$
$$x_{58} = 98.174770424681$$
$$x_{59} = 10.2101761241668$$
$$x_{60} = -65.9734457663932$$
$$x_{61} = -50.2654824968965$$
$$x_{62} = 19.8967534727354$$
$$x_{63} = -90.0589894029074$$
$$x_{64} = -21.9911485866033$$
$$x_{65} = 74.3510261349584$$
$$x_{66} = 77.4926187885482$$
$$x_{67} = -49.4800842940392$$
$$x_{68} = -61.7846555205993$$
$$x_{69} = 4.18879020478639$$
$$x_{70} = 94.2477796093546$$
$$x_{71} = -43.9822971751915$$
$$x_{72} = -99.7455667514759$$
$$x_{73} = -46.0766922526503$$
$$x_{74} = -17.8023583703422$$
$$x_{75} = 43.9822971688428$$
$$x_{76} = -33.7721210260903$$
$$x_{77} = 76.1836218495525$$
$$x_{78} = 18.0641577581413$$
$$x_{79} = 21.9911485850485$$
$$x_{80} = 0$$
$$x_{81} = 30.3687289847013$$
$$x_{82} = 1276.27201552085$$
$$x_{83} = -79.5870138909414$$
$$x_{84} = -85.870199198121$$
$$x_{85} = -11.7809724509617$$
$$x_{86} = -28.2743339229441$$
$$x_{87} = 87.9645943334654$$
$$x_{88} = -91.8915851175014$$
$$x_{89} = 32.2013246992954$$
$$x_{90} = 63.6172512351933$$
$$x_{91} = -25.9181393921158$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sin{\left(x \right)} \cos{\left(2 x \right)} \sin{\left(3 x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(x \right)} \cos{\left(2 x \right)} \sin{\left(3 x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to -\infty}\left(\sin{\left(x \right)} \cos{\left(2 x \right)} \sin{\left(3 x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(x \right)} \cos{\left(2 x \right)} \sin{\left(3 x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (sin(x)*cos(2*x))*sin(3*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \sin{\left(3 x \right)} \cos{\left(2 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \sin{\left(3 x \right)} \cos{\left(2 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \sin{\left(3 x \right)} \cos{\left(2 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \sin{\left(3 x \right)} \cos{\left(2 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
$$\sin{\left(x \right)} \cos{\left(2 x \right)} \sin{\left(3 x \right)} = \sin{\left(x \right)} \sin{\left(3 x \right)} \cos{\left(2 x \right)}$$
- No
$$\sin{\left(x \right)} \cos{\left(2 x \right)} \sin{\left(3 x \right)} = - \sin{\left(x \right)} \sin{\left(3 x \right)} \cos{\left(2 x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sin{\left(x \right)} \cos{\left(2 x \right)} \sin{\left(3 x \right)} = \sin{\left(x \right)} \sin{\left(3 x \right)} \cos{\left(2 x \right)}$$
- No
$$\sin{\left(x \right)} \cos{\left(2 x \right)} \sin{\left(3 x \right)} = - \sin{\left(x \right)} \sin{\left(3 x \right)} \cos{\left(2 x \right)}$$
- No
so, the function
not is
neither even, nor odd