Mister Exam
Graphing y = tan2x+sin2x
The solution
You have entered
[src]
f(x) = tan(2*x) + sin(2*x)
$$f{\left(x \right)} = \sin{\left(2 x \right)} + \tan{\left(2 x \right)}$$
f = sin(2*x) + tan(2*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:
$$\sin{\left(2 x \right)} + \tan{\left(2 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
$$x_{4} = \pi$$
Numerical solution
$$x_{1} = 12.5663706143592$$
$$x_{2} = 78.5398163397448$$
$$x_{3} = -65.9734457253857$$
$$x_{4} = -15.707963267949$$
$$x_{5} = -34.5575191894877$$
$$x_{6} = -69.1150383789755$$
$$x_{7} = 50.2654824574367$$
$$x_{8} = 81.6814089933346$$
$$x_{9} = 51.836329140843$$
$$x_{10} = -56.5486677646163$$
$$x_{11} = -25.1327412287183$$
$$x_{12} = -100.530964914873$$
$$x_{13} = -91.106186954104$$
$$x_{14} = 18.8495559215388$$
$$x_{15} = -75.398223686155$$
$$x_{16} = 7.85402696220054$$
$$x_{17} = 42.4114601529741$$
$$x_{18} = -80.1105774908369$$
$$x_{19} = 56.5486677646163$$
$$x_{20} = -51.8362622743528$$
$$x_{21} = -67.5442938266249$$
$$x_{22} = 75.398223686155$$
$$x_{23} = 14.1371750231145$$
$$x_{24} = -14.1371241329818$$
$$x_{25} = 15.707963267949$$
$$x_{26} = -59.6902604182061$$
$$x_{27} = 21.9911485751286$$
$$x_{28} = 6.28318530717959$$
$$x_{29} = 64.4026109201766$$
$$x_{30} = 62.8318530717959$$
$$x_{31} = -1.57084140427182$$
$$x_{32} = -87.9645943005142$$
$$x_{33} = 9.42477796076938$$
$$x_{34} = -23.5619921981979$$
$$x_{35} = -28.2743338823081$$
$$x_{36} = -9.42477796076938$$
$$x_{37} = 59.6902604182061$$
$$x_{38} = 28.2743338823081$$
$$x_{39} = -47.1238898038469$$
$$x_{40} = -3.14159265358979$$
$$x_{41} = 31.4159265358979$$
$$x_{42} = 94.2477796076938$$
$$x_{43} = -72.2566310325652$$
$$x_{44} = 53.4070751110265$$
$$x_{45} = -97.3893722612836$$
$$x_{46} = 37.6991118430775$$
$$x_{47} = -45.5531430049699$$
$$x_{48} = -50.2654824574367$$
$$x_{49} = -94.2477796076938$$
$$x_{50} = 29.8451780521704$$
$$x_{51} = -58.1194263552532$$
$$x_{52} = -37.6991118430775$$
$$x_{53} = -36.1282752374094$$
$$x_{54} = 58.1194602352783$$
$$x_{55} = 20.4203093788003$$
$$x_{56} = -12.5663706143592$$
$$x_{57} = 86.3937616823144$$
$$x_{58} = -7.85396913241537$$
$$x_{59} = -81.6814089933346$$
$$x_{60} = 80.1106033207563$$
$$x_{61} = -73.8274107880694$$
$$x_{62} = 43.9822971502571$$
$$x_{63} = -78.5398163397448$$
$$x_{64} = 97.3893722612836$$
$$x_{65} = -31.4159265358979$$
$$x_{66} = -29.8451149451132$$
$$x_{67} = 0$$
$$x_{68} = -21.9911485751286$$
$$x_{69} = 100.530964914873$$
$$x_{70} = 34.5575191894877$$
$$x_{71} = 65.9734457253857$$
$$x_{72} = -95.8185601488805$$
$$x_{73} = 36.1283178447041$$
$$x_{74} = 73.8274802320386$$
$$x_{75} = -53.4070751110265$$
$$x_{76} = 40.8407044966673$$
$$x_{77} = -6.28318530717959$$
$$x_{78} = -89.5354446652553$$
$$x_{79} = -43.9822971502571$$
$$x_{80} = 84.8230016469244$$
$$x_{81} = 95.8186313295079$$
$$x_{82} = 72.2566310325652$$
$$x_{83} = 87.9645943005142$$
so we need to solve the equation:
$$\sin{\left(2 x \right)} + \tan{\left(2 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
$$x_{4} = \pi$$
Numerical solution
$$x_{1} = 12.5663706143592$$
$$x_{2} = 78.5398163397448$$
$$x_{3} = -65.9734457253857$$
$$x_{4} = -15.707963267949$$
$$x_{5} = -34.5575191894877$$
$$x_{6} = -69.1150383789755$$
$$x_{7} = 50.2654824574367$$
$$x_{8} = 81.6814089933346$$
$$x_{9} = 51.836329140843$$
$$x_{10} = -56.5486677646163$$
$$x_{11} = -25.1327412287183$$
$$x_{12} = -100.530964914873$$
$$x_{13} = -91.106186954104$$
$$x_{14} = 18.8495559215388$$
$$x_{15} = -75.398223686155$$
$$x_{16} = 7.85402696220054$$
$$x_{17} = 42.4114601529741$$
$$x_{18} = -80.1105774908369$$
$$x_{19} = 56.5486677646163$$
$$x_{20} = -51.8362622743528$$
$$x_{21} = -67.5442938266249$$
$$x_{22} = 75.398223686155$$
$$x_{23} = 14.1371750231145$$
$$x_{24} = -14.1371241329818$$
$$x_{25} = 15.707963267949$$
$$x_{26} = -59.6902604182061$$
$$x_{27} = 21.9911485751286$$
$$x_{28} = 6.28318530717959$$
$$x_{29} = 64.4026109201766$$
$$x_{30} = 62.8318530717959$$
$$x_{31} = -1.57084140427182$$
$$x_{32} = -87.9645943005142$$
$$x_{33} = 9.42477796076938$$
$$x_{34} = -23.5619921981979$$
$$x_{35} = -28.2743338823081$$
$$x_{36} = -9.42477796076938$$
$$x_{37} = 59.6902604182061$$
$$x_{38} = 28.2743338823081$$
$$x_{39} = -47.1238898038469$$
$$x_{40} = -3.14159265358979$$
$$x_{41} = 31.4159265358979$$
$$x_{42} = 94.2477796076938$$
$$x_{43} = -72.2566310325652$$
$$x_{44} = 53.4070751110265$$
$$x_{45} = -97.3893722612836$$
$$x_{46} = 37.6991118430775$$
$$x_{47} = -45.5531430049699$$
$$x_{48} = -50.2654824574367$$
$$x_{49} = -94.2477796076938$$
$$x_{50} = 29.8451780521704$$
$$x_{51} = -58.1194263552532$$
$$x_{52} = -37.6991118430775$$
$$x_{53} = -36.1282752374094$$
$$x_{54} = 58.1194602352783$$
$$x_{55} = 20.4203093788003$$
$$x_{56} = -12.5663706143592$$
$$x_{57} = 86.3937616823144$$
$$x_{58} = -7.85396913241537$$
$$x_{59} = -81.6814089933346$$
$$x_{60} = 80.1106033207563$$
$$x_{61} = -73.8274107880694$$
$$x_{62} = 43.9822971502571$$
$$x_{63} = -78.5398163397448$$
$$x_{64} = 97.3893722612836$$
$$x_{65} = -31.4159265358979$$
$$x_{66} = -29.8451149451132$$
$$x_{67} = 0$$
$$x_{68} = -21.9911485751286$$
$$x_{69} = 100.530964914873$$
$$x_{70} = 34.5575191894877$$
$$x_{71} = 65.9734457253857$$
$$x_{72} = -95.8185601488805$$
$$x_{73} = 36.1283178447041$$
$$x_{74} = 73.8274802320386$$
$$x_{75} = -53.4070751110265$$
$$x_{76} = 40.8407044966673$$
$$x_{77} = -6.28318530717959$$
$$x_{78} = -89.5354446652553$$
$$x_{79} = -43.9822971502571$$
$$x_{80} = 84.8230016469244$$
$$x_{81} = 95.8186313295079$$
$$x_{82} = 72.2566310325652$$
$$x_{83} = 87.9645943005142$$
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:
$$y = \lim_{x \to -\infty}\left(\sin{\left(2 x \right)} + \tan{\left(2 x \right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\sin{\left(2 x \right)} + \tan{\left(2 x \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\sin{\left(2 x \right)} + \tan{\left(2 x \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\sin{\left(2 x \right)} + \tan{\left(2 x \right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(2*x) + sin(2*x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} + \tan{\left(2 x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} + \tan{\left(2 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} + \tan{\left(2 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} + \tan{\left(2 x \right)}}{x}\right)$$
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:
$$\sin{\left(2 x \right)} + \tan{\left(2 x \right)} = - \sin{\left(2 x \right)} - \tan{\left(2 x \right)}$$
- No
$$\sin{\left(2 x \right)} + \tan{\left(2 x \right)} = \sin{\left(2 x \right)} + \tan{\left(2 x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sin{\left(2 x \right)} + \tan{\left(2 x \right)} = - \sin{\left(2 x \right)} - \tan{\left(2 x \right)}$$
- No
$$\sin{\left(2 x \right)} + \tan{\left(2 x \right)} = \sin{\left(2 x \right)} + \tan{\left(2 x \right)}$$
- No
so, the function
not is
neither even, nor odd