Mister Exam
Graphing y = tan(tan(x))/sin(sin(x))
The solution
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tan(tan(x))
f(x) = -----------
sin(sin(x))
$$f{\left(x \right)} = \frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}}$$
f = tan(tan(x))/sin(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:
$$\frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 92.3688142097829$$
$$x_{2} = -89.8435596984251$$
$$x_{3} = 246.923192377915$$
$$x_{4} = -86.0856289026033$$
$$x_{5} = 158.571018876118$$
$$x_{6} = 73.669596169072$$
$$x_{7} = 8.16215070509047$$
$$x_{8} = -83.5603743912455$$
$$x_{9} = -64.0944803274748$$
$$x_{10} = -95.9242837308692$$
$$x_{11} = 79.9527814762516$$
$$x_{12} = -17.5869286658598$$
$$x_{13} = -57.8112950202952$$
$$x_{14} = 51.7727026025533$$
$$x_{15} = -70.5280035154822$$
$$x_{16} = 23.2537758308075$$
$$x_{17} = -20.1121831772177$$
$$x_{18} = -1.87896539791088$$
$$x_{19} = -315.421892614658$$
$$x_{20} = -42.7196698945782$$
$$x_{21} = 20.4004605044607$$
$$x_{22} = -95.5104068633727$$
$$x_{23} = 111.218370131322$$
$$x_{24} = 42.253669633174$$
$$x_{25} = -35.8201464451666$$
$$x_{26} = -95.5104068633728$$
$$x_{27} = -10.8377430972761$$
$$x_{28} = 14.4453360122701$$
$$x_{29} = -92.5191520906107$$
$$x_{30} = -30.0029613993912$$
$$x_{31} = -42.3584988627726$$
$$x_{32} = -48.85251732093$$
$$x_{33} = -7.748273837594$$
$$x_{34} = 48.3865170595258$$
$$x_{35} = -42.1033317523462$$
$$x_{36} = -900.374464324592$$
$$x_{37} = 95.7549997528104$$
$$x_{38} = 96.1267450056047$$
$$x_{39} = -70.9940037768863$$
$$x_{40} = 1.87896539791088$$
$$x_{41} = -39.5780772409884$$
$$x_{42} = -13.8289978700381$$
$$x_{43} = -79.8024435954237$$
$$x_{44} = -73.9852585496483$$
$$x_{45} = 51.9419865806121$$
$$x_{46} = 64.2448182083026$$
$$x_{47} = 10.6874052164483$$
$$x_{48} = -74.1355964304761$$
$$x_{49} = 61.1032255547128$$
$$x_{50} = 54.6697023667054$$
$$x_{51} = 80.4187817376557$$
$$x_{52} = -23.6676526983039$$
$$x_{53} = -140.109042155862$$
$$x_{54} = 57.8112950202952$$
$$x_{55} = 86.0856289026033$$
$$x_{56} = 58.4276331625272$$
$$x_{57} = 102.409930312784$$
$$x_{58} = 83.146497523749$$
$$x_{59} = 36.2861467065708$$
$$x_{60} = -51.5281097131156$$
$$x_{61} = 4.40421990926871$$
$$x_{62} = 23.6676526983039$$
$$x_{63} = 35.8201464451666$$
$$x_{64} = -60.952887673885$$
$$x_{65} = 26.3953684843973$$
$$x_{66} = 70.3776656346544$$
so we need to solve the equation:
$$\frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 92.3688142097829$$
$$x_{2} = -89.8435596984251$$
$$x_{3} = 246.923192377915$$
$$x_{4} = -86.0856289026033$$
$$x_{5} = 158.571018876118$$
$$x_{6} = 73.669596169072$$
$$x_{7} = 8.16215070509047$$
$$x_{8} = -83.5603743912455$$
$$x_{9} = -64.0944803274748$$
$$x_{10} = -95.9242837308692$$
$$x_{11} = 79.9527814762516$$
$$x_{12} = -17.5869286658598$$
$$x_{13} = -57.8112950202952$$
$$x_{14} = 51.7727026025533$$
$$x_{15} = -70.5280035154822$$
$$x_{16} = 23.2537758308075$$
$$x_{17} = -20.1121831772177$$
$$x_{18} = -1.87896539791088$$
$$x_{19} = -315.421892614658$$
$$x_{20} = -42.7196698945782$$
$$x_{21} = 20.4004605044607$$
$$x_{22} = -95.5104068633727$$
$$x_{23} = 111.218370131322$$
$$x_{24} = 42.253669633174$$
$$x_{25} = -35.8201464451666$$
$$x_{26} = -95.5104068633728$$
$$x_{27} = -10.8377430972761$$
$$x_{28} = 14.4453360122701$$
$$x_{29} = -92.5191520906107$$
$$x_{30} = -30.0029613993912$$
$$x_{31} = -42.3584988627726$$
$$x_{32} = -48.85251732093$$
$$x_{33} = -7.748273837594$$
$$x_{34} = 48.3865170595258$$
$$x_{35} = -42.1033317523462$$
$$x_{36} = -900.374464324592$$
$$x_{37} = 95.7549997528104$$
$$x_{38} = 96.1267450056047$$
$$x_{39} = -70.9940037768863$$
$$x_{40} = 1.87896539791088$$
$$x_{41} = -39.5780772409884$$
$$x_{42} = -13.8289978700381$$
$$x_{43} = -79.8024435954237$$
$$x_{44} = -73.9852585496483$$
$$x_{45} = 51.9419865806121$$
$$x_{46} = 64.2448182083026$$
$$x_{47} = 10.6874052164483$$
$$x_{48} = -74.1355964304761$$
$$x_{49} = 61.1032255547128$$
$$x_{50} = 54.6697023667054$$
$$x_{51} = 80.4187817376557$$
$$x_{52} = -23.6676526983039$$
$$x_{53} = -140.109042155862$$
$$x_{54} = 57.8112950202952$$
$$x_{55} = 86.0856289026033$$
$$x_{56} = 58.4276331625272$$
$$x_{57} = 102.409930312784$$
$$x_{58} = 83.146497523749$$
$$x_{59} = 36.2861467065708$$
$$x_{60} = -51.5281097131156$$
$$x_{61} = 4.40421990926871$$
$$x_{62} = 23.6676526983039$$
$$x_{63} = 35.8201464451666$$
$$x_{64} = -60.952887673885$$
$$x_{65} = 26.3953684843973$$
$$x_{66} = 70.3776656346544$$
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:
$$y = \lim_{x \to -\infty}\left(\frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(tan(x))/sin(sin(x)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(\tan{\left(x \right)} \right)}}{x \sin{\left(\sin{\left(x \right)} \right)}}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\tan{\left(x \right)} \right)}}{x \sin{\left(\sin{\left(x \right)} \right)}}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(\tan{\left(x \right)} \right)}}{x \sin{\left(\sin{\left(x \right)} \right)}}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\tan{\left(x \right)} \right)}}{x \sin{\left(\sin{\left(x \right)} \right)}}\right)$$
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:
$$\frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}} = \frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}}$$
- No
$$\frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}} = - \frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}} = \frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}}$$
- No
$$\frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}} = - \frac{\tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(\sin{\left(x \right)} \right)}}$$
- No
so, the function
not is
neither even, nor odd