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arctg(tg(x/sqrt(2)))
Plotting a function graph/ arctg(tg(x/sqrt(2)))

Graphing y = arctg(tg(x/sqrt(2)))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
           /   /  x  \\
f(x) = atan|tan|-----||
           |   |  ___||
           \   \\/ 2 //
f(x)=atan⁡(tan⁡(x2))f{\left(x \right)} = \operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)} 
f = atan(tan(x/(sqrt(2))))
The graph of the function
05-30-25-20-15-10-510152025305-5
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:
atan⁡(tan⁡(x2))=0\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−31.1001805671086x_{1} = -31.1001805671086
x2=−13.3286488144751x_{2} = -13.3286488144751
x3=31.1001805671086x_{3} = 31.1001805671086
x4=102.186307577642x_{4} = 102.186307577642
x5=−48.871712319742x_{5} = -48.871712319742
x6=−62.2003611342171x_{6} = -62.2003611342171
x7=62.2003611342171x_{7} = 62.2003611342171
x8=−93.3005417013257x_{8} = -93.3005417013257
x9=0x_{9} = 0
x10=−53.3145952579004x_{10} = -53.3145952579004
x11=79.9718928868506x_{11} = 79.9718928868506
x12=97.7434246394841x_{12} = 97.7434246394841
x13=−44.4288293815837x_{13} = -44.4288293815837
x14=−75.5290099486922x_{14} = -75.5290099486922
x15=−26.6572976289502x_{15} = -26.6572976289502
x16=57.7574781960588x_{16} = 57.7574781960588
x17=−97.7434246394841x_{17} = -97.7434246394841
x18=8.88576587631673x_{18} = 8.88576587631673
x19=53.3145952579004x_{19} = 53.3145952579004
x20=−71.0861270105339x_{20} = -71.0861270105339
x21=88.8576587631673x_{21} = 88.8576587631673
x22=−8.88576587631673x_{22} = -8.88576587631673
x23=−66.6432440723755x_{23} = -66.6432440723755
x24=−79.9718928868506x_{24} = -79.9718928868506
x25=−4.44288293815837x_{25} = -4.44288293815837
x26=17.7715317526335x_{26} = 17.7715317526335
x27=39.9859464434253x_{27} = 39.9859464434253
x28=48.871712319742x_{28} = 48.871712319742
x29=93.3005417013257x_{29} = 93.3005417013257
x30=−17.7715317526335x_{30} = -17.7715317526335
x31=44.4288293815837x_{31} = 44.4288293815837
x32=−35.5430635052669x_{32} = -35.5430635052669
x33=−84.414775825009x_{33} = -84.414775825009
x34=−22.2144146907918x_{34} = -22.2144146907918
x35=35.5430635052669x_{35} = 35.5430635052669
x36=−88.8576587631673x_{36} = -88.8576587631673
x37=71.0861270105339x_{37} = 71.0861270105339
x38=−57.7574781960588x_{38} = -57.7574781960588
x39=75.5290099486922x_{39} = 75.5290099486922
x40=−39.9859464434253x_{40} = -39.9859464434253
x41=66.6432440723755x_{41} = 66.6432440723755
x42=13.3286488144751x_{42} = 13.3286488144751
x43=4.44288293815837x_{43} = 4.44288293815837
x44=26.6572976289502x_{44} = 26.6572976289502
x45=84.414775825009x_{45} = 84.414775825009
x46=22.2144146907918x_{46} = 22.2144146907918
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to atan(tan(x/(sqrt(2)))).
atan⁡(tan⁡(02))\operatorname{atan}{\left(\tan{\left(\frac{0}{\sqrt{2}} \right)} \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
22=0\frac{\sqrt{2}}{2} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
0=00 = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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:
atan⁡(tan⁡(x2))=−atan⁡(tan⁡(22x))\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)} = - \operatorname{atan}{\left(\tan{\left(\frac{\sqrt{2}}{2} x \right)} \right)}
- No
atan⁡(tan⁡(x2))=atan⁡(tan⁡(22x))\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)} = \operatorname{atan}{\left(\tan{\left(\frac{\sqrt{2}}{2} x \right)} \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = arctg(tg(x/sqrt(2)))
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