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  • Graphing y =:
  • -x^2+4x+5
  • x^2+3x+3
  • x^2-3x-4
  • x^2-2x-2

  • Identical expressions

  • sqrt(cot(x/ two)^ two)
  • square root of ( cotangent of (x divide by 2) squared )
  • square root of ( cotangent of (x divide by two) to the power of two)
  • √(cot(x/2)^2)
  • sqrt(cot(x/2)2)
  • sqrtcotx/22
  • sqrt(cot(x/2)²)
  • sqrt(cot(x/2) to the power of 2)
  • sqrtcotx/2^2
  • sqrt(cot(x divide by 2)^2)
Plotting a function graph/ sqrt(cot(x/2)^2)

Graphing y = sqrt(cot(x/2)^2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
           _________
          /    2/x\ 
f(x) =   /  cot |-| 
       \/       \2/
f(x)=cot⁡2(x2)f{\left(x \right)} = \sqrt{\cot^{2}{\left(\frac{x}{2} \right)}} 
f = sqrt(cot(x/2)^2)
The graph of the function
02468-8-6-4-2-101002000
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:
cot⁡2(x2)=0\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=πx_{1} = \pi
Numerical solution
x1=53.4070751110265x_{1} = 53.4070751110265
x2=−97.3893722612836x_{2} = -97.3893722612836
x3=97.3893722612836x_{3} = 97.3893722612836
x4=78.5398163397448x_{4} = 78.5398163397448
x5=−59.6902604182061x_{5} = -59.6902604182061
x6=−65.9734457253857x_{6} = -65.9734457253857
x7=21.9911485751286x_{7} = 21.9911485751286
x8=−21.9911485751286x_{8} = -21.9911485751286
x9=−15.707963267949x_{9} = -15.707963267949
x10=−34.5575191894877x_{10} = -34.5575191894877
x11=−40.8407044966673x_{11} = -40.8407044966673
x12=9.42477796076938x_{12} = 9.42477796076938
x13=34.5575191894877x_{13} = 34.5575191894877
x14=65.9734457253857x_{14} = 65.9734457253857
x15=−28.2743338823081x_{15} = -28.2743338823081
x16=−53.4070751110265x_{16} = -53.4070751110265
x17=−9.42477796076938x_{17} = -9.42477796076938
x18=40.8407044966673x_{18} = 40.8407044966673
x19=−91.106186954104x_{19} = -91.106186954104
x20=59.6902604182061x_{20} = 59.6902604182061
x21=47.1238898038469x_{21} = 47.1238898038469
x22=91.106186954104x_{22} = 91.106186954104
x23=28.2743338823081x_{23} = 28.2743338823081
x24=−47.1238898038469x_{24} = -47.1238898038469
x25=−3.14159265358979x_{25} = -3.14159265358979
x26=−72.2566310325652x_{26} = -72.2566310325652
x27=−84.8230016469244x_{27} = -84.8230016469244
x28=84.8230016469244x_{28} = 84.8230016469244
x29=72.2566310325652x_{29} = 72.2566310325652
x30=−78.5398163397448x_{30} = -78.5398163397448
x31=15.707963267949x_{31} = 15.707963267949
x32=3.14159265358979x_{32} = 3.14159265358979
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(cot(x/2)^2).
cot⁡2(02)\sqrt{\cot^{2}{\left(\frac{0}{2} \right)}}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
(−cot⁡2(x2)−1)cot⁡2(x2)2cot⁡(x2)=0\frac{\left(- \cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}}{2 \cot{\left(\frac{x}{2} \right)}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞cot⁡2(x2)y = \lim_{x \to -\infty} \sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞cot⁡2(x2)y = \lim_{x \to \infty} \sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(cot(x/2)^2), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(cot⁡2(x2)x)y = x \lim_{x \to -\infty}\left(\frac{\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡2(x2)x)y = x \lim_{x \to \infty}\left(\frac{\sqrt{\cot^{2}{\left(\frac{x}{2} \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:
cot⁡2(x2)=∣cot⁡(x2)∣\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}} = \left|{\cot{\left(\frac{x}{2} \right)}}\right|
- No
cot⁡2(x2)=−∣cot⁡(x2)∣\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}} = - \left|{\cot{\left(\frac{x}{2} \right)}}\right|
- No
so, the function
not is
neither even, nor odd
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