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tan(x/2)
Plotting a function graph/ tan(x/2)

Graphing y = tan(x/2)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /x\
f(x) = tan|-|
          \2/
f(x)=tan⁡(x2)f{\left(x \right)} = \tan{\left(\frac{x}{2} \right)} 
f = tan(x/2)
The graph of the function
0-30-20-10102030405060708090-2000020000
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:
tan⁡(x2)=0\tan{\left(\frac{x}{2} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=94.2477796076938x_{1} = 94.2477796076938
x2=−37.6991118430775x_{2} = -37.6991118430775
x3=−31.4159265358979x_{3} = -31.4159265358979
x4=−81.6814089933346x_{4} = -81.6814089933346
x5=6.28318530717959x_{5} = 6.28318530717959
x6=62.8318530717959x_{6} = 62.8318530717959
x7=81.6814089933346x_{7} = 81.6814089933346
x8=100.530964914873x_{8} = 100.530964914873
x9=50.2654824574367x_{9} = 50.2654824574367
x10=−56.5486677646163x_{10} = -56.5486677646163
x11=25.1327412287183x_{11} = 25.1327412287183
x12=−87.9645943005142x_{12} = -87.9645943005142
x13=−6.28318530717959x_{13} = -6.28318530717959
x14=12.5663706143592x_{14} = 12.5663706143592
x15=75.398223686155x_{15} = 75.398223686155
x16=56.5486677646163x_{16} = 56.5486677646163
x17=−12.5663706143592x_{17} = -12.5663706143592
x18=18.8495559215388x_{18} = 18.8495559215388
x19=−94.2477796076938x_{19} = -94.2477796076938
x20=−25.1327412287183x_{20} = -25.1327412287183
x21=−75.398223686155x_{21} = -75.398223686155
x22=31.4159265358979x_{22} = 31.4159265358979
x23=43.9822971502571x_{23} = 43.9822971502571
x24=−50.2654824574367x_{24} = -50.2654824574367
x25=−43.9822971502571x_{25} = -43.9822971502571
x26=69.1150383789755x_{26} = 69.1150383789755
x27=−18.8495559215388x_{27} = -18.8495559215388
x28=0x_{28} = 0
x29=−62.8318530717959x_{29} = -62.8318530717959
x30=87.9645943005142x_{30} = 87.9645943005142
x31=37.6991118430775x_{31} = 37.6991118430775
x32=−100.530964914873x_{32} = -100.530964914873
x33=−69.1150383789755x_{33} = -69.1150383789755
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(x/2).
tan⁡(02)\tan{\left(\frac{0}{2} \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
tan⁡2(x2)2+12=0\frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{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
(tan⁡2(x2)+1)tan⁡(x2)2=0\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[0,∞)\left[0, \infty\right)
Convex at the intervals
(−∞,0]\left(-\infty, 0\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞tan⁡(x2)=⟨−∞,∞⟩\lim_{x \to -\infty} \tan{\left(\frac{x}{2} \right)} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞tan⁡(x2)=⟨−∞,∞⟩\lim_{x \to \infty} \tan{\left(\frac{x}{2} \right)} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x/2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(tan⁡(x2)x)=lim⁡x→−∞(tan⁡(x2)x)\lim_{x \to -\infty}\left(\frac{\tan{\left(\frac{x}{2} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(\frac{x}{2} \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(tan⁡(x2)x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(\frac{x}{2} \right)}}{x}\right)
lim⁡x→∞(tan⁡(x2)x)=lim⁡x→∞(tan⁡(x2)x)\lim_{x \to \infty}\left(\frac{\tan{\left(\frac{x}{2} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(\frac{x}{2} \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(x2)x)y = x \lim_{x \to \infty}\left(\frac{\tan{\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:
tan⁡(x2)=−tan⁡(x2)\tan{\left(\frac{x}{2} \right)} = - \tan{\left(\frac{x}{2} \right)}
- No
tan⁡(x2)=tan⁡(x2)\tan{\left(\frac{x}{2} \right)} = \tan{\left(\frac{x}{2} \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = tan(x/2)
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