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Plotting a function graph/ sinx/(1-2cosx)

Graphing y = sinx/(1-2cosx)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          sin(x)   
f(x) = ------------
       1 - 2*cos(x)
f(x)=sin⁡(x)1−2cos⁡(x)f{\left(x \right)} = \frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} 
f = sin(x)/(1 - 2*cos(x))
The graph of the function
0.00.51.01.52.02.53.03.54.04.55.05.56.0-50005000
The domain of the function
The points at which the function is not precisely defined:
x1=1.0471975511966x_{1} = 1.0471975511966
x2=5.23598775598299x_{2} = 5.23598775598299
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⁡(x)1−2cos⁡(x)=0\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=72.2566310325652x_{1} = 72.2566310325652
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=3.14159265358979x_{3} = 3.14159265358979
x4=−43.9822971502571x_{4} = -43.9822971502571
x5=81.6814089933346x_{5} = 81.6814089933346
x6=−100.530964914873x_{6} = -100.530964914873
x7=28.2743338823081x_{7} = 28.2743338823081
x8=65.9734457253857x_{8} = 65.9734457253857
x9=−31.4159265358979x_{9} = -31.4159265358979
x10=−9.42477796076938x_{10} = -9.42477796076938
x11=40.8407044966673x_{11} = 40.8407044966673
x12=56.5486677646163x_{12} = 56.5486677646163
x13=−56.5486677646163x_{13} = -56.5486677646163
x14=12.5663706143592x_{14} = 12.5663706143592
x15=43.9822971502571x_{15} = 43.9822971502571
x16=100.530964914873x_{16} = 100.530964914873
x17=−3.14159265358979x_{17} = -3.14159265358979
x18=−15.707963267949x_{18} = -15.707963267949
x19=59.6902604182061x_{19} = 59.6902604182061
x20=6.28318530717959x_{20} = 6.28318530717959
x21=9.42477796076938x_{21} = 9.42477796076938
x22=−53.4070751110265x_{22} = -53.4070751110265
x23=−47.1238898038469x_{23} = -47.1238898038469
x24=−87.9645943005142x_{24} = -87.9645943005142
x25=69.1150383789755x_{25} = 69.1150383789755
x26=21.9911485751286x_{26} = 21.9911485751286
x27=87.9645943005142x_{27} = 87.9645943005142
x28=18.8495559215388x_{28} = 18.8495559215388
x29=−84.8230016469244x_{29} = -84.8230016469244
x30=−72.2566310325652x_{30} = -72.2566310325652
x31=25.1327412287183x_{31} = 25.1327412287183
x32=37.6991118430775x_{32} = 37.6991118430775
x33=−25.1327412287183x_{33} = -25.1327412287183
x34=0x_{34} = 0
x35=50.2654824574367x_{35} = 50.2654824574367
x36=−6.28318530717959x_{36} = -6.28318530717959
x37=−65.9734457253857x_{37} = -65.9734457253857
x38=−21.9911485751286x_{38} = -21.9911485751286
x39=−62.8318530717959x_{39} = -62.8318530717959
x40=75.398223686155x_{40} = 75.398223686155
x41=84.8230016469244x_{41} = 84.8230016469244
x42=53.4070751110265x_{42} = 53.4070751110265
x43=34.5575191894877x_{43} = 34.5575191894877
x44=−28.2743338823081x_{44} = -28.2743338823081
x45=15.707963267949x_{45} = 15.707963267949
x46=−91.106186954104x_{46} = -91.106186954104
x47=47.1238898038469x_{47} = 47.1238898038469
x48=97.3893722612836x_{48} = 97.3893722612836
x49=−69.1150383789755x_{49} = -69.1150383789755
x50=94.2477796076938x_{50} = 94.2477796076938
x51=−18.8495559215388x_{51} = -18.8495559215388
x52=−50.2654824574367x_{52} = -50.2654824574367
x53=−37.6991118430775x_{53} = -37.6991118430775
x54=−81.6814089933346x_{54} = -81.6814089933346
x55=62.8318530717959x_{55} = 62.8318530717959
x56=78.5398163397448x_{56} = 78.5398163397448
x57=31.4159265358979x_{57} = 31.4159265358979
x58=−78.5398163397448x_{58} = -78.5398163397448
x59=−40.8407044966673x_{59} = -40.8407044966673
x60=−97.3893722612836x_{60} = -97.3893722612836
x61=−75.398223686155x_{61} = -75.398223686155
x62=91.106186954104x_{62} = 91.106186954104
x63=−12.5663706143592x_{63} = -12.5663706143592
x64=−94.2477796076938x_{64} = -94.2477796076938
x65=−34.5575191894877x_{65} = -34.5575191894877
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)/(1 - 2*cos(x)).
sin⁡(0)1−2cos⁡(0)\frac{\sin{\left(0 \right)}}{1 - 2 \cos{\left(0 \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
cos⁡(x)1−2cos⁡(x)−2sin⁡2(x)(1−2cos⁡(x))2=0\frac{\cos{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} - \frac{2 \sin^{2}{\left(x \right)}}{\left(1 - 2 \cos{\left(x \right)}\right)^{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
(−2(cos⁡(x)+4sin⁡2(x)2cos⁡(x)−1)2cos⁡(x)−1+1−4cos⁡(x)2cos⁡(x)−1)sin⁡(x)2cos⁡(x)−1=0\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=1.0471975511966x_{1} = 1.0471975511966
x2=5.23598775598299x_{2} = 5.23598775598299

lim⁡x→1.0471975511966−((−2(cos⁡(x)+4sin⁡2(x)2cos⁡(x)−1)2cos⁡(x)−1+1−4cos⁡(x)2cos⁡(x)−1)sin⁡(x)2cos⁡(x)−1)=4.74636579600418⋅1047\lim_{x \to 1.0471975511966^-}\left(\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) = 4.74636579600418 \cdot 10^{47}
lim⁡x→1.0471975511966+((−2(cos⁡(x)+4sin⁡2(x)2cos⁡(x)−1)2cos⁡(x)−1+1−4cos⁡(x)2cos⁡(x)−1)sin⁡(x)2cos⁡(x)−1)=4.74636579600418⋅1047\lim_{x \to 1.0471975511966^+}\left(\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) = 4.74636579600418 \cdot 10^{47}
- limits are equal, then skip the corresponding point
lim⁡x→5.23598775598299−((−2(cos⁡(x)+4sin⁡2(x)2cos⁡(x)−1)2cos⁡(x)−1+1−4cos⁡(x)2cos⁡(x)−1)sin⁡(x)2cos⁡(x)−1)=4.74636579600418⋅1047\lim_{x \to 5.23598775598299^-}\left(\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) = 4.74636579600418 \cdot 10^{47}
lim⁡x→5.23598775598299+((−2(cos⁡(x)+4sin⁡2(x)2cos⁡(x)−1)2cos⁡(x)−1+1−4cos⁡(x)2cos⁡(x)−1)sin⁡(x)2cos⁡(x)−1)=4.74636579600418⋅1047\lim_{x \to 5.23598775598299^+}\left(\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) = 4.74636579600418 \cdot 10^{47}
- limits are equal, then skip the corresponding point

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(-\infty, 0\right]
Convex at the intervals
[0,∞)\left[0, \infty\right)
Vertical asymptotes
Have:
x1=1.0471975511966x_{1} = 1.0471975511966
x2=5.23598775598299x_{2} = 5.23598775598299
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→−∞(sin⁡(x)1−2cos⁡(x))y = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(sin⁡(x)1−2cos⁡(x))y = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)/(1 - 2*cos(x)), 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→−∞(sin⁡(x)x(1−2cos⁡(x)))y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(1 - 2 \cos{\left(x \right)}\right)}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(sin⁡(x)x(1−2cos⁡(x)))y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(1 - 2 \cos{\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:
sin⁡(x)1−2cos⁡(x)=−sin⁡(x)1−2cos⁡(x)\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} = - \frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}
- No
sin⁡(x)1−2cos⁡(x)=sin⁡(x)1−2cos⁡(x)\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} = \frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}
- No
so, the function
not is
neither even, nor odd
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