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sin(x)/sqrt(1-cos(x))
Plotting a function graph/ sin(x)/sqrt(1-cos(x))

Graphing y = sin(x)/sqrt(1-cos(x))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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           sin(x)    
f(x) = --------------
         ____________
       \/ 1 - cos(x)
f(x)=sin⁡(x)−cos⁡(x)+1f{\left(x \right)} = \frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} 
f = sin(x)/(sqrt(1 - cos(x)))
The graph of the function
0-70-60-50-40-30-20-10105-5
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
x2=6.28318530717959x_{2} = 6.28318530717959
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)−cos⁡(x)+1=0\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=πx_{1} = \pi
Numerical solution
x1=−47.1238898038469x_{1} = -47.1238898038469
x2=−72.2566310325652x_{2} = -72.2566310325652
x3=−21.9911485751286x_{3} = -21.9911485751286
x4=84.8230016469244x_{4} = 84.8230016469244
x5=53.4070751110265x_{5} = 53.4070751110265
x6=−59.6902604182061x_{6} = -59.6902604182061
x7=72.2566310325652x_{7} = 72.2566310325652
x8=−9.42477796076938x_{8} = -9.42477796076938
x9=−1674.46888436336x_{9} = -1674.46888436336
x10=−78.5398163397448x_{10} = -78.5398163397448
x11=78.5398163397448x_{11} = 78.5398163397448
x12=34.5575191894877x_{12} = 34.5575191894877
x13=65.9734457253857x_{13} = 65.9734457253857
x14=91.106186954104x_{14} = 91.106186954104
x15=−3.14159265358979x_{15} = -3.14159265358979
x16=−34.5575191894877x_{16} = -34.5575191894877
x17=47.1238898038469x_{17} = 47.1238898038469
x18=9.42477796076938x_{18} = 9.42477796076938
x19=−53.4070751110265x_{19} = -53.4070751110265
x20=21.9911485751286x_{20} = 21.9911485751286
x21=40.8407044966673x_{21} = 40.8407044966673
x22=3.14159265358979x_{22} = 3.14159265358979
x23=−91.106186954104x_{23} = -91.106186954104
x24=103.672557568463x_{24} = 103.672557568463
x25=−28.2743338823081x_{25} = -28.2743338823081
x26=−9861.45933961836x_{26} = -9861.45933961836
x27=97.3893722612836x_{27} = 97.3893722612836
x28=−65.9734457253857x_{28} = -65.9734457253857
x29=28.2743338823081x_{29} = 28.2743338823081
x30=59.6902604182061x_{30} = 59.6902604182061
x31=−84.8230016469244x_{31} = -84.8230016469244
x32=−122.522113490002x_{32} = -122.522113490002
x33=−15.707963267949x_{33} = -15.707963267949
x34=15.707963267949x_{34} = 15.707963267949
x35=−97.3893722612836x_{35} = -97.3893722612836
x36=−40.8407044966673x_{36} = -40.8407044966673
x37=−166.504410640259x_{37} = -166.504410640259
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)/(sqrt(1 - cos(x))).
sin⁡(0)−cos⁡(0)+1\frac{\sin{\left(0 \right)}}{\sqrt{- \cos{\left(0 \right)} + 1}}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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)−cos⁡(x)+1−sin⁡2(x)2(−cos⁡(x)+1)32=0\frac{\cos{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} - \frac{\sin^{2}{\left(x \right)}}{2 \left(- \cos{\left(x \right)} + 1\right)^{\frac{3}{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
−(1+2cos⁡(x)+3sin⁡2(x)cos⁡(x)−14⋅(−cos⁡(x)+1)+cos⁡(x)−cos⁡(x)+1)sin⁡(x)−cos⁡(x)+1=0- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} = 0
Solve this equation
The roots of this equation
x1=πx_{1} = \pi
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=0x_{1} = 0
x2=6.28318530717959x_{2} = 6.28318530717959

lim⁡x→0−(−(1+2cos⁡(x)+3sin⁡2(x)cos⁡(x)−14⋅(−cos⁡(x)+1)+cos⁡(x)−cos⁡(x)+1)sin⁡(x)−cos⁡(x)+1)=24\lim_{x \to 0^-}\left(- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = \frac{\sqrt{2}}{4}
Let's take the limit
lim⁡x→0+(−(1+2cos⁡(x)+3sin⁡2(x)cos⁡(x)−14⋅(−cos⁡(x)+1)+cos⁡(x)−cos⁡(x)+1)sin⁡(x)−cos⁡(x)+1)=−24\lim_{x \to 0^+}\left(- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = - \frac{\sqrt{2}}{4}
Let's take the limit
- the limits are not equal, so
x1=0x_{1} = 0
- is an inflection point
lim⁡x→6.28318530717959−(−(1+2cos⁡(x)+3sin⁡2(x)cos⁡(x)−14⋅(−cos⁡(x)+1)+cos⁡(x)−cos⁡(x)+1)sin⁡(x)−cos⁡(x)+1)=−∞\lim_{x \to 6.28318530717959^-}\left(- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = -\infty
Let's take the limit
lim⁡x→6.28318530717959+(−(1+2cos⁡(x)+3sin⁡2(x)cos⁡(x)−14⋅(−cos⁡(x)+1)+cos⁡(x)−cos⁡(x)+1)sin⁡(x)−cos⁡(x)+1)=∞i\lim_{x \to 6.28318530717959^+}\left(- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = \infty i
Let's take the limit
- the limits are not equal, so
x2=6.28318530717959x_{2} = 6.28318530717959
- is an inflection 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
[π,∞)\left[\pi, \infty\right)
Convex at the intervals
(−∞,π]\left(-\infty, \pi\right]
Vertical asymptotes
Have:
x1=0x_{1} = 0
x2=6.28318530717959x_{2} = 6.28318530717959
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡(x)−cos⁡(x)+1)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\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→∞(sin⁡(x)−cos⁡(x)+1)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\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 sin(x)/(sqrt(1 - cos(x))), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)x−cos⁡(x)+1)=lim⁡x→−∞(sin⁡(x)x−cos⁡(x)+1)\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right) = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(sin⁡(x)x−cos⁡(x)+1)y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right)
lim⁡x→∞(sin⁡(x)x−cos⁡(x)+1)=lim⁡x→∞(sin⁡(x)x−cos⁡(x)+1)\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right) = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(sin⁡(x)x−cos⁡(x)+1)y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\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)−cos⁡(x)+1=−sin⁡(x)−cos⁡(x)+1\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} = - \frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}
- No
sin⁡(x)−cos⁡(x)+1=sin⁡(x)−cos⁡(x)+1\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} = \frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sin(x)/sqrt(1-cos(x))
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