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sin(x)/(cos(x)-2)
Plotting a function graph/ sin(x)/(cos(x)-2)

Graphing y = sin(x)/(cos(x)-2)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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         sin(x)  
f(x) = ----------
       cos(x) - 2
f(x)=sin⁡(x)cos⁡(x)−2f{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} - 2} 
f = sin(x)/(cos(x) - 1*2)
The graph of the function
01002003004005006007008009001-1
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)−2=0\frac{\sin{\left(x \right)}}{\cos{\left(x \right)} - 2} = 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=94.2477796076938x_{2} = 94.2477796076938
x3=914.20346219463x_{3} = 914.20346219463
x4=−3.14159265358979x_{4} = -3.14159265358979
x5=40.8407044966673x_{5} = 40.8407044966673
x6=−91.106186954104x_{6} = -91.106186954104
x7=−15.707963267949x_{7} = -15.707963267949
x8=−37.6991118430775x_{8} = -37.6991118430775
x9=−31.4159265358979x_{9} = -31.4159265358979
x10=−81.6814089933346x_{10} = -81.6814089933346
x11=6.28318530717959x_{11} = 6.28318530717959
x12=62.8318530717959x_{12} = 62.8318530717959
x13=53.4070751110265x_{13} = 53.4070751110265
x14=81.6814089933346x_{14} = 81.6814089933346
x15=100.530964914873x_{15} = 100.530964914873
x16=50.2654824574367x_{16} = 50.2654824574367
x17=84.8230016469244x_{17} = 84.8230016469244
x18=−47.1238898038469x_{18} = -47.1238898038469
x19=−56.5486677646163x_{19} = -56.5486677646163
x20=−53.4070751110265x_{20} = -53.4070751110265
x21=3.14159265358979x_{21} = 3.14159265358979
x22=−9.42477796076938x_{22} = -9.42477796076938
x23=543.495529071034x_{23} = 543.495529071034
x24=−87.9645943005142x_{24} = -87.9645943005142
x25=15.707963267949x_{25} = 15.707963267949
x26=91.106186954104x_{26} = 91.106186954104
x27=97.3893722612836x_{27} = 97.3893722612836
x28=78.5398163397448x_{28} = 78.5398163397448
x29=128.805298797182x_{29} = 128.805298797182
x30=9.42477796076938x_{30} = 9.42477796076938
x31=−6.28318530717959x_{31} = -6.28318530717959
x32=75.398223686155x_{32} = 75.398223686155
x33=−392.699081698724x_{33} = -392.699081698724
x34=12.5663706143592x_{34} = 12.5663706143592
x35=56.5486677646163x_{35} = 56.5486677646163
x36=−12.5663706143592x_{36} = -12.5663706143592
x37=−34.5575191894877x_{37} = -34.5575191894877
x38=−28.2743338823081x_{38} = -28.2743338823081
x39=18.8495559215388x_{39} = 18.8495559215388
x40=−78.5398163397448x_{40} = -78.5398163397448
x41=−94.2477796076938x_{41} = -94.2477796076938
x42=−25.1327412287183x_{42} = -25.1327412287183
x43=59.6902604182061x_{43} = 59.6902604182061
x44=−21.9911485751286x_{44} = -21.9911485751286
x45=−72.2566310325652x_{45} = -72.2566310325652
x46=−75.398223686155x_{46} = -75.398223686155
x47=31.4159265358979x_{47} = 31.4159265358979
x48=43.9822971502571x_{48} = 43.9822971502571
x49=28.2743338823081x_{49} = 28.2743338823081
x50=−40.8407044966673x_{50} = -40.8407044966673
x51=−50.2654824574367x_{51} = -50.2654824574367
x52=47.1238898038469x_{52} = 47.1238898038469
x53=−84.8230016469244x_{53} = -84.8230016469244
x54=−97.3893722612836x_{54} = -97.3893722612836
x55=−43.9822971502571x_{55} = -43.9822971502571
x56=65.9734457253857x_{56} = 65.9734457253857
x57=−65.9734457253857x_{57} = -65.9734457253857
x58=34.5575191894877x_{58} = 34.5575191894877
x59=0x_{59} = 0
x60=−59.6902604182061x_{60} = -59.6902604182061
x61=21.9911485751286x_{61} = 21.9911485751286
x62=87.9645943005142x_{62} = 87.9645943005142
x63=37.6991118430775x_{63} = 37.6991118430775
x64=−100.530964914873x_{64} = -100.530964914873
x65=−69.1150383789755x_{65} = -69.1150383789755
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)/(cos(x) - 1*2).
sin⁡(0)(−1)2+cos⁡(0)\frac{\sin{\left(0 \right)}}{\left(-1\right) 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)cos⁡(x)−2+sin⁡2(x)(cos⁡(x)−2)2=0\frac{\cos{\left(x \right)}}{\cos{\left(x \right)} - 2} + \frac{\sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 2\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=−π3x_{1} = - \frac{\pi}{3}
x2=π3x_{2} = \frac{\pi}{3}
The values of the extrema at the points:
            ___    
 -pi     -\/ 3     
(----, -----------)
  3    2*(1/2 - 2) 

          ___    
 pi     \/ 3     
(--, -----------)
 3   2*(1/2 - 2)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=π3x_{1} = \frac{\pi}{3}
Maxima of the function at points:
x1=−π3x_{1} = - \frac{\pi}{3}
Decreasing at intervals
(−∞,−π3]∪[π3,∞)\left(-\infty, - \frac{\pi}{3}\right] \cup \left[\frac{\pi}{3}, \infty\right)
Increasing at intervals
[−π3,π3]\left[- \frac{\pi}{3}, \frac{\pi}{3}\right]
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+cos⁡(x)+2sin⁡2(x)cos⁡(x)−2cos⁡(x)−2+2cos⁡(x)cos⁡(x)−2)sin⁡(x)cos⁡(x)−2=0\frac{\left(-1 + \frac{\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 2}}{\cos{\left(x \right)} - 2} + \frac{2 \cos{\left(x \right)}}{\cos{\left(x \right)} - 2}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 2} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi

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→−∞(sin⁡(x)cos⁡(x)−2)=⟨−1,1⟩\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)} - 2}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞(sin⁡(x)cos⁡(x)−2)=⟨−1,1⟩\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)} - 2}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)/(cos(x) - 1*2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)x(cos⁡(x)−2))=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(\cos{\left(x \right)} - 2\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)x(cos⁡(x)−2))=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(\cos{\left(x \right)} - 2\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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)−2=−sin⁡(x)cos⁡(x)−2\frac{\sin{\left(x \right)}}{\cos{\left(x \right)} - 2} = - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} - 2}
- No
sin⁡(x)cos⁡(x)−2=sin⁡(x)cos⁡(x)−2\frac{\sin{\left(x \right)}}{\cos{\left(x \right)} - 2} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} - 2}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sin(x)/(cos(x)-2)
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