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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) + 2)
The graph of the function
02468-8-6-4-2-10101-1
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=−59.6902604182061x_{2} = -59.6902604182061
x3=81.6814089933346x_{3} = 81.6814089933346
x4=−43.9822971502571x_{4} = -43.9822971502571
x5=−100.530964914873x_{5} = -100.530964914873
x6=34.5575191894877x_{6} = 34.5575191894877
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=−483.805268652828x_{18} = -483.805268652828
x19=−15.707963267949x_{19} = -15.707963267949
x20=59.6902604182061x_{20} = 59.6902604182061
x21=6.28318530717959x_{21} = 6.28318530717959
x22=9.42477796076938x_{22} = 9.42477796076938
x23=−53.4070751110265x_{23} = -53.4070751110265
x24=−47.1238898038469x_{24} = -47.1238898038469
x25=−87.9645943005142x_{25} = -87.9645943005142
x26=69.1150383789755x_{26} = 69.1150383789755
x27=21.9911485751286x_{27} = 21.9911485751286
x28=87.9645943005142x_{28} = 87.9645943005142
x29=18.8495559215388x_{29} = 18.8495559215388
x30=−84.8230016469244x_{30} = -84.8230016469244
x31=−72.2566310325652x_{31} = -72.2566310325652
x32=25.1327412287183x_{32} = 25.1327412287183
x33=37.6991118430775x_{33} = 37.6991118430775
x34=−25.1327412287183x_{34} = -25.1327412287183
x35=0x_{35} = 0
x36=50.2654824574367x_{36} = 50.2654824574367
x37=−6.28318530717959x_{37} = -6.28318530717959
x38=−65.9734457253857x_{38} = -65.9734457253857
x39=−21.9911485751286x_{39} = -21.9911485751286
x40=−62.8318530717959x_{40} = -62.8318530717959
x41=−1388.58395288669x_{41} = -1388.58395288669
x42=75.398223686155x_{42} = 75.398223686155
x43=53.4070751110265x_{43} = 53.4070751110265
x44=84.8230016469244x_{44} = 84.8230016469244
x45=−28.2743338823081x_{45} = -28.2743338823081
x46=15.707963267949x_{46} = 15.707963267949
x47=−91.106186954104x_{47} = -91.106186954104
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=−97.3893722612836x_{59} = -97.3893722612836
x60=−75.398223686155x_{60} = -75.398223686155
x61=−12.5663706143592x_{61} = -12.5663706143592
x62=−94.2477796076938x_{62} = -94.2477796076938
x63=−34.5575191894877x_{63} = -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)/(cos(x) + 2).
sin⁡(0)cos⁡(0)+2\frac{\sin{\left(0 \right)}}{\cos{\left(0 \right)} + 2}
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=−2π3x_{1} = - \frac{2 \pi}{3}
x2=2π3x_{2} = \frac{2 \pi}{3}
The values of the extrema at the points:
           ___  
 -2*pi  -\/ 3   
(-----, -------)
   3       3    

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


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=−2π3x_{1} = - \frac{2 \pi}{3}
Maxima of the function at points:
x1=2π3x_{1} = \frac{2 \pi}{3}
Decreasing at intervals
[−2π3,2π3]\left[- \frac{2 \pi}{3}, \frac{2 \pi}{3}\right]
Increasing at intervals
(−∞,−2π3]∪[2π3,∞)\left(-\infty, - \frac{2 \pi}{3}\right] \cup \left[\frac{2 \pi}{3}, \infty\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

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:
Have no bends at the whole real axis
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) + 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
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