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Plotting a function graph/ (7+2cosx)sinx

Graphing y = (7+2cosx)sinx

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = (7 + 2*cos(x))*sin(x)
f(x)=(2cos⁡(x)+7)sin⁡(x)f{\left(x \right)} = \left(2 \cos{\left(x \right)} + 7\right) \sin{\left(x \right)} 
f = (2*cos(x) + 7)*sin(x)
The graph of the function
02468-8-6-4-2-1010-2020
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:
(2cos⁡(x)+7)sin⁡(x)=0\left(2 \cos{\left(x \right)} + 7\right) \sin{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−18.8495559215388x_{1} = -18.8495559215388
x2=−53.4070751110265x_{2} = -53.4070751110265
x3=−37.6991118430775x_{3} = -37.6991118430775
x4=−59.6902604182061x_{4} = -59.6902604182061
x5=−15.707963267949x_{5} = -15.707963267949
x6=9698.09652163169x_{6} = 9698.09652163169
x7=−56.5486677646163x_{7} = -56.5486677646163
x8=12.5663706143592x_{8} = 12.5663706143592
x9=3.14159265358979x_{9} = 3.14159265358979
x10=−31.4159265358979x_{10} = -31.4159265358979
x11=84.8230016469244x_{11} = 84.8230016469244
x12=−81.6814089933346x_{12} = -81.6814089933346
x13=94.2477796076938x_{13} = 94.2477796076938
x14=21.9911485751286x_{14} = 21.9911485751286
x15=0x_{15} = 0
x16=−87.9645943005142x_{16} = -87.9645943005142
x17=81.6814089933346x_{17} = 81.6814089933346
x18=40.8407044966673x_{18} = 40.8407044966673
x19=−75.398223686155x_{19} = -75.398223686155
x20=−78.5398163397448x_{20} = -78.5398163397448
x21=62.8318530717959x_{21} = 62.8318530717959
x22=100.530964914873x_{22} = 100.530964914873
x23=−21.9911485751286x_{23} = -21.9911485751286
x24=47.1238898038469x_{24} = 47.1238898038469
x25=91.106186954104x_{25} = 91.106186954104
x26=75.398223686155x_{26} = 75.398223686155
x27=28.2743338823081x_{27} = 28.2743338823081
x28=34.5575191894877x_{28} = 34.5575191894877
x29=6.28318530717959x_{29} = 6.28318530717959
x30=78.5398163397448x_{30} = 78.5398163397448
x31=72.2566310325652x_{31} = 72.2566310325652
x32=−6.28318530717959x_{32} = -6.28318530717959
x33=15.707963267949x_{33} = 15.707963267949
x34=31.4159265358979x_{34} = 31.4159265358979
x35=−47.1238898038469x_{35} = -47.1238898038469
x36=25.1327412287183x_{36} = 25.1327412287183
x37=18.8495559215388x_{37} = 18.8495559215388
x38=−94.2477796076938x_{38} = -94.2477796076938
x39=−3.14159265358979x_{39} = -3.14159265358979
x40=−40.8407044966673x_{40} = -40.8407044966673
x41=56.5486677646163x_{41} = 56.5486677646163
x42=−25.1327412287183x_{42} = -25.1327412287183
x43=53.4070751110265x_{43} = 53.4070751110265
x44=−402.123859659494x_{44} = -402.123859659494
x45=−28.2743338823081x_{45} = -28.2743338823081
x46=−9.42477796076938x_{46} = -9.42477796076938
x47=87.9645943005142x_{47} = 87.9645943005142
x48=−50.2654824574367x_{48} = -50.2654824574367
x49=−100.530964914873x_{49} = -100.530964914873
x50=−43.9822971502571x_{50} = -43.9822971502571
x51=50.2654824574367x_{51} = 50.2654824574367
x52=−97.3893722612836x_{52} = -97.3893722612836
x53=69.1150383789755x_{53} = 69.1150383789755
x54=59.6902604182061x_{54} = 59.6902604182061
x55=97.3893722612836x_{55} = 97.3893722612836
x56=−62.8318530717959x_{56} = -62.8318530717959
x57=−72.2566310325652x_{57} = -72.2566310325652
x58=−91.106186954104x_{58} = -91.106186954104
x59=−12.5663706143592x_{59} = -12.5663706143592
x60=−69.1150383789755x_{60} = -69.1150383789755
x61=37.6991118430775x_{61} = 37.6991118430775
x62=9.42477796076938x_{62} = 9.42477796076938
x63=65.9734457253857x_{63} = 65.9734457253857
x64=−65.9734457253857x_{64} = -65.9734457253857
x65=−84.8230016469244x_{65} = -84.8230016469244
x66=−34.5575191894877x_{66} = -34.5575191894877
x67=43.9822971502571x_{67} = 43.9822971502571
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (7 + 2*cos(x))*sin(x).
(2cos⁡(0)+7)sin⁡(0)\left(2 \cos{\left(0 \right)} + 7\right) \sin{\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
(2cos⁡(x)+7)cos⁡(x)−2sin⁡2(x)=0\left(2 \cos{\left(x \right)} + 7\right) \cos{\left(x \right)} - 2 \sin^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−2atan⁡(155)x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}
x2=2atan⁡(155)x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}
The values of the extrema at the points:
        /  ____\   /         /      /  ____\\\    /      /  ____\\ 
        |\/ 15 |   |         |      |\/ 15 |||    |      |\/ 15 || 
(-2*atan|------|, -|7 + 2*cos|2*atan|------|||*sin|2*atan|------||)
        \  5   /   \         \      \  5   ///    \      \  5   // 

       /  ____\  /         /      /  ____\\\    /      /  ____\\ 
       |\/ 15 |  |         |      |\/ 15 |||    |      |\/ 15 || 
(2*atan|------|, |7 + 2*cos|2*atan|------|||*sin|2*atan|------||)
       \  5   /  \         \      \  5   ///    \      \  5   //


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=−2atan⁡(155)x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}
Maxima of the function at points:
x1=2atan⁡(155)x_{1} = 2 \operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}
Decreasing at intervals
[−2atan⁡(155),2atan⁡(155)]\left[- 2 \operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}, 2 \operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}\right]
Increasing at intervals
(−∞,−2atan⁡(155)]∪[2atan⁡(155),∞)\left(-\infty, - 2 \operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}\right] \cup \left[2 \operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}, \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
−(8cos⁡(x)+7)sin⁡(x)=0- \left(8 \cos{\left(x \right)} + 7\right) \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−2atan⁡(15)x_{2} = - 2 \operatorname{atan}{\left(\sqrt{15} \right)}
x3=2atan⁡(15)x_{3} = 2 \operatorname{atan}{\left(\sqrt{15} \right)}

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
[−2atan⁡(15),0]∪[2atan⁡(15),∞)\left[- 2 \operatorname{atan}{\left(\sqrt{15} \right)}, 0\right] \cup \left[2 \operatorname{atan}{\left(\sqrt{15} \right)}, \infty\right)
Convex at the intervals
(−∞,−2atan⁡(15)]∪[0,2atan⁡(15)]\left(-\infty, - 2 \operatorname{atan}{\left(\sqrt{15} \right)}\right] \cup \left[0, 2 \operatorname{atan}{\left(\sqrt{15} \right)}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞((2cos⁡(x)+7)sin⁡(x))=⟨−9,9⟩\lim_{x \to -\infty}\left(\left(2 \cos{\left(x \right)} + 7\right) \sin{\left(x \right)}\right) = \left\langle -9, 9\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−9,9⟩y = \left\langle -9, 9\right\rangle
lim⁡x→∞((2cos⁡(x)+7)sin⁡(x))=⟨−9,9⟩\lim_{x \to \infty}\left(\left(2 \cos{\left(x \right)} + 7\right) \sin{\left(x \right)}\right) = \left\langle -9, 9\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−9,9⟩y = \left\langle -9, 9\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (7 + 2*cos(x))*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((2cos⁡(x)+7)sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\left(2 \cos{\left(x \right)} + 7\right) \sin{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞((2cos⁡(x)+7)sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\left(2 \cos{\left(x \right)} + 7\right) \sin{\left(x \right)}}{x}\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:
(2cos⁡(x)+7)sin⁡(x)=−(2cos⁡(x)+7)sin⁡(x)\left(2 \cos{\left(x \right)} + 7\right) \sin{\left(x \right)} = - \left(2 \cos{\left(x \right)} + 7\right) \sin{\left(x \right)}
- No
(2cos⁡(x)+7)sin⁡(x)=(2cos⁡(x)+7)sin⁡(x)\left(2 \cos{\left(x \right)} + 7\right) \sin{\left(x \right)} = \left(2 \cos{\left(x \right)} + 7\right) \sin{\left(x \right)}
- Yes
so, the function
is
odd
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