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Plotting a function graph/ cos(x)*sin(x)

Graphing y = cos(x)*sin(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cos(x)*sin(x)
f(x)=sin⁡(x)cos⁡(x)f{\left(x \right)} = \sin{\left(x \right)} \cos{\left(x \right)} 
f = sin(x)*cos(x)
The graph of the function
0.00.51.01.52.02.53.03.54.04.55.05.56.01-1
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)=0\sin{\left(x \right)} \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Numerical solution
x1=48.6946861306418x_{1} = 48.6946861306418
x2=81.6814089933346x_{2} = 81.6814089933346
x3=−14.1371669411541x_{3} = -14.1371669411541
x4=−86.3937979737193x_{4} = -86.3937979737193
x5=−1.5707963267949x_{5} = -1.5707963267949
x6=−483.805268652828x_{6} = -483.805268652828
x7=23.5619449019235x_{7} = 23.5619449019235
x8=59.6902604182061x_{8} = 59.6902604182061
x9=73.8274273593601x_{9} = 73.8274273593601
x10=4.71238898038469x_{10} = 4.71238898038469
x11=34.5575191894877x_{11} = 34.5575191894877
x12=−21.9911485751286x_{12} = -21.9911485751286
x13=−20.4203522483337x_{13} = -20.4203522483337
x14=15.707963267949x_{14} = 15.707963267949
x15=−95.8185759344887x_{15} = -95.8185759344887
x16=26.7035375555132x_{16} = 26.7035375555132
x17=−81.6814089933346x_{17} = -81.6814089933346
x18=31.4159265358979x_{18} = 31.4159265358979
x19=−40.8407044966673x_{19} = -40.8407044966673
x20=20.4203522483337x_{20} = 20.4203522483337
x21=−94.2477796076938x_{21} = -94.2477796076938
x22=67.5442420521806x_{22} = 67.5442420521806
x23=−59.6902604182061x_{23} = -59.6902604182061
x24=36.1283155162826x_{24} = 36.1283155162826
x25=−43.9822971502571x_{25} = -43.9822971502571
x26=58.1194640914112x_{26} = 58.1194640914112
x27=−29.845130209103x_{27} = -29.845130209103
x28=−31.4159265358979x_{28} = -31.4159265358979
x29=12.5663706143592x_{29} = 12.5663706143592
x30=43.9822971502571x_{30} = 43.9822971502571
x31=7.85398163397448x_{31} = 7.85398163397448
x32=−15.707963267949x_{32} = -15.707963267949
x33=0x_{33} = 0
x34=89.5353906273091x_{34} = 89.5353906273091
x35=−65.9734457253857x_{35} = -65.9734457253857
x36=−28.2743338823081x_{36} = -28.2743338823081
x37=51.8362787842316x_{37} = 51.8362787842316
x38=70.6858347057703x_{38} = 70.6858347057703
x39=−50.2654824574367x_{39} = -50.2654824574367
x40=80.1106126665397x_{40} = 80.1106126665397
x41=−75.398223686155x_{41} = -75.398223686155
x42=45.553093477052x_{42} = 45.553093477052
x43=−119.380520836412x_{43} = -119.380520836412
x44=14.1371669411541x_{44} = 14.1371669411541
x45=28.2743338823081x_{45} = 28.2743338823081
x46=65.9734457253857x_{46} = 65.9734457253857
x47=−67.5442420521806x_{47} = -67.5442420521806
x48=42.4115008234622x_{48} = 42.4115008234622
x49=−45.553093477052x_{49} = -45.553093477052
x50=113.097335529233x_{50} = 113.097335529233
x51=−58.1194640914112x_{51} = -58.1194640914112
x52=−87.9645943005142x_{52} = -87.9645943005142
x53=−6.28318530717959x_{53} = -6.28318530717959
x54=−83.2522053201295x_{54} = -83.2522053201295
x55=−97.3893722612836x_{55} = -97.3893722612836
x56=94.2477796076938x_{56} = 94.2477796076938
x57=590.619418874881x_{57} = 590.619418874881
x58=−17.2787595947439x_{58} = -17.2787595947439
x59=95.8185759344887x_{59} = 95.8185759344887
x60=−39.2699081698724x_{60} = -39.2699081698724
x61=72.2566310325652x_{61} = 72.2566310325652
x62=−36.1283155162826x_{62} = -36.1283155162826
x63=1.5707963267949x_{63} = 1.5707963267949
x64=−9.42477796076938x_{64} = -9.42477796076938
x65=−64.4026493985908x_{65} = -64.4026493985908
x66=56.5486677646163x_{66} = 56.5486677646163
x67=92.6769832808989x_{67} = 92.6769832808989
x68=−51.8362787842316x_{68} = -51.8362787842316
x69=100.530964914873x_{69} = 100.530964914873
x70=−89.5353906273091x_{70} = -89.5353906273091
x71=6.28318530717959x_{71} = 6.28318530717959
x72=−61.261056745001x_{72} = -61.261056745001
x73=−53.4070751110265x_{73} = -53.4070751110265
x74=−73.8274273593601x_{74} = -73.8274273593601
x75=21.9911485751286x_{75} = 21.9911485751286
x76=29.845130209103x_{76} = 29.845130209103
x77=87.9645943005142x_{77} = 87.9645943005142
x78=−72.2566310325652x_{78} = -72.2566310325652
x79=37.6991118430775x_{79} = 37.6991118430775
x80=50.2654824574367x_{80} = 50.2654824574367
x81=86.3937979737193x_{81} = 86.3937979737193
x82=64.4026493985908x_{82} = 64.4026493985908
x83=−48.6946861306418x_{83} = -48.6946861306418
x84=−37.6991118430775x_{84} = -37.6991118430775
x85=−23.5619449019235x_{85} = -23.5619449019235
x86=78.5398163397448x_{86} = 78.5398163397448
x87=−42.4115008234622x_{87} = -42.4115008234622
x88=−80.1106126665397x_{88} = -80.1106126665397
x89=−7.85398163397448x_{89} = -7.85398163397448
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)*sin(x).
sin⁡(0)cos⁡(0)\sin{\left(0 \right)} \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
−sin⁡2(x)+cos⁡2(x)=0- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\pi}{4}
x2=π4x_{2} = \frac{\pi}{4}
The values of the extrema at the points:
 -pi        
(----, -1/2)
  4         

 pi      
(--, 1/2)
 4


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=−π4x_{1} = - \frac{\pi}{4}
Maxima of the function at points:
x1=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
[−π4,π4]\left[- \frac{\pi}{4}, \frac{\pi}{4}\right]
Increasing at intervals
(−∞,−π4]∪[π4,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[\frac{\pi}{4}, \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
−4sin⁡(x)cos⁡(x)=0- 4 \sin{\left(x \right)} \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}

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
[−π2,0]∪[π2,∞)\left[- \frac{\pi}{2}, 0\right] \cup \left[\frac{\pi}{2}, \infty\right)
Convex at the intervals
(−∞,−π2]∪[0,π2]\left(-\infty, - \frac{\pi}{2}\right] \cup \left[0, \frac{\pi}{2}\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))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} \cos{\left(x \right)}\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))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} \cos{\left(x \right)}\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 cos(x)*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cos{\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:
sin⁡(x)cos⁡(x)=−sin⁡(x)cos⁡(x)\sin{\left(x \right)} \cos{\left(x \right)} = - \sin{\left(x \right)} \cos{\left(x \right)}
- No
sin⁡(x)cos⁡(x)=sin⁡(x)cos⁡(x)\sin{\left(x \right)} \cos{\left(x \right)} = \sin{\left(x \right)} \cos{\left(x \right)}
- Yes
so, the function
is
odd
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