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  • Graphing y =:
  • (x-1)^2*(x+2)
  • (9x^2-1)/x
  • 9^(1/(x-3))
  • 5x^2-3x-1

  • Identical expressions

  • cos(x)(sin(x)^ two)
  • co sinus of e of (x)( sinus of (x) squared )
  • co sinus of e of (x)( sinus of (x) to the power of two)
  • cos(x)(sin(x)2)
  • cosxsinx2
  • cos(x)(sin(x)²)
  • cos(x)(sin(x) to the power of 2)
  • cosxsinx^2

  • Similar expressions

  • cosx(sinx^2)
Plotting a function graph/ cos(x)(sin(x)^2)

Graphing y = cos(x)(sin(x)^2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                 2   
f(x) = cos(x)*sin (x)
f(x)=sin⁡2(x)cos⁡(x)f{\left(x \right)} = \sin^{2}{\left(x \right)} \cos{\left(x \right)} 
f = sin(x)^2*cos(x)
The graph of the function
02468-8-6-4-2-10101.0-1.0
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⁡2(x)cos⁡(x)=0\sin^{2}{\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=−34.5575191076725x_{1} = -34.5575191076725
x2=−97.389372410446x_{2} = -97.389372410446
x3=29.845130209103x_{3} = 29.845130209103
x4=14.1371669411541x_{4} = 14.1371669411541
x5=−80.1106126665397x_{5} = -80.1106126665397
x6=−7.85398163397448x_{6} = -7.85398163397448
x7=−37.6991118769198x_{7} = -37.6991118769198
x8=−15.7079632963762x_{8} = -15.7079632963762
x9=−87.9645943590963x_{9} = -87.9645943590963
x10=−25.1327411700478x_{10} = -25.1327411700478
x11=78.5398162040055x_{11} = 78.5398162040055
x12=72.2566310277219x_{12} = 72.2566310277219
x13=43.9822971001043x_{13} = 43.9822971001043
x14=−58.1194640914112x_{14} = -58.1194640914112
x15=95.8185759344887x_{15} = 95.8185759344887
x16=−31.4159266812001x_{16} = -31.4159266812001
x17=80.1106126665397x_{17} = 80.1106126665397
x18=12.5663704724455x_{18} = 12.5663704724455
x19=−43.982296876345x_{19} = -43.982296876345
x20=−14.1371669411541x_{20} = -14.1371669411541
x21=7.85398163397448x_{21} = 7.85398163397448
x22=−29.845130209103x_{22} = -29.845130209103
x23=84.8230015887783x_{23} = 84.8230015887783
x24=31.4159265728873x_{24} = 31.4159265728873
x25=−65.9734457652028x_{25} = -65.9734457652028
x26=−50.2654823143599x_{26} = -50.2654823143599
x27=−12.5663705453118x_{27} = -12.5663705453118
x28=97.38937222667x_{28} = 97.38937222667
x29=37.6991119937168x_{29} = 37.6991119937168
x30=42.4115008234622x_{30} = 42.4115008234622
x31=−47.1238897080294x_{31} = -47.1238897080294
x32=0x_{32} = 0
x33=−67.5442420521806x_{33} = -67.5442420521806
x34=56.5486676266806x_{34} = 56.5486676266806
x35=43.9822971693493x_{35} = 43.9822971693493
x36=62.8318530302311x_{36} = 62.8318530302311
x37=28.2743338652459x_{37} = 28.2743338652459
x38=−45.553093477052x_{38} = -45.553093477052
x39=34.5575190494922x_{39} = 34.5575190494922
x40=100.530964781462x_{40} = 100.530964781462
x41=94.2477796093527x_{41} = 94.2477796093527
x42=−23.5619449019235x_{42} = -23.5619449019235
x43=58.1194640914112x_{43} = 58.1194640914112
x44=65.9734451192804x_{44} = 65.9734451192804
x45=−56.5486676717583x_{45} = -56.5486676717583
x46=−28.2743341715057x_{46} = -28.2743341715057
x47=59.6902605703693x_{47} = 59.6902605703693
x48=91.1061875857656x_{48} = 91.1061875857656
x49=−207.34511550934x_{49} = -207.34511550934
x50=−36.1283155162826x_{50} = -36.1283155162826
x51=−51.8362787842316x_{51} = -51.8362787842316
x52=−73.8274273593601x_{52} = -73.8274273593601
x53=64.4026493985908x_{53} = 64.4026493985908
x54=−72.2566308917313x_{54} = -72.2566308917313
x55=87.964594335453x_{55} = 87.964594335453
x56=75.3982236793524x_{56} = 75.3982236793524
x57=−21.9911486312213x_{57} = -21.9911486312213
x58=−43.9822971746609x_{58} = -43.9822971746609
x59=−95.8185759344887x_{59} = -95.8185759344887
x60=6.28318528433976x_{60} = 6.28318528433976
x61=53.4070751278617x_{61} = 53.4070751278617
x62=65.9734457527245x_{62} = 65.9734457527245
x63=−21.9911485864718x_{63} = -21.9911485864718
x64=20.4203522483337x_{64} = 20.4203522483337
x65=9.42477806710815x_{65} = 9.42477806710815
x66=−75.398223834204x_{66} = -75.398223834204
x67=9.42477801500462x_{67} = 9.42477801500462
x68=−1.5707963267949x_{68} = -1.5707963267949
x69=81.6814091468681x_{69} = 81.6814091468681
x70=−9.42477810445402x_{70} = -9.42477810445402
x71=86.3937979737193x_{71} = 86.3937979737193
x72=18.8495559220487x_{72} = 18.8495559220487
x73=50.2654819973737x_{73} = 50.2654819973737
x74=−69.1150382393654x_{74} = -69.1150382393654
x75=−6.28318516003462x_{75} = -6.28318516003462
x76=−59.6902604573056x_{76} = -59.6902604573056
x77=−3.14159262638665x_{77} = -3.14159262638665
x78=50.2654824463642x_{78} = 50.2654824463642
x79=36.1283155162826x_{79} = 36.1283155162826
x80=51.8362787842316x_{80} = 51.8362787842316
x81=−28.2743337371269x_{81} = -28.2743337371269
x82=73.8274273593601x_{82} = 73.8274273593601
x83=15.7079634169143x_{83} = 15.7079634169143
x84=−91.1061867632284x_{84} = -91.1061867632284
x85=−100.530964804106x_{85} = -100.530964804106
x86=−53.407075257786x_{86} = -53.407075257786
x87=−89.5353906273091x_{87} = -89.5353906273091
x88=21.9911485851767x_{88} = 21.9911485851767
x89=40.8407044744738x_{89} = 40.8407044744738
x90=−81.6814090375457x_{90} = -81.6814090375457
x91=−94.2477794692392x_{91} = -94.2477794692392
x92=−78.5398162373076x_{92} = -78.5398162373076
x93=−65.9734453602046x_{93} = -65.9734453602046
x94=94.2477792514877x_{94} = 94.2477792514877
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)^2.
sin⁡2(0)cos⁡(0)\sin^{2}{\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⁡3(x)+2sin⁡(x)cos⁡2(x)=0- \sin^{3}{\left(x \right)} + 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−2atan⁡(2−3)x_{2} = - 2 \operatorname{atan}{\left(\sqrt{2 - \sqrt{3}} \right)}
x3=2atan⁡(2−3)x_{3} = 2 \operatorname{atan}{\left(\sqrt{2 - \sqrt{3}} \right)}
x4=−2atan⁡(3+2)x_{4} = - 2 \operatorname{atan}{\left(\sqrt{\sqrt{3} + 2} \right)}
x5=2atan⁡(3+2)x_{5} = 2 \operatorname{atan}{\left(\sqrt{\sqrt{3} + 2} \right)}
The values of the extrema at the points:
(0, 0)

        /   ___________\      /      /   ___________\\    /      /   ___________\\ 
        |  /       ___ |     2|      |  /       ___ ||    |      |  /       ___ || 
(-2*atan\\/  2 - \/ 3  /, sin \2*atan\\/  2 - \/ 3  //*cos\2*atan\\/  2 - \/ 3  //)

       /   ___________\      /      /   ___________\\    /      /   ___________\\ 
       |  /       ___ |     2|      |  /       ___ ||    |      |  /       ___ || 
(2*atan\\/  2 - \/ 3  /, sin \2*atan\\/  2 - \/ 3  //*cos\2*atan\\/  2 - \/ 3  //)

        /   ___________\      /      /   ___________\\    /      /   ___________\\ 
        |  /       ___ |     2|      |  /       ___ ||    |      |  /       ___ || 
(-2*atan\\/  2 + \/ 3  /, sin \2*atan\\/  2 + \/ 3  //*cos\2*atan\\/  2 + \/ 3  //)

       /   ___________\      /      /   ___________\\    /      /   ___________\\ 
       |  /       ___ |     2|      |  /       ___ ||    |      |  /       ___ || 
(2*atan\\/  2 + \/ 3  /, sin \2*atan\\/  2 + \/ 3  //*cos\2*atan\\/  2 + \/ 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=0x_{1} = 0
x2=−2atan⁡(3+2)x_{2} = - 2 \operatorname{atan}{\left(\sqrt{\sqrt{3} + 2} \right)}
x3=2atan⁡(3+2)x_{3} = 2 \operatorname{atan}{\left(\sqrt{\sqrt{3} + 2} \right)}
Maxima of the function at points:
x3=−2atan⁡(2−3)x_{3} = - 2 \operatorname{atan}{\left(\sqrt{2 - \sqrt{3}} \right)}
x3=2atan⁡(2−3)x_{3} = 2 \operatorname{atan}{\left(\sqrt{2 - \sqrt{3}} \right)}
Decreasing at intervals
[2atan⁡(3+2),∞)\left[2 \operatorname{atan}{\left(\sqrt{\sqrt{3} + 2} \right)}, \infty\right)
Increasing at intervals
(−∞,−2atan⁡(3+2)]\left(-\infty, - 2 \operatorname{atan}{\left(\sqrt{\sqrt{3} + 2} \right)}\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
−(7sin⁡2(x)−2cos⁡2(x))cos⁡(x)=0- \left(7 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
x3=−2atan⁡(8−37)x_{3} = - 2 \operatorname{atan}{\left(\sqrt{8 - 3 \sqrt{7}} \right)}
x4=2atan⁡(8−37)x_{4} = 2 \operatorname{atan}{\left(\sqrt{8 - 3 \sqrt{7}} \right)}
x5=−2atan⁡(37+8)x_{5} = - 2 \operatorname{atan}{\left(\sqrt{3 \sqrt{7} + 8} \right)}
x6=2atan⁡(37+8)x_{6} = 2 \operatorname{atan}{\left(\sqrt{3 \sqrt{7} + 8} \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
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Convex at the intervals
(−∞,−2atan⁡(37+8)]\left(-\infty, - 2 \operatorname{atan}{\left(\sqrt{3 \sqrt{7} + 8} \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→−∞(sin⁡2(x)cos⁡(x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin^{2}{\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⁡2(x)cos⁡(x))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin^{2}{\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)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡2(x)cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\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⁡2(x)cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\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⁡2(x)cos⁡(x)=sin⁡2(x)cos⁡(x)\sin^{2}{\left(x \right)} \cos{\left(x \right)} = \sin^{2}{\left(x \right)} \cos{\left(x \right)}
- Yes
sin⁡2(x)cos⁡(x)=−sin⁡2(x)cos⁡(x)\sin^{2}{\left(x \right)} \cos{\left(x \right)} = - \sin^{2}{\left(x \right)} \cos{\left(x \right)}
- No
so, the function
is
even
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