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  • How to use it?

  • Graphing y =:
  • x^3-x^2-2x
  • |x+3|-|x-1|+x+2
  • (x-3)sqrt(x)
  • |x|-3

  • Identical expressions

  • three *sin(x)*(cos(x))^ two
  • 3 multiply by sinus of (x) multiply by ( co sinus of e of (x)) squared
  • three multiply by sinus of (x) multiply by ( co sinus of e of (x)) to the power of two
  • 3*sin(x)*(cos(x))2
  • 3*sinx*cosx2
  • 3*sin(x)*(cos(x))²
  • 3*sin(x)*(cos(x)) to the power of 2
  • 3sin(x)(cos(x))^2
  • 3sin(x)(cos(x))2
  • 3sinxcosx2
  • 3sinxcosx^2

  • Similar expressions

  • 3*sinx*(cosx)^2
Plotting a function graph/ 3*sin(x)*(cos(x))^2

Graphing y = 3*sin(x)*(cos(x))^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                   2   
f(x) = 3*sin(x)*cos (x)
f(x)=3sin⁡(x)cos⁡2(x)f{\left(x \right)} = 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)} 
f = (3*sin(x))*cos(x)^2
The graph of the function
02468-8-6-4-2-10102.5-2.5
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:
3sin⁡(x)cos⁡2(x)=03 \sin{\left(x \right)} \cos^{2}{\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=−50.2654824574367x_{1} = -50.2654824574367
x2=−1.57079642505341x_{2} = -1.57079642505341
x3=−61.261056881309x_{3} = -61.261056881309
x4=7.85398164444075x_{4} = 7.85398164444075
x5=36.1283160593477x_{5} = 36.1283160593477
x6=−7.85398150264842x_{6} = -7.85398150264842
x7=72.2566310325652x_{7} = 72.2566310325652
x8=−43.9822971502571x_{8} = -43.9822971502571
x9=23.5619450555027x_{9} = 23.5619450555027
x10=−37.6991118430775x_{10} = -37.6991118430775
x11=45.5530936288414x_{11} = 45.5530936288414
x12=34.5575191894877x_{12} = 34.5575191894877
x13=−94.2477796076938x_{13} = -94.2477796076938
x14=−21.9911485751286x_{14} = -21.9911485751286
x15=50.2654824574367x_{15} = 50.2654824574367
x16=−42.4115006663339x_{16} = -42.4115006663339
x17=−15.707963267949x_{17} = -15.707963267949
x18=−51.8362786906154x_{18} = -51.8362786906154
x19=21.9911485751286x_{19} = 21.9911485751286
x20=−87.9645943005142x_{20} = -87.9645943005142
x21=−72.2566310325652x_{21} = -72.2566310325652
x22=20.4203521537986x_{22} = 20.4203521537986
x23=70.6858345559153x_{23} = 70.6858345559153
x24=59.6902604182061x_{24} = 59.6902604182061
x25=95.8185760508519x_{25} = 95.8185760508519
x26=−83.2522054524035x_{26} = -83.2522054524035
x27=−80.1106125824842x_{27} = -80.1106125824842
x28=89.5353907744432x_{28} = 89.5353907744432
x29=−29.8451300981866x_{29} = -29.8451300981866
x30=−9.42477796076938x_{30} = -9.42477796076938
x31=9.42477796076938x_{31} = 9.42477796076938
x32=80.1106131546315x_{32} = 80.1106131546315
x33=−14.13716684381x_{33} = -14.13716684381
x34=−65.9734457253857x_{34} = -65.9734457253857
x35=15.707963267949x_{35} = 15.707963267949
x36=−36.128315423197x_{36} = -36.128315423197
x37=−45.5530935824522x_{37} = -45.5530935824522
x38=67.5442422018325x_{38} = 67.5442422018325
x39=28.2743338823081x_{39} = 28.2743338823081
x40=94.2477796076938x_{40} = 94.2477796076938
x41=64.4026493118058x_{41} = 64.4026493118058
x42=37.6991118430775x_{42} = 37.6991118430775
x43=251.327412287183x_{43} = 251.327412287183
x44=6.28318530717959x_{44} = 6.28318530717959
x45=−70.6858349962623x_{45} = -70.6858349962623
x46=−86.3937978155375x_{46} = -86.3937978155375
x47=−53.4070751110265x_{47} = -53.4070751110265
x48=−6.28318530717959x_{48} = -6.28318530717959
x49=−89.5353907394375x_{49} = -89.5353907394375
x50=−73.8274272804402x_{50} = -73.8274272804402
x51=−67.5442421609972x_{51} = -67.5442421609972
x52=−75.398223686155x_{52} = -75.398223686155
x53=−64.4026492408158x_{53} = -64.4026492408158
x54=0x_{54} = 0
x55=−58.1194640027517x_{55} = -58.1194640027517
x56=4.71238883532779x_{56} = 4.71238883532779
x57=−17.278759737384x_{57} = -17.278759737384
x58=−97.3893722612836x_{58} = -97.3893722612836
x59=73.8274274722061x_{59} = 73.8274274722061
x60=−23.561945003804x_{60} = -23.561945003804
x61=14.1371670924752x_{61} = 14.1371670924752
x62=−28.2743338823081x_{62} = -28.2743338823081
x63=81.6814089933346x_{63} = 81.6814089933346
x64=29.8451303144929x_{64} = 29.8451303144929
x65=100.530964914873x_{65} = 100.530964914873
x66=−39.2699083096144x_{66} = -39.2699083096144
x67=51.8362788934209x_{67} = 51.8362788934209
x68=78.5398163397448x_{68} = 78.5398163397448
x69=86.3937978909611x_{69} = 86.3937978909611
x70=−20.4203520921076x_{70} = -20.4203520921076
x71=1.57079648184495x_{71} = 1.57079648184495
x72=12.5663706143592x_{72} = 12.5663706143592
x73=87.9645943005142x_{73} = 87.9645943005142
x74=43.9822971502571x_{74} = 43.9822971502571
x75=−31.4159265358979x_{75} = -31.4159265358979
x76=−95.818575585294x_{76} = -95.818575585294
x77=−81.6814089933346x_{77} = -81.6814089933346
x78=56.5486677646163x_{78} = 56.5486677646163
x79=26.7035374084741x_{79} = 26.7035374084741
x80=42.4115007327518x_{80} = 42.4115007327518
x81=92.6769831301454x_{81} = 92.6769831301454
x82=−95.8185758682892x_{82} = -95.8185758682892
x83=−59.6902604182061x_{83} = -59.6902604182061
x84=65.9734457253857x_{84} = 65.9734457253857
x85=7.85398173541774x_{85} = 7.85398173541774
x86=48.6946859820148x_{86} = 48.6946859820148
x87=−92.6769836764771x_{87} = -92.6769836764771
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (3*sin(x))*cos(x)^2.
3sin⁡(0)cos⁡2(0)3 \sin{\left(0 \right)} \cos^{2}{\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
−6sin⁡2(x)cos⁡(x)+3cos⁡3(x)=0- 6 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 3 \cos^{3}{\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⁡(5−26)x_{3} = - 2 \operatorname{atan}{\left(\sqrt{5 - 2 \sqrt{6}} \right)}
x4=2atan⁡(5−26)x_{4} = 2 \operatorname{atan}{\left(\sqrt{5 - 2 \sqrt{6}} \right)}
x5=−2atan⁡(26+5)x_{5} = - 2 \operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \right)}
x6=2atan⁡(26+5)x_{6} = 2 \operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \right)}
The values of the extrema at the points:
 -pi     
(----, 0)
  2      

 pi    
(--, 0)
 2     

        /   _____________\         /      /   _____________\\    /      /   _____________\\ 
        |  /         ___ |        2|      |  /         ___ ||    |      |  /         ___ || 
(-2*atan\\/  5 - 2*\/ 6  /, -3*cos \2*atan\\/  5 - 2*\/ 6  //*sin\2*atan\\/  5 - 2*\/ 6  //)

       /   _____________\        /      /   _____________\\    /      /   _____________\\ 
       |  /         ___ |       2|      |  /         ___ ||    |      |  /         ___ || 
(2*atan\\/  5 - 2*\/ 6  /, 3*cos \2*atan\\/  5 - 2*\/ 6  //*sin\2*atan\\/  5 - 2*\/ 6  //)

        /   _____________\         /      /   _____________\\    /      /   _____________\\ 
        |  /         ___ |        2|      |  /         ___ ||    |      |  /         ___ || 
(-2*atan\\/  5 + 2*\/ 6  /, -3*cos \2*atan\\/  5 + 2*\/ 6  //*sin\2*atan\\/  5 + 2*\/ 6  //)

       /   _____________\        /      /   _____________\\    /      /   _____________\\ 
       |  /         ___ |       2|      |  /         ___ ||    |      |  /         ___ || 
(2*atan\\/  5 + 2*\/ 6  /, 3*cos \2*atan\\/  5 + 2*\/ 6  //*sin\2*atan\\/  5 + 2*\/ 6  //)


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=π2x_{1} = \frac{\pi}{2}
x2=−2atan⁡(5−26)x_{2} = - 2 \operatorname{atan}{\left(\sqrt{5 - 2 \sqrt{6}} \right)}
x3=−2atan⁡(26+5)x_{3} = - 2 \operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \right)}
Maxima of the function at points:
x3=−π2x_{3} = - \frac{\pi}{2}
x3=2atan⁡(5−26)x_{3} = 2 \operatorname{atan}{\left(\sqrt{5 - 2 \sqrt{6}} \right)}
x3=2atan⁡(26+5)x_{3} = 2 \operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \right)}
Decreasing at intervals
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
(−∞,−2atan⁡(26+5)]\left(-\infty, - 2 \operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \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
3(2sin⁡2(x)−7cos⁡2(x))sin⁡(x)=03 \left(2 \sin^{2}{\left(x \right)} - 7 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−2atan⁡(711−627)x_{2} = - 2 \operatorname{atan}{\left(\frac{\sqrt{7} \sqrt{11 - 6 \sqrt{2}}}{7} \right)}
x3=2atan⁡(711−627)x_{3} = 2 \operatorname{atan}{\left(\frac{\sqrt{7} \sqrt{11 - 6 \sqrt{2}}}{7} \right)}
x4=−2atan⁡(762+117)x_{4} = - 2 \operatorname{atan}{\left(\frac{\sqrt{7} \sqrt{6 \sqrt{2} + 11}}{7} \right)}
x5=2atan⁡(762+117)x_{5} = 2 \operatorname{atan}{\left(\frac{\sqrt{7} \sqrt{6 \sqrt{2} + 11}}{7} \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⁡(711−627),∞)\left[2 \operatorname{atan}{\left(\frac{\sqrt{7} \sqrt{11 - 6 \sqrt{2}}}{7} \right)}, \infty\right)
Convex at the intervals
(−∞,−2atan⁡(711−627)]\left(-\infty, - 2 \operatorname{atan}{\left(\frac{\sqrt{7} \sqrt{11 - 6 \sqrt{2}}}{7} \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→−∞(3sin⁡(x)cos⁡2(x))=⟨−3,3⟩\lim_{x \to -\infty}\left(3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
lim⁡x→∞(3sin⁡(x)cos⁡2(x))=⟨−3,3⟩\lim_{x \to \infty}\left(3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (3*sin(x))*cos(x)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3sin⁡(x)cos⁡2(x)x)=0\lim_{x \to -\infty}\left(\frac{3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(3sin⁡(x)cos⁡2(x)x)=0\lim_{x \to \infty}\left(\frac{3 \sin{\left(x \right)} \cos^{2}{\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:
3sin⁡(x)cos⁡2(x)=−3sin⁡(x)cos⁡2(x)3 \sin{\left(x \right)} \cos^{2}{\left(x \right)} = - 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}
- No
3sin⁡(x)cos⁡2(x)=3sin⁡(x)cos⁡2(x)3 \sin{\left(x \right)} \cos^{2}{\left(x \right)} = 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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