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  • Graphing y =:
  • -x^2+5x-4
  • |x^2+8x+12|
  • (x^2-4)/(2x+5)
  • -x^2-2x

  • Identical expressions

  • sin(x)^ six *cos(x)
  • sinus of (x) to the power of 6 multiply by co sinus of e of (x)
  • sinus of (x) to the power of six multiply by co sinus of e of (x)
  • sin(x)6*cos(x)
  • sinx6*cosx
  • sin(x)⁶*cos(x)
  • sin(x)^6cos(x)
  • sin(x)6cos(x)
  • sinx6cosx
  • sinx^6cosx

  • Similar expressions

  • sinx^6*cosx
Plotting a function graph/ sin(x)^6*cos(x)

Graphing y = sin(x)^6*cos(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          6          
f(x) = sin (x)*cos(x)
f(x)=sin⁡6(x)cos⁡(x)f{\left(x \right)} = \sin^{6}{\left(x \right)} \cos{\left(x \right)} 
f = sin(x)^6*cos(x)
The graph of the function
02468-8-6-4-2-10100.5-0.5
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:
sin⁡6(x)cos⁡(x)=0\sin^{6}{\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=7.85398163397448x_{1} = 7.85398163397448
x2=−73.8274273593601x_{2} = -73.8274273593601
x3=−21.9913952130885x_{3} = -21.9913952130885
x4=50.2651022915147x_{4} = 50.2651022915147
x5=73.8274273593601x_{5} = 73.8274273593601
x6=28.2737364436143x_{6} = 28.2737364436143
x7=6.28237069462517x_{7} = 6.28237069462517
x8=0x_{8} = 0
x9=−87.9655807165942x_{9} = -87.9655807165942
x10=−1.5707963267949x_{10} = -1.5707963267949
x11=−95.8185759344887x_{11} = -95.8185759344887
x12=−53.4149735399443x_{12} = -53.4149735399443
x13=14.1371669411541x_{13} = 14.1371669411541
x14=−72.2498706463197x_{14} = -72.2498706463197
x15=58.1194640914112x_{15} = 58.1194640914112
x16=−31.4236461391621x_{16} = -31.4236461391621
x17=100.524354683131x_{17} = 100.524354683131
x18=−36.1283155162826x_{18} = -36.1283155162826
x19=−37.7003603220942x_{19} = -37.7003603220942
x20=−43.9827903906253x_{20} = -43.9827903906253
x21=87.9655804270211x_{21} = 87.9655804270211
x22=12.5590139806329x_{22} = 12.5590139806329
x23=65.9741854777245x_{23} = 65.9741854777245
x24=72.2564681995667x_{24} = 72.2564681995667
x25=43.9827903871798x_{25} = 43.9827903871798
x26=94.2478341327998x_{26} = 94.2478341327998
x27=59.698211455972x_{27} = 59.698211455972
x28=36.1283155162826x_{28} = 36.1283155162826
x29=−65.9741855218327x_{29} = -65.9741855218327
x30=−58.1194640914112x_{30} = -58.1194640914112
x31=−81.6830903260566x_{31} = -81.6830903260566
x32=−7.85398163397448x_{32} = -7.85398163397448
x33=86.3937979737193x_{33} = 86.3937979737193
x34=21.9913952130384x_{34} = 21.9913952130384
x35=51.8362787842316x_{35} = 51.8362787842316
x36=34.5503466421936x_{36} = 34.5503466421936
x37=−94.2412263246354x_{37} = -94.2412263246354
x38=15.7155274169849x_{38} = 15.7155274169849
x39=−14.1371669411541x_{39} = -14.1371669411541
x40=80.1106126665397x_{40} = 80.1106126665397
x41=95.8185759344887x_{41} = 95.8185759344887
x42=−15.7089949109591x_{42} = -15.7089949109591
x43=37.7068706961535x_{43} = 37.7068706961535
x44=−9.432316956092x_{44} = -9.432316956092
x45=−97.397622880584x_{45} = -97.397622880584
x46=78.5330170231139x_{46} = 78.5330170231139
x47=−28.267166652441x_{47} = -28.267166652441
x48=56.5416809995023x_{48} = 56.5416809995023
x49=20.4203522483337x_{49} = 20.4203522483337
x50=−6.27581837004495x_{50} = -6.27581837004495
x51=−80.1106126665397x_{51} = -80.1106126665397
x52=−65.9811894209038x_{52} = -65.9811894209038
x53=64.4026493985908x_{53} = 64.4026493985908
x54=−23.5619449019235x_{54} = -23.5619449019235
x55=29.845130209103x_{55} = 29.845130209103
x56=42.4115008234622x_{56} = 42.4115008234622
x57=−51.8362787842316x_{57} = -51.8362787842316
x58=−29.845130209103x_{58} = -29.845130209103
x59=−50.2585174163754x_{59} = -50.2585174163754
x60=−75.4062991299145x_{60} = -75.4062991299145
x61=81.6895496856672x_{61} = 81.6895496856672
x62=−59.6917254813975x_{62} = -59.6917254813975
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)^6*cos(x).
sin⁡6(0)cos⁡(0)\sin^{6}{\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⁡7(x)+6sin⁡5(x)cos⁡2(x)=0- \sin^{7}{\left(x \right)} + 6 \sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−2atan⁡(34−73)x_{2} = - 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{4 - \sqrt{7}}}{3} \right)}
x3=2atan⁡(34−73)x_{3} = 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{4 - \sqrt{7}}}{3} \right)}
x4=−2atan⁡(37+43)x_{4} = - 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{\sqrt{7} + 4}}{3} \right)}
x5=2atan⁡(37+43)x_{5} = 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{\sqrt{7} + 4}}{3} \right)}
The values of the extrema at the points:
(0, 0)

        /         ___________\      /      /         ___________\\    /      /         ___________\\ 
        |  ___   /       ___ |      |      |  ___   /       ___ ||    |      |  ___   /       ___ || 
        |\/ 3 *\/  4 - \/ 7  |     6|      |\/ 3 *\/  4 - \/ 7  ||    |      |\/ 3 *\/  4 - \/ 7  || 
(-2*atan|--------------------|, sin |2*atan|--------------------||*cos|2*atan|--------------------||)
        \         3          /      \      \         3          //    \      \         3          // 

       /         ___________\      /      /         ___________\\    /      /         ___________\\ 
       |  ___   /       ___ |      |      |  ___   /       ___ ||    |      |  ___   /       ___ || 
       |\/ 3 *\/  4 - \/ 7  |     6|      |\/ 3 *\/  4 - \/ 7  ||    |      |\/ 3 *\/  4 - \/ 7  || 
(2*atan|--------------------|, sin |2*atan|--------------------||*cos|2*atan|--------------------||)
       \         3          /      \      \         3          //    \      \         3          // 

        /         ___________\      /      /         ___________\\    /      /         ___________\\ 
        |  ___   /       ___ |      |      |  ___   /       ___ ||    |      |  ___   /       ___ || 
        |\/ 3 *\/  4 + \/ 7  |     6|      |\/ 3 *\/  4 + \/ 7  ||    |      |\/ 3 *\/  4 + \/ 7  || 
(-2*atan|--------------------|, sin |2*atan|--------------------||*cos|2*atan|--------------------||)
        \         3          /      \      \         3          //    \      \         3          // 

       /         ___________\      /      /         ___________\\    /      /         ___________\\ 
       |  ___   /       ___ |      |      |  ___   /       ___ ||    |      |  ___   /       ___ || 
       |\/ 3 *\/  4 + \/ 7  |     6|      |\/ 3 *\/  4 + \/ 7  ||    |      |\/ 3 *\/  4 + \/ 7  || 
(2*atan|--------------------|, sin |2*atan|--------------------||*cos|2*atan|--------------------||)
       \         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=0x_{1} = 0
x2=−2atan⁡(37+43)x_{2} = - 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{\sqrt{7} + 4}}{3} \right)}
x3=2atan⁡(37+43)x_{3} = 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{\sqrt{7} + 4}}{3} \right)}
Maxima of the function at points:
x3=−2atan⁡(34−73)x_{3} = - 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{4 - \sqrt{7}}}{3} \right)}
x3=2atan⁡(34−73)x_{3} = 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{4 - \sqrt{7}}}{3} \right)}
Decreasing at intervals
[2atan⁡(37+43),∞)\left[2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{\sqrt{7} + 4}}{3} \right)}, \infty\right)
Increasing at intervals
(−∞,−2atan⁡(37+43)]\left(-\infty, - 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{\sqrt{7} + 4}}{3} \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
−(19sin⁡2(x)−30cos⁡2(x))sin⁡4(x)cos⁡(x)=0- \left(19 \sin^{2}{\left(x \right)} - 30 \cos^{2}{\left(x \right)}\right) \sin^{4}{\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}
x4=−2atan⁡(1534−71915)x_{4} = - 2 \operatorname{atan}{\left(\frac{\sqrt{15} \sqrt{34 - 7 \sqrt{19}}}{15} \right)}
x5=2atan⁡(1534−71915)x_{5} = 2 \operatorname{atan}{\left(\frac{\sqrt{15} \sqrt{34 - 7 \sqrt{19}}}{15} \right)}
x6=−2atan⁡(15719+3415)x_{6} = - 2 \operatorname{atan}{\left(\frac{\sqrt{15} \sqrt{7 \sqrt{19} + 34}}{15} \right)}
x7=2atan⁡(15719+3415)x_{7} = 2 \operatorname{atan}{\left(\frac{\sqrt{15} \sqrt{7 \sqrt{19} + 34}}{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
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Convex at the intervals
(−∞,−2atan⁡(15719+3415)]\left(-\infty, - 2 \operatorname{atan}{\left(\frac{\sqrt{15} \sqrt{7 \sqrt{19} + 34}}{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→−∞(sin⁡6(x)cos⁡(x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin^{6}{\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⁡6(x)cos⁡(x))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin^{6}{\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 sin(x)^6*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡6(x)cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{6}{\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⁡6(x)cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{6}{\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⁡6(x)cos⁡(x)=sin⁡6(x)cos⁡(x)\sin^{6}{\left(x \right)} \cos{\left(x \right)} = \sin^{6}{\left(x \right)} \cos{\left(x \right)}
- Yes
sin⁡6(x)cos⁡(x)=−sin⁡6(x)cos⁡(x)\sin^{6}{\left(x \right)} \cos{\left(x \right)} = - \sin^{6}{\left(x \right)} \cos{\left(x \right)}
- No
so, the function
is
even
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