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cos(x)^4*sin(x)

  • How to use it?

  • Graphing y =:
  • (x^2-x-6)/(x-2)
  • x^2+6x+10
  • -x^2+4x
  • x^2+4x+5

  • Identical expressions

  • cos(x)^ four *sin(x)
  • co sinus of e of (x) to the power of 4 multiply by sinus of (x)
  • co sinus of e of (x) to the power of four multiply by sinus of (x)
  • cos(x)4*sin(x)
  • cosx4*sinx
  • cos(x)⁴*sin(x)
  • cos(x)^4sin(x)
  • cos(x)4sin(x)
  • cosx4sinx
  • cosx^4sinx

  • Similar expressions

  • cosx^4*sinx
Plotting a function graph/ cos(x)^4*sin(x)

Graphing y = cos(x)^4*sin(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          4          
f(x) = cos (x)*sin(x)
f(x)=sin⁡(x)cos⁡4(x)f{\left(x \right)} = \sin{\left(x \right)} \cos^{4}{\left(x \right)} 
f = sin(x)*cos(x)^4
The graph of the function
0-20-101020304050607080900.5-0.5
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⁡(x)cos⁡4(x)=0\sin{\left(x \right)} \cos^{4}{\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=πx_{3} = \pi
x4=3π2x_{4} = \frac{3 \pi}{2}
Numerical solution
x1=72.2566310325652x_{1} = 72.2566310325652
x2=94.2477796076938x_{2} = 94.2477796076938
x3=−23.5624592548809x_{3} = -23.5624592548809
x4=80.1097338230489x_{4} = 80.1097338230489
x5=48.6936384544703x_{5} = 48.6936384544703
x6=−15.707963267949x_{6} = -15.707963267949
x7=−37.6991118430775x_{7} = -37.6991118430775
x8=95.8191542378595x_{8} = 95.8191542378595
x9=51.8368083468028x_{9} = 51.8368083468028
x10=64.402225324733x_{10} = 64.402225324733
x11=−29.8448010550122x_{11} = -29.8448010550122
x12=−81.6814089933346x_{12} = -81.6814089933346
x13=−51.83599882721x_{13} = -51.83599882721
x14=14.1377544028184x_{14} = 14.1377544028184
x15=6.28318530717959x_{15} = 6.28318530717959
x16=−1.57128845156238x_{16} = -1.57128845156238
x17=50.2654824574367x_{17} = 50.2654824574367
x18=−86.3942476952527x_{18} = -86.3942476952527
x19=−73.8271873469416x_{19} = -73.8271873469416
x20=42.4110538406422x_{20} = 42.4110538406422
x21=−36.1278886571281x_{21} = -36.1278886571281
x22=−14.1367137375758x_{22} = -14.1367137375758
x23=7.85446073245193x_{23} = 7.85446073245193
x24=−80.1102398600213x_{24} = -80.1102398600213
x25=36.12756931678x_{25} = 36.12756931678
x26=29.8456347584674x_{26} = 29.8456347584674
x27=58.1200321773123x_{27} = 58.1200321773123
x28=−89.5359702676269x_{28} = -89.5359702676269
x29=−87.9645943005142x_{29} = -87.9645943005142
x30=15.707963267949x_{30} = 15.707963267949
x31=92.6759646251107x_{31} = 92.6759646251107
x32=−67.5448001650043x_{32} = -67.5448001650043
x33=70.6848013733878x_{33} = 70.6848013733878
x34=73.8279815042637x_{34} = 73.8279815042637
x35=−45.5536298267302x_{35} = -45.5536298267302
x36=−58.1190640304595x_{36} = -58.1190640304595
x37=−94.2477796076938x_{37} = -94.2477796076938
x38=−21.9911485751286x_{38} = -21.9911485751286
x39=−7.85359154756817x_{39} = -7.85359154756817
x40=−72.2566310325652x_{40} = -72.2566310325652
x41=87.9645943005142x_{41} = 87.9645943005142
x42=86.3933970305611x_{42} = 86.3933970305611
x43=43.9822971502571x_{43} = 43.9822971502571
x44=28.2743338823081x_{44} = 28.2743338823081
x45=−86.392754643863x_{45} = -86.392754643863
x46=−43.9822971502571x_{46} = -43.9822971502571
x47=−20.4207948804258x_{47} = -20.4207948804258
x48=36.1277548171785x_{48} = 36.1277548171785
x49=−95.8183697328683x_{49} = -95.8183697328683
x50=65.9734457253857x_{50} = 65.9734457253857
x51=80.1114818379005x_{51} = 80.1114818379005
x52=−65.9734457253857x_{52} = -65.9734457253857
x53=−64.4015912708019x_{53} = -64.4015912708019
x54=0x_{54} = 0
x55=20.4198825810465x_{55} = 20.4198825810465
x56=−59.6902604182061x_{56} = -59.6902604182061
x57=36.1288287787813x_{57} = 36.1288287787813
x58=21.9911485751286x_{58} = 21.9911485751286
x59=58.1198687108489x_{59} = 58.1198687108489
x60=14.1376257184341x_{60} = 14.1376257184341
x61=26.7024758752468x_{61} = 26.7024758752468
x62=−100.530964914873x_{62} = -100.530964914873
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)^4*sin(x).
sin⁡(0)cos⁡4(0)\sin{\left(0 \right)} \cos^{4}{\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
−4sin⁡2(x)cos⁡3(x)+cos⁡5(x)=0- 4 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)} + \cos^{5}{\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⁡(2−5)x_{3} = - 2 \operatorname{atan}{\left(2 - \sqrt{5} \right)}
x4=2atan⁡(2−5)x_{4} = 2 \operatorname{atan}{\left(2 - \sqrt{5} \right)}
x5=−2atan⁡(2+5)x_{5} = - 2 \operatorname{atan}{\left(2 + \sqrt{5} \right)}
x6=2atan⁡(2+5)x_{6} = 2 \operatorname{atan}{\left(2 + \sqrt{5} \right)}
The values of the extrema at the points:
 -pi     
(----, 0)
  2      

 pi    
(--, 0)
 2     

        /      ___\      4/      /      ___\\    /      /      ___\\ 
(-2*atan\2 - \/ 5 /, -cos \2*atan\2 - \/ 5 //*sin\2*atan\2 - \/ 5 //)

       /      ___\     4/      /      ___\\    /      /      ___\\ 
(2*atan\2 - \/ 5 /, cos \2*atan\2 - \/ 5 //*sin\2*atan\2 - \/ 5 //)

        /      ___\      4/      /      ___\\    /      /      ___\\ 
(-2*atan\2 + \/ 5 /, -cos \2*atan\2 + \/ 5 //*sin\2*atan\2 + \/ 5 //)

       /      ___\     4/      /      ___\\    /      /      ___\\ 
(2*atan\2 + \/ 5 /, cos \2*atan\2 + \/ 5 //*sin\2*atan\2 + \/ 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=π2x_{1} = \frac{\pi}{2}
x2=2atan⁡(2−5)x_{2} = 2 \operatorname{atan}{\left(2 - \sqrt{5} \right)}
x3=−2atan⁡(2+5)x_{3} = - 2 \operatorname{atan}{\left(2 + \sqrt{5} \right)}
Maxima of the function at points:
x3=−π2x_{3} = - \frac{\pi}{2}
x3=−2atan⁡(2−5)x_{3} = - 2 \operatorname{atan}{\left(2 - \sqrt{5} \right)}
x3=2atan⁡(2+5)x_{3} = 2 \operatorname{atan}{\left(2 + \sqrt{5} \right)}
Decreasing at intervals
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
(−∞,−2atan⁡(2+5)]\left(-\infty, - 2 \operatorname{atan}{\left(2 + \sqrt{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
(12sin⁡2(x)−13cos⁡2(x))sin⁡(x)cos⁡2(x)=0\left(12 \sin^{2}{\left(x \right)} - 13 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}
x3=πx_{3} = \pi
x4=3π2x_{4} = \frac{3 \pi}{2}
x5=−atan⁡(396)x_{5} = - \operatorname{atan}{\left(\frac{\sqrt{39}}{6} \right)}
x6=atan⁡(396)x_{6} = \operatorname{atan}{\left(\frac{\sqrt{39}}{6} \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
[π,∞)\left[\pi, \infty\right)
Convex at the intervals
(−∞,−atan⁡(396)]∪[0,atan⁡(396)]\left(-\infty, - \operatorname{atan}{\left(\frac{\sqrt{39}}{6} \right)}\right] \cup \left[0, \operatorname{atan}{\left(\frac{\sqrt{39}}{6} \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⁡(x)cos⁡4(x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} \cos^{4}{\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⁡4(x))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} \cos^{4}{\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)^4*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)cos⁡4(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cos^{4}{\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⁡4(x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cos^{4}{\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⁡4(x)=−sin⁡(x)cos⁡4(x)\sin{\left(x \right)} \cos^{4}{\left(x \right)} = - \sin{\left(x \right)} \cos^{4}{\left(x \right)}
- No
sin⁡(x)cos⁡4(x)=sin⁡(x)cos⁡4(x)\sin{\left(x \right)} \cos^{4}{\left(x \right)} = \sin{\left(x \right)} \cos^{4}{\left(x \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = cos(x)^4*sin(x)
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