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  • Graphing y =:
  • x^3-3x+1 x^3-3x+1
  • x*arctgx x*arctgx
  • x+1
  • 3^x
  • Integral of d{x}:
  • sin(x)^4 sin(x)^4
  • Derivative of:
  • sin(x)^4 sin(x)^4

  • Identical expressions

  • sin(x)^ four
  • sinus of (x) to the power of 4
  • sinus of (x) to the power of four
  • sin(x)4
  • sinx4
  • sin(x)⁴
  • sinx^4

  • Similar expressions

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

Graphing y = sin(x)^4

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          4   
f(x) = sin (x)
f(x)=sin⁡4(x)f{\left(x \right)} = \sin^{4}{\left(x \right)} 
f = sin(x)^4
The graph of the function
02468-8-6-4-2-101002
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⁡4(x)=0\sin^{4}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=−75.399135702077x_{1} = -75.399135702077
x2=12.5655386528631x_{2} = 12.5655386528631
x3=−100.530131852098x_{3} = -100.530131852098
x4=−31.4167988656097x_{4} = -31.4167988656097
x5=−53.4079673694574x_{5} = -53.4079673694574
x6=56.5478768483803x_{6} = 56.5478768483803
x7=−97.3903038626381x_{7} = -97.3903038626381
x8=100.530215717383x_{8} = 100.530215717383
x9=−15.7080840961257x_{9} = -15.7080840961257
x10=−94.2470163876969x_{10} = -94.2470163876969
x11=−28.2735039765954x_{11} = -28.2735039765954
x12=94.247786003435x_{12} = 94.247786003435
x13=−50.2646745128884x_{13} = -50.2646745128884
x14=−6.28233370088717x_{14} = -6.28233370088717
x15=59.691175606583x_{15} = 59.691175606583
x16=−65.9735393756143x_{16} = -65.9735393756143
x17=72.2566119325769x_{17} = 72.2566119325769
x18=50.2654378689882x_{18} = 50.2654378689882
x19=−87.9647198526024x_{19} = -87.9647198526024
x20=−9.42563019128159x_{20} = -9.42563019128159
x21=97.3891235063633x_{21} = 97.3891235063633
x22=21.991179696756x_{22} = 21.991179696756
x23=6.283089833683x_{23} = 6.283089833683
x24=−6.28329572483107x_{24} = -6.28329572483107
x25=−78.5408115487035x_{25} = -78.5408115487035
x26=−21.9911797014772x_{26} = -21.9911797014772
x27=78.5390461988625x_{27} = 78.5390461988625
x28=34.556707666247x_{28} = 34.556707666247
x29=81.682344881864x_{29} = 81.682344881864
x30=0x_{30} = 0
x31=37.7000060867127x_{31} = 37.7000060867127
x32=−37.6992579404428x_{32} = -37.6992579404428
x33=−59.6904316816828x_{33} = -59.6904316816828
x34=−276.460396846785x_{34} = -276.460396846785
x35=28.2742638291305x_{35} = 28.2742638291305
x36=43.982359365339x_{36} = 43.982359365339
x37=87.964718403403x_{37} = 87.964718403403
x38=65.9735389544475x_{38} = 65.9735389544475
x39=15.7088363188216x_{39} = 15.7088363188216
x40=−72.2558453147736x_{40} = -72.2558453147736
x41=−43.98235944366x_{41} = -43.98235944366
x42=97.3894444037214x_{42} = 97.3894444037214
x43=−81.681605304824x_{43} = -81.681605304824
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)^4.
sin⁡4(0)\sin^{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⁡3(x)cos⁡(x)=04 \sin^{3}{\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}
The values of the extrema at the points:
(0, 0)

 -pi     
(----, 1)
  2      

 pi    
(--, 1)
 2


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
Maxima of the function at points:
x1=−π2x_{1} = - \frac{\pi}{2}
x1=π2x_{1} = \frac{\pi}{2}
Decreasing at intervals
(−∞,−π2]∪[0,∞)\left(-\infty, - \frac{\pi}{2}\right] \cup \left[0, \infty\right)
Increasing at intervals
(−∞,0]∪[π2,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{2}, \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
4(−sin⁡2(x)+3cos⁡2(x))sin⁡2(x)=04 \left(- \sin^{2}{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−2π3x_{2} = - \frac{2 \pi}{3}
x3=−π3x_{3} = - \frac{\pi}{3}
x4=π3x_{4} = \frac{\pi}{3}
x5=2π3x_{5} = \frac{2 \pi}{3}

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
[2π3,∞)\left[\frac{2 \pi}{3}, \infty\right)
Convex at the intervals
(−∞,−π3]∪[π3,2π3]\left(-\infty, - \frac{\pi}{3}\right] \cup \left[\frac{\pi}{3}, \frac{2 \pi}{3}\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⁡4(x)=⟨0,1⟩\lim_{x \to -\infty} \sin^{4}{\left(x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
lim⁡x→∞sin⁡4(x)=⟨0,1⟩\lim_{x \to \infty} \sin^{4}{\left(x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)^4, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡4(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{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⁡4(x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{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⁡4(x)=sin⁡4(x)\sin^{4}{\left(x \right)} = \sin^{4}{\left(x \right)}
- Yes
sin⁡4(x)=−sin⁡4(x)\sin^{4}{\left(x \right)} = - \sin^{4}{\left(x \right)}
- No
so, the function
is
even
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