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sin(x)-1/2sin(2x)

  • How to use it?

  • Graphing y =:
  • x^3+x^2-x+1
  • -x^2+5x-4
  • -x^2+4x
  • -x^2-2x

  • Identical expressions

  • sin(x)- one /2sin(2x)
  • sinus of (x) minus 1 divide by 2 sinus of (2x)
  • sinus of (x) minus one divide by 2 sinus of (2x)
  • sinx-1/2sin2x
  • sin(x)-1 divide by 2sin(2x)

  • Similar expressions

  • sin(x)+1/2sin(2x)
  • sinx-1/2sin(2x)
Plotting a function graph/ sin(x)-1/2sin(2x)

Graphing y = sin(x)-1/2sin(2x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                sin(2*x)
f(x) = sin(x) - --------
                   2
f(x)=sin⁡(x)−sin⁡(2x)2f{\left(x \right)} = \sin{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2} 
f = sin(x) - sin(2*x)/2
The graph of the function
0-80-60-40-2020406002
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)−sin⁡(2x)2=0\sin{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2} = 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=72.2566310325652x_{1} = 72.2566310325652
x2=−87.9646059734436x_{2} = -87.9646059734436
x3=−3.14159265358979x_{3} = -3.14159265358979
x4=−31.4160020636547x_{4} = -31.4160020636547
x5=6.28317667998233x_{5} = 6.28317667998233
x6=40.8407044966673x_{6} = 40.8407044966673
x7=94.2477801894829x_{7} = 94.2477801894829
x8=−91.106186954104x_{8} = -91.106186954104
x9=−15.707963267949x_{9} = -15.707963267949
x10=31.4160561497334x_{10} = 31.4160561497334
x11=−6.28311070964686x_{11} = -6.28311070964686
x12=53.4070751110265x_{12} = 53.4070751110265
x13=84.8230016469244x_{13} = 84.8230016469244
x14=−47.1238898038469x_{14} = -47.1238898038469
x15=−53.4070751110265x_{15} = -53.4070751110265
x16=−100.530843957291x_{16} = -100.530843957291
x17=−94.2477125450604x_{17} = -94.2477125450604
x18=12.5662986015402x_{18} = 12.5662986015402
x19=−9.42477796076938x_{19} = -9.42477796076938
x20=−37.6991527323202x_{20} = -37.6991527323202
x21=37.6991667998317x_{21} = 37.6991667998317
x22=37.699190106891x_{22} = 37.699190106891
x23=15.707963267949x_{23} = 15.707963267949
x24=75.3983560849494x_{24} = 75.3983560849494
x25=−75.398302687768x_{25} = -75.398302687768
x26=97.3893722612836x_{26} = 97.3893722612836
x27=−25.1328674095127x_{27} = -25.1328674095127
x28=78.5398163397448x_{28} = 78.5398163397448
x29=87.9646063176306x_{29} = 87.9646063176306
x30=9.42477796076938x_{30} = 9.42477796076938
x31=62.8317326282647x_{31} = 62.8317326282647
x32=100.53090005808x_{32} = 100.53090005808
x33=−34.5575191894877x_{33} = -34.5575191894877
x34=−28.2743338823081x_{34} = -28.2743338823081
x35=56.5485993084863x_{35} = 56.5485993084863
x36=−78.5398163397448x_{36} = -78.5398163397448
x37=59.6902604182061x_{37} = 59.6902604182061
x38=−12.566243836043x_{38} = -12.566243836043
x39=43.9823032538019x_{39} = 43.9823032538019
x40=−37.6991249685283x_{40} = -37.6991249685283
x41=−21.9911485751286x_{41} = -21.9911485751286
x42=−81.6814156709145x_{42} = -81.6814156709145
x43=18.8494325865513x_{43} = 18.8494325865513
x44=−72.2566310325652x_{44} = -72.2566310325652
x45=75.3983371038378x_{45} = 75.3983371038378
x46=28.2743338823081x_{46} = 28.2743338823081
x47=6.28310686972697x_{47} = 6.28310686972697
x48=81.6814908613415x_{48} = 81.6814908613415
x49=50.2653660953425x_{49} = 50.2653660953425
x50=−97.3893722612836x_{50} = -97.3893722612836
x51=43.982408931344x_{51} = 43.982408931344
x52=65.9734457253857x_{52} = 65.9734457253857
x53=87.9646679208845x_{53} = 87.9646679208845
x54=−65.9734457253857x_{54} = -65.9734457253857
x55=−56.5485438718624x_{55} = -56.5485438718624
x56=0x_{56} = 0
x57=−59.6902604182061x_{57} = -59.6902604182061
x58=34.5575191894877x_{58} = 34.5575191894877
x59=−69.1151673590674x_{59} = -69.1151673590674
x60=50.2654784088459x_{60} = 50.2654784088459
x61=−43.9823032324142x_{61} = -43.9823032324142
x62=21.9911485751286x_{62} = 21.9911485751286
x63=−50.2654115920793x_{63} = -50.2654115920793
x64=−81.6814265277516x_{64} = -81.6814265277516
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x) - sin(2*x)/2.
sin⁡(0)−sin⁡(2⋅0)2\sin{\left(0 \right)} - \frac{\sin{\left(2 \cdot 0 \right)}}{2}
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
cos⁡(x)−cos⁡(2x)=0\cos{\left(x \right)} - \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−4π3x_{2} = - \frac{4 \pi}{3}
x3=−2π3x_{3} = - \frac{2 \pi}{3}
x4=2π3x_{4} = \frac{2 \pi}{3}
x5=4π3x_{5} = \frac{4 \pi}{3}
x6=2πx_{6} = 2 \pi
The values of the extrema at the points:
(0, 0)

            ___ 
 -4*pi  3*\/ 3  
(-----, -------)
   3       4    

             ___ 
 -2*pi  -3*\/ 3  
(-----, --------)
   3       4     

           ___ 
 2*pi  3*\/ 3  
(----, -------)
  3       4    

            ___ 
 4*pi  -3*\/ 3  
(----, --------)
  3       4     

(2*pi, 0)


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=−2π3x_{1} = - \frac{2 \pi}{3}
x2=4π3x_{2} = \frac{4 \pi}{3}
Maxima of the function at points:
x2=−4π3x_{2} = - \frac{4 \pi}{3}
x2=2π3x_{2} = \frac{2 \pi}{3}
Decreasing at intervals
[4π3,∞)\left[\frac{4 \pi}{3}, \infty\right)
Increasing at intervals
(−∞,−2π3]∪[2π3,4π3]\left(-\infty, - \frac{2 \pi}{3}\right] \cup \left[\frac{2 \pi}{3}, \frac{4 \pi}{3}\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
−sin⁡(x)+2sin⁡(2x)=0- \sin{\left(x \right)} + 2 \sin{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
x3=−ilog⁡(14−15i4)x_{3} = - i \log{\left(\frac{1}{4} - \frac{\sqrt{15} i}{4} \right)}
x4=−ilog⁡(14+15i4)x_{4} = - i \log{\left(\frac{1}{4} + \frac{\sqrt{15} i}{4} \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
(−∞,0]∪[atan⁡(15),π]\left(-\infty, 0\right] \cup \left[\operatorname{atan}{\left(\sqrt{15} \right)}, \pi\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)−sin⁡(2x)2)=⟨−32,32⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2}\right) = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−32,32⟩y = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
lim⁡x→∞(sin⁡(x)−sin⁡(2x)2)=⟨−32,32⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2}\right) = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−32,32⟩y = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x) - sin(2*x)/2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)−sin⁡(2x)2x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)−sin⁡(2x)2x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2}}{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)−sin⁡(2x)2=−sin⁡(x)+sin⁡(2x)2\sin{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2} = - \sin{\left(x \right)} + \frac{\sin{\left(2 x \right)}}{2}
- No
sin⁡(x)−sin⁡(2x)2=sin⁡(x)−sin⁡(2x)2\sin{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2} = \sin{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sin(x)-1/2sin(2x)
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