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Plotting a function graph/ sin(y)

Graphing y = sin(y)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(y) = sin(y)
f(y)=sin⁡(y)f{\left(y \right)} = \sin{\left(y \right)} 
f = sin(y)
The graph of the function
02468-8-6-4-2-10102-2
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis Y at f = 0
so we need to solve the equation:
sin⁡(y)=0\sin{\left(y \right)} = 0
Solve this equation
The points of intersection with the axis Y:

Analytical solution
y1=0y_{1} = 0
y2=πy_{2} = \pi
Numerical solution
y1=59.6902604182061y_{1} = 59.6902604182061
y2=−100.530964914873y_{2} = -100.530964914873
y3=−15.707963267949y_{3} = -15.707963267949
y4=−267.035375555132y_{4} = -267.035375555132
y5=84.8230016469244y_{5} = 84.8230016469244
y6=0y_{6} = 0
y7=−25.1327412287183y_{7} = -25.1327412287183
y8=−75.398223686155y_{8} = -75.398223686155
y9=−50.2654824574367y_{9} = -50.2654824574367
y10=97.3893722612836y_{10} = 97.3893722612836
y11=81.6814089933346y_{11} = 81.6814089933346
y12=−72.2566310325652y_{12} = -72.2566310325652
y13=91.106186954104y_{13} = 91.106186954104
y14=50.2654824574367y_{14} = 50.2654824574367
y15=−43.9822971502571y_{15} = -43.9822971502571
y16=−37.6991118430775y_{16} = -37.6991118430775
y17=25.1327412287183y_{17} = 25.1327412287183
y18=−2642.07942166902y_{18} = -2642.07942166902
y19=−65.9734457253857y_{19} = -65.9734457253857
y20=−53.4070751110265y_{20} = -53.4070751110265
y21=−18.8495559215388y_{21} = -18.8495559215388
y22=−59.6902604182061y_{22} = -59.6902604182061
y23=15.707963267949y_{23} = 15.707963267949
y24=9.42477796076938y_{24} = 9.42477796076938
y25=18.8495559215388y_{25} = 18.8495559215388
y26=−56.5486677646163y_{26} = -56.5486677646163
y27=−232.477856365645y_{27} = -232.477856365645
y28=−6.28318530717959y_{28} = -6.28318530717959
y29=−62.8318530717959y_{29} = -62.8318530717959
y30=12.5663706143592y_{30} = 12.5663706143592
y31=56.5486677646163y_{31} = 56.5486677646163
y32=40.8407044966673y_{32} = 40.8407044966673
y33=3.14159265358979y_{33} = 3.14159265358979
y34=−21.9911485751286y_{34} = -21.9911485751286
y35=−84.8230016469244y_{35} = -84.8230016469244
y36=6.28318530717959y_{36} = 6.28318530717959
y37=69.1150383789755y_{37} = 69.1150383789755
y38=72.2566310325652y_{38} = 72.2566310325652
y39=−78.5398163397448y_{39} = -78.5398163397448
y40=37.6991118430775y_{40} = 37.6991118430775
y41=21.9911485751286y_{41} = 21.9911485751286
y42=47.1238898038469y_{42} = 47.1238898038469
y43=34.5575191894877y_{43} = 34.5575191894877
y44=−97.3893722612836y_{44} = -97.3893722612836
y45=−31.4159265358979y_{45} = -31.4159265358979
y46=100.530964914873y_{46} = 100.530964914873
y47=−47.1238898038469y_{47} = -47.1238898038469
y48=28.2743338823081y_{48} = 28.2743338823081
y49=94.2477796076938y_{49} = 94.2477796076938
y50=−40.8407044966673y_{50} = -40.8407044966673
y51=−12.5663706143592y_{51} = -12.5663706143592
y52=−34.5575191894877y_{52} = -34.5575191894877
y53=−28.2743338823081y_{53} = -28.2743338823081
y54=78.5398163397448y_{54} = 78.5398163397448
y55=−94.2477796076938y_{55} = -94.2477796076938
y56=−91.106186954104y_{56} = -91.106186954104
y57=43.9822971502571y_{57} = 43.9822971502571
y58=75.398223686155y_{58} = 75.398223686155
y59=62.8318530717959y_{59} = 62.8318530717959
y60=−3.14159265358979y_{60} = -3.14159265358979
y61=87.9645943005142y_{61} = 87.9645943005142
y62=53.4070751110265y_{62} = 53.4070751110265
y63=−113.097335529233y_{63} = -113.097335529233
y64=−81.6814089933346y_{64} = -81.6814089933346
y65=−87.9645943005142y_{65} = -87.9645943005142
y66=65.9734457253857y_{66} = 65.9734457253857
y67=−69.1150383789755y_{67} = -69.1150383789755
y68=31.4159265358979y_{68} = 31.4159265358979
y69=−9.42477796076938y_{69} = -9.42477796076938
The points of intersection with the Y axis coordinate
The graph crosses Y axis when y equals 0:
substitute y = 0 to sin(y).
sin⁡(0)\sin{\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
ddyf(y)=0\frac{d}{d y} f{\left(y \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddyf(y)=\frac{d}{d y} f{\left(y \right)} =
the first derivative
cos⁡(y)=0\cos{\left(y \right)} = 0
Solve this equation
The roots of this equation
y1=π2y_{1} = \frac{\pi}{2}
y2=3π2y_{2} = \frac{3 \pi}{2}
The values of the extrema at the points:
 pi    
(--, 1)
 2     

 3*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:
y1=3π2y_{1} = \frac{3 \pi}{2}
Maxima of the function at points:
y1=π2y_{1} = \frac{\pi}{2}
Decreasing at intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right)
Increasing at intervals
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dy2f(y)=0\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dy2f(y)=\frac{d^{2}}{d y^{2}} f{\left(y \right)} =
the second derivative
−sin⁡(y)=0- \sin{\left(y \right)} = 0
Solve this equation
The roots of this equation
y1=0y_{1} = 0
y2=πy_{2} = \pi

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
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \infty\right)
Convex at the intervals
[0,π]\left[0, \pi\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at y->+oo and y->-oo
lim⁡y→−∞sin⁡(y)=⟨−1,1⟩\lim_{y \to -\infty} \sin{\left(y \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⁡y→∞sin⁡(y)=⟨−1,1⟩\lim_{y \to \infty} \sin{\left(y \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(y), divided by y at y->+oo and y ->-oo
lim⁡y→−∞(sin⁡(y)y)=0\lim_{y \to -\infty}\left(\frac{\sin{\left(y \right)}}{y}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡y→∞(sin⁡(y)y)=0\lim_{y \to \infty}\left(\frac{\sin{\left(y \right)}}{y}\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(-y) и f = -f(-y).
So, check:
sin⁡(y)=−sin⁡(y)\sin{\left(y \right)} = - \sin{\left(y \right)}
- No
sin⁡(y)=sin⁡(y)\sin{\left(y \right)} = \sin{\left(y \right)}
- Yes
so, the function
is
odd
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