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Plotting a function graph/ sin

Graphing y = sin

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(x)
f(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} 
f = sin(x)
The graph of the function
055010152025303540452-2
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)=0\sin{\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=31.4159265358979x_{1} = 31.4159265358979
x2=3.14159265358979x_{2} = 3.14159265358979
x3=−267.035375555132x_{3} = -267.035375555132
x4=−47.1238898038469x_{4} = -47.1238898038469
x5=−12.5663706143592x_{5} = -12.5663706143592
x6=−34.5575191894877x_{6} = -34.5575191894877
x7=−69.1150383789755x_{7} = -69.1150383789755
x8=−2642.07942166902x_{8} = -2642.07942166902
x9=−65.9734457253857x_{9} = -65.9734457253857
x10=75.398223686155x_{10} = 75.398223686155
x11=−56.5486677646163x_{11} = -56.5486677646163
x12=−50.2654824574367x_{12} = -50.2654824574367
x13=59.6902604182061x_{13} = 59.6902604182061
x14=72.2566310325652x_{14} = 72.2566310325652
x15=91.106186954104x_{15} = 91.106186954104
x16=−91.106186954104x_{16} = -91.106186954104
x17=−62.8318530717959x_{17} = -62.8318530717959
x18=−6.28318530717959x_{18} = -6.28318530717959
x19=−232.477856365645x_{19} = -232.477856365645
x20=62.8318530717959x_{20} = 62.8318530717959
x21=6.28318530717959x_{21} = 6.28318530717959
x22=−25.1327412287183x_{22} = -25.1327412287183
x23=94.2477796076938x_{23} = 94.2477796076938
x24=−9.42477796076938x_{24} = -9.42477796076938
x25=−37.6991118430775x_{25} = -37.6991118430775
x26=65.9734457253857x_{26} = 65.9734457253857
x27=−100.530964914873x_{27} = -100.530964914873
x28=−43.9822971502571x_{28} = -43.9822971502571
x29=25.1327412287183x_{29} = 25.1327412287183
x30=21.9911485751286x_{30} = 21.9911485751286
x31=87.9645943005142x_{31} = 87.9645943005142
x32=−40.8407044966673x_{32} = -40.8407044966673
x33=−97.3893722612836x_{33} = -97.3893722612836
x34=43.9822971502571x_{34} = 43.9822971502571
x35=−53.4070751110265x_{35} = -53.4070751110265
x36=97.3893722612836x_{36} = 97.3893722612836
x37=100.530964914873x_{37} = 100.530964914873
x38=−94.2477796076938x_{38} = -94.2477796076938
x39=−31.4159265358979x_{39} = -31.4159265358979
x40=18.8495559215388x_{40} = 18.8495559215388
x41=78.5398163397448x_{41} = 78.5398163397448
x42=−18.8495559215388x_{42} = -18.8495559215388
x43=53.4070751110265x_{43} = 53.4070751110265
x44=47.1238898038469x_{44} = 47.1238898038469
x45=12.5663706143592x_{45} = 12.5663706143592
x46=81.6814089933346x_{46} = 81.6814089933346
x47=34.5575191894877x_{47} = 34.5575191894877
x48=−75.398223686155x_{48} = -75.398223686155
x49=−15.707963267949x_{49} = -15.707963267949
x50=50.2654824574367x_{50} = 50.2654824574367
x51=−81.6814089933346x_{51} = -81.6814089933346
x52=−3.14159265358979x_{52} = -3.14159265358979
x53=−59.6902604182061x_{53} = -59.6902604182061
x54=−28.2743338823081x_{54} = -28.2743338823081
x55=−87.9645943005142x_{55} = -87.9645943005142
x56=9.42477796076938x_{56} = 9.42477796076938
x57=−21.9911485751286x_{57} = -21.9911485751286
x58=−113.097335529233x_{58} = -113.097335529233
x59=56.5486677646163x_{59} = 56.5486677646163
x60=15.707963267949x_{60} = 15.707963267949
x61=84.8230016469244x_{61} = 84.8230016469244
x62=−78.5398163397448x_{62} = -78.5398163397448
x63=37.6991118430775x_{63} = 37.6991118430775
x64=−72.2566310325652x_{64} = -72.2566310325652
x65=−84.8230016469244x_{65} = -84.8230016469244
x66=69.1150383789755x_{66} = 69.1150383789755
x67=0x_{67} = 0
x68=28.2743338823081x_{68} = 28.2743338823081
x69=40.8407044966673x_{69} = 40.8407044966673
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⁡(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
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)=0\cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{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:
x1=3π2x_{1} = \frac{3 \pi}{2}
Maxima of the function at points:
x1=π2x_{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
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)=0- \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{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 x->+oo and x->-oo
lim⁡x→−∞sin⁡(x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(x \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)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(x \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), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\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)x)=0\lim_{x \to \infty}\left(\frac{\sin{\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)=−sin⁡(x)\sin{\left(x \right)} = - \sin{\left(x \right)}
- No
sin⁡(x)=sin⁡(x)\sin{\left(x \right)} = \sin{\left(x \right)}
- Yes
so, the function
is
odd
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