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sin(x+pi/3)
Plotting a function graph/ sin(x+pi/3)

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sin(x+pi/3)

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sin((x + pi)/3)
 
sin(x + pi/3)
 
sin(x + pi/3)

Graphing y = sin(x+pi/3)

v

The graph:

from to

Intersection points:

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Piecewise:

The solution

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          /    pi\
f(x) = sin|x + --|
          \    3 /
f(x)=sin⁡(x+π3)f{\left(x \right)} = \sin{\left(x + \frac{\pi}{3} \right)} 
f = sin(x + pi/3)
The graph of the function
0-80-60-40-202040608002
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+π3)=0\sin{\left(x + \frac{\pi}{3} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π3x_{1} = - \frac{\pi}{3}
x2=2π3x_{2} = \frac{2 \pi}{3}
Numerical solution
x1=93.2005820564972x_{1} = 93.2005820564972
x2=5.23598775598299x_{2} = 5.23598775598299
x3=−85.870199198121x_{3} = -85.870199198121
x4=20.943951023932x_{4} = 20.943951023932
x5=−7.33038285837618x_{5} = -7.33038285837618
x6=58.6430628670095x_{6} = 58.6430628670095
x7=−95.2949771588904x_{7} = -95.2949771588904
x8=−67.0206432765823x_{8} = -67.0206432765823
x9=−70.162235930172x_{9} = -70.162235930172
x10=2.0943951023932x_{10} = 2.0943951023932
x11=30.3687289847013x_{11} = 30.3687289847013
x12=42.9350995990605x_{12} = 42.9350995990605
x13=−4939.63084899435x_{13} = -4939.63084899435
x14=80.634211442138x_{14} = 80.634211442138
x15=−79.5870138909414x_{15} = -79.5870138909414
x16=−63.8790506229925x_{16} = -63.8790506229925
x17=−48.1710873550435x_{17} = -48.1710873550435
x18=86.9173967493176x_{18} = 86.9173967493176
x19=−13.6135681655558x_{19} = -13.6135681655558
x20=−26.1799387799149x_{20} = -26.1799387799149
x21=−82.7286065445312x_{21} = -82.7286065445312
x22=74.3510261349584x_{22} = 74.3510261349584
x23=96.342174710087x_{23} = 96.342174710087
x24=24.0855436775217x_{24} = 24.0855436775217
x25=64.9262481741891x_{25} = 64.9262481741891
x26=71.2094334813686x_{26} = 71.2094334813686
x27=52.3598775598299x_{27} = 52.3598775598299
x28=−92.1533845053006x_{28} = -92.1533845053006
x29=99.4837673636768x_{29} = 99.4837673636768
x30=39.7935069454707x_{30} = 39.7935069454707
x31=27.2271363311115x_{31} = 27.2271363311115
x32=−32.4631240870945x_{32} = -32.4631240870945
x33=−29.3215314335047x_{33} = -29.3215314335047
x34=68.0678408277789x_{34} = 68.0678408277789
x35=−23.0383461263252x_{35} = -23.0383461263252
x36=8.37758040957278x_{36} = 8.37758040957278
x37=17.8023583703422x_{37} = 17.8023583703422
x38=−57.5958653158129x_{38} = -57.5958653158129
x39=33.5103216382911x_{39} = 33.5103216382911
x40=−45.0294947014537x_{40} = -45.0294947014537
x41=11.5191730631626x_{41} = 11.5191730631626
x42=83.7758040957278x_{42} = 83.7758040957278
x43=−38.7463093942741x_{43} = -38.7463093942741
x44=−60.7374579694027x_{44} = -60.7374579694027
x45=90.0589894029074x_{45} = 90.0589894029074
x46=−89.0117918517108x_{46} = -89.0117918517108
x47=−76.4454212373516x_{47} = -76.4454212373516
x48=−19.8967534727354x_{48} = -19.8967534727354
x49=77.4926187885482x_{49} = 77.4926187885482
x50=36.6519142918809x_{50} = 36.6519142918809
x51=−98.4365698124802x_{51} = -98.4365698124802
x52=−41.8879020478639x_{52} = -41.8879020478639
x53=49.2182849062401x_{53} = 49.2182849062401
x54=−73.3038285837618x_{54} = -73.3038285837618
x55=−10.471975511966x_{55} = -10.471975511966
x56=61.7846555205993x_{56} = 61.7846555205993
x57=−4.18879020478639x_{57} = -4.18879020478639
x58=102.625360017267x_{58} = 102.625360017267
x59=−35.6047167406843x_{59} = -35.6047167406843
x60=14.6607657167524x_{60} = 14.6607657167524
x61=−16.7551608191456x_{61} = -16.7551608191456
x62=−51.3126800086333x_{62} = -51.3126800086333
x63=−54.4542726622231x_{63} = -54.4542726622231
x64=−154.985237577096x_{64} = -154.985237577096
x65=46.0766922526503x_{65} = 46.0766922526503
x66=−1.0471975511966x_{66} = -1.0471975511966
x67=55.5014702134197x_{67} = 55.5014702134197
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x + pi/3).
sin⁡(0+π3)\sin{\left(0 + \frac{\pi}{3} \right)}
The result:
f(0)=32f{\left(0 \right)} = \frac{\sqrt{3}}{2}
The point:
(0, sqrt(3)/2)
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+π3)=0\cos{\left(x + \frac{\pi}{3} \right)} = 0
Solve this equation
The roots of this equation
x1=π6x_{1} = \frac{\pi}{6}
x2=7π6x_{2} = \frac{7 \pi}{6}
The values of the extrema at the points:
 pi    
(--, 1)
 6     

 7*pi     
(----, -1)
  6


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=7π6x_{1} = \frac{7 \pi}{6}
Maxima of the function at points:
x1=π6x_{1} = \frac{\pi}{6}
Decreasing at intervals
(−∞,π6]∪[7π6,∞)\left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{7 \pi}{6}, \infty\right)
Increasing at intervals
[π6,7π6]\left[\frac{\pi}{6}, \frac{7 \pi}{6}\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+π3)=0- \sin{\left(x + \frac{\pi}{3} \right)} = 0
Solve this equation
The roots of this equation
x1=−π3x_{1} = - \frac{\pi}{3}
x2=2π3x_{2} = \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
(−∞,−π3]∪[2π3,∞)\left(-\infty, - \frac{\pi}{3}\right] \cup \left[\frac{2 \pi}{3}, \infty\right)
Convex at the intervals
[−π3,2π3]\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⁡(x+π3)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(x + \frac{\pi}{3} \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+π3)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(x + \frac{\pi}{3} \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 + pi/3), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x+π3)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x + \frac{\pi}{3} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x+π3)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x + \frac{\pi}{3} \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+π3)=cos⁡(x+π6)\sin{\left(x + \frac{\pi}{3} \right)} = \cos{\left(x + \frac{\pi}{6} \right)}
- No
sin⁡(x+π3)=−cos⁡(x+π6)\sin{\left(x + \frac{\pi}{3} \right)} = - \cos{\left(x + \frac{\pi}{6} \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sin(x+pi/3)
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