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

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cos(x+pi/3)

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cos((x + pi)/3)
 
cos(x + pi/3)
 
cos(x + pi/3)

Graphing y = cos(x+pi/3)

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The graph:

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Intersection points:

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Piecewise:

The solution

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          /    pi\
f(x) = cos|x + --|
          \    3 /
f(x)=cos⁡(x+π3)f{\left(x \right)} = \cos{\left(x + \frac{\pi}{3} \right)} 
f = cos(x + pi/3)
The graph of the function
05-65-60-55-50-45-40-35-30-25-20-15-10-5102-2
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:
cos⁡(x+π3)=0\cos{\left(x + \frac{\pi}{3} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π6x_{1} = \frac{\pi}{6}
x2=7π6x_{2} = \frac{7 \pi}{6}
Numerical solution
x1=−65.4498469497874x_{1} = -65.4498469497874
x2=−18.3259571459405x_{2} = -18.3259571459405
x3=−62.3082542961976x_{3} = -62.3082542961976
x4=−68.5914396033772x_{4} = -68.5914396033772
x5=−46.6002910282486x_{5} = -46.6002910282486
x6=119.90411961201x_{6} = 119.90411961201
x7=50.789081233035x_{7} = 50.789081233035
x8=60.2138591938044x_{8} = 60.2138591938044
x9=−12.0427718387609x_{9} = -12.0427718387609
x10=927.293431584587x_{10} = 927.293431584587
x11=−5.75958653158129x_{11} = -5.75958653158129
x12=3.66519142918809x_{12} = 3.66519142918809
x13=−24.60914245312x_{13} = -24.60914245312
x14=31.9395253114962x_{14} = 31.9395253114962
x15=−87.4409955249159x_{15} = -87.4409955249159
x16=−2.61799387799149x_{16} = -2.61799387799149
x17=101.054563690472x_{17} = 101.054563690472
x18=−100.007366139275x_{18} = -100.007366139275
x19=16.2315620435473x_{19} = 16.2315620435473
x20=53.9306738866248x_{20} = 53.9306738866248
x21=−71.733032256967x_{21} = -71.733032256967
x22=−15.1843644923507x_{22} = -15.1843644923507
x23=19.3731546971371x_{23} = 19.3731546971371
x24=25.6563400043166x_{24} = 25.6563400043166
x25=63.3554518473942x_{25} = 63.3554518473942
x26=88.4881930761125x_{26} = 88.4881930761125
x27=−96.8657734856853x_{27} = -96.8657734856853
x28=0.523598775598299x_{28} = 0.523598775598299
x29=38.2227106186758x_{29} = 38.2227106186758
x30=75.9218224617533x_{30} = 75.9218224617533
x31=28.7979326579064x_{31} = 28.7979326579064
x32=−84.2994028713261x_{32} = -84.2994028713261
x33=57.0722665402146x_{33} = 57.0722665402146
x34=79.0634151153431x_{34} = 79.0634151153431
x35=47.6474885794452x_{35} = 47.6474885794452
x36=44.5058959258554x_{36} = 44.5058959258554
x37=−52.8834763354282x_{37} = -52.8834763354282
x38=−59.1666616426078x_{38} = -59.1666616426078
x39=−93.7241808320955x_{39} = -93.7241808320955
x40=−40.317105721069x_{40} = -40.317105721069
x41=91.6297857297023x_{41} = 91.6297857297023
x42=−8.90117918517108x_{42} = -8.90117918517108
x43=−21.4675497995303x_{43} = -21.4675497995303
x44=85.3466004225227x_{44} = 85.3466004225227
x45=69.6386371545737x_{45} = 69.6386371545737
x46=6.80678408277789x_{46} = 6.80678408277789
x47=−78.0162175641465x_{47} = -78.0162175641465
x48=82.2050077689329x_{48} = 82.2050077689329
x49=−74.8746249105567x_{49} = -74.8746249105567
x50=41.3643032722656x_{50} = 41.3643032722656
x51=94.7713783832921x_{51} = 94.7713783832921
x52=97.9129710368819x_{52} = 97.9129710368819
x53=13.0899693899575x_{53} = 13.0899693899575
x54=−90.5825881785057x_{54} = -90.5825881785057
x55=−34.0339204138894x_{55} = -34.0339204138894
x56=−49.7418836818384x_{56} = -49.7418836818384
x57=−37.1755130674792x_{57} = -37.1755130674792
x58=−56.025068989018x_{58} = -56.025068989018
x59=66.497044500984x_{59} = 66.497044500984
x60=−81.1578102177363x_{60} = -81.1578102177363
x61=−2509.60893144265x_{61} = -2509.60893144265
x62=9.94837673636768x_{62} = 9.94837673636768
x63=72.7802298081635x_{63} = 72.7802298081635
x64=22.5147473507269x_{64} = 22.5147473507269
x65=−30.8923277602996x_{65} = -30.8923277602996
x66=−43.4586983746588x_{66} = -43.4586983746588
x67=35.081117965086x_{67} = 35.081117965086
x68=−27.7507351067098x_{68} = -27.7507351067098
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x + pi/3).
cos⁡(0+π3)\cos{\left(0 + \frac{\pi}{3} \right)}
The result:
f(0)=12f{\left(0 \right)} = \frac{1}{2}
The point:
(0, 1/2)
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
−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}
The values of the extrema at the points:
 -pi      /  pi   pi\ 
(----, cos|- -- + --|)
  3       \  3    3 / 

 2*pi     
(----, -1)
  3


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}
Maxima of the function at points:
x1=−π3x_{1} = - \frac{\pi}{3}
Decreasing at intervals
(−∞,−π3]∪[2π3,∞)\left(-\infty, - \frac{\pi}{3}\right] \cup \left[\frac{2 \pi}{3}, \infty\right)
Increasing at intervals
[−π3,2π3]\left[- \frac{\pi}{3}, \frac{2 \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
−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}

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
[π6,7π6]\left[\frac{\pi}{6}, \frac{7 \pi}{6}\right]
Convex at the intervals
(−∞,π6]∪[7π6,∞)\left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{7 \pi}{6}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞cos⁡(x+π3)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\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→∞cos⁡(x+π3)=⟨−1,1⟩\lim_{x \to \infty} \cos{\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 cos(x + pi/3), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x+π3)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\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→∞(cos⁡(x+π3)x)=0\lim_{x \to \infty}\left(\frac{\cos{\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:
cos⁡(x+π3)=sin⁡(x+π6)\cos{\left(x + \frac{\pi}{3} \right)} = \sin{\left(x + \frac{\pi}{6} \right)}
- No
cos⁡(x+π3)=−sin⁡(x+π6)\cos{\left(x + \frac{\pi}{3} \right)} = - \sin{\left(x + \frac{\pi}{6} \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = cos(x+pi/3)
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