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log(cos(x)^2)

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  • Graphing y =:
  • -x^2+4x
  • x^2-10x+27
  • x-8
  • x^3/(x+1)^2

  • Identical expressions

  • log(cos(x)^ two)
  • logarithm of ( co sinus of e of (x) squared )
  • logarithm of ( co sinus of e of (x) to the power of two)
  • log(cos(x)2)
  • logcosx2
  • log(cos(x)²)
  • log(cos(x) to the power of 2)
  • logcosx^2

  • Similar expressions

  • log(cosx^2)
Plotting a function graph/ log(cos(x)^2)

Graphing y = log(cos(x)^2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /   2   \
f(x) = log\cos (x)/
f(x)=log⁡(cos⁡2(x))f{\left(x \right)} = \log{\left(\cos^{2}{\left(x \right)} \right)} 
f = log(cos(x)^2)
The graph of the function
0-70-60-50-40-30-20-101020304050-1010
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:
log⁡(cos⁡2(x))=0\log{\left(\cos^{2}{\left(x \right)} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
x3=2πx_{3} = 2 \pi
Numerical solution
x1=−50.2654822771894x_{1} = -50.2654822771894
x2=−72.256630857317x_{2} = -72.256630857317
x3=53.4070754913975x_{3} = 53.4070754913975
x4=40.8407041550563x_{4} = 40.8407041550563
x5=3.14159207244778x_{5} = 3.14159207244778
x6=−15.7079632966706x_{6} = -15.7079632966706
x7=−75.3982226597999x_{7} = -75.3982226597999
x8=28.2743338651582x_{8} = 28.2743338651582
x9=91.106187597873x_{9} = 91.106187597873
x10=78.5398161731332x_{10} = 78.5398161731332
x11=37.6991120433529x_{11} = 37.6991120433529
x12=34.5575190133278x_{12} = 34.5575190133278
x13=−94.2477794374461x_{13} = -94.2477794374461
x14=−56.5486688343165x_{14} = -56.5486688343165
x15=−9.42477796310118x_{15} = -9.42477796310118
x16=72.2566310277163x_{16} = 72.2566310277163
x17=−62.8318536803612x_{17} = -62.8318536803612
x18=75.39822407273x_{18} = 75.39822407273
x19=−18.8495565116576x_{19} = -18.8495565116576
x20=−34.5575202359721x_{20} = -34.5575202359721
x21=−87.9645943584596x_{21} = -87.9645943584596
x22=100.530964753022x_{22} = 100.530964753022
x23=47.1238892401961x_{23} = 47.1238892401961
x24=21.9911485852153x_{24} = 21.9911485852153
x25=−47.1238887896178x_{25} = -47.1238887896178
x26=−3.14159159553391x_{26} = -3.14159159553391
x27=−43.9822971744998x_{27} = -43.9822971744998
x28=−84.8230022649727x_{28} = -84.8230022649727
x29=−40.8407050959251x_{29} = -40.8407050959251
x30=56.5486675932357x_{30} = 56.5486675932357
x31=50.2654824463311x_{31} = 50.2654824463311
x32=−34.5575188333352x_{32} = -34.5575188333352
x33=−6.28318511692891x_{33} = -6.28318511692891
x34=59.6902606235069x_{34} = 59.6902606235069
x35=31.4159269101267x_{35} = 31.4159269101267
x36=−3.14159300683281x_{36} = -3.14159300683281
x37=62.8318542034359x_{37} = 62.8318542034359
x38=9.42477834784266x_{38} = 9.42477834784266
x39=−37.6991118773736x_{39} = -37.6991118773736
x40=18.8495570029843x_{40} = 18.8495570029843
x41=−78.5398174338057x_{41} = -78.5398174338057
x42=−47.1238901689402x_{42} = -47.1238901689402
x43=91.1061864073649x_{43} = 91.1061864073649
x44=−12.5663716386669x_{44} = -12.5663716386669
x45=−9.42477814652397x_{45} = -9.42477814652397
x46=−69.1150373853363x_{46} = -69.1150373853363
x47=−31.4159260171396x_{47} = -31.4159260171396
x48=−75.3982238864105x_{48} = -75.3982238864105
x49=6.28318528416623x_{49} = 6.28318528416623
x50=25.1327406563971x_{50} = 25.1327406563971
x51=−31.4159267264704x_{51} = -31.4159267264704
x52=12.5663704334084x_{52} = 12.5663704334084
x53=−78.5398159953694x_{53} = -78.5398159953694
x54=−53.4070753064322x_{54} = -53.4070753064322
x55=−56.5486674143785x_{55} = -56.5486674143785
x56=−53.4070742959952x_{56} = -53.4070742959952
x57=−91.1061873312798x_{57} = -91.1061873312798
x58=69.1150378238503x_{58} = 69.1150378238503
x59=18.8495555741382x_{59} = 18.8495555741382
x60=−84.8230010779785x_{60} = -84.8230010779785
x61=−25.1327401930409x_{61} = -25.1327401930409
x62=−21.991148586432x_{62} = -21.991148586432
x63=94.2477796093522x_{63} = 94.2477796093522
x64=15.7079634632083x_{64} = 15.7079634632083
x65=−18.8495553258088x_{65} = -18.8495553258088
x66=81.681409203672x_{66} = 81.681409203672
x67=9.42477832891555x_{67} = 9.42477832891555
x68=0x_{68} = 0
x69=84.82300131674x_{69} = 84.82300131674
x70=−12.5663702522378x_{70} = -12.5663702522378
x71=69.1150390127643x_{71} = 69.1150390127643
x72=−28.2743336970608x_{72} = -28.2743336970608
x73=−59.6902604579606x_{73} = -59.6902604579606
x74=40.8407056026057x_{74} = 40.8407056026057
x75=−25.1327415878584x_{75} = -25.1327415878584
x76=−65.9734457648386x_{76} = -65.9734457648386
x77=−97.3893724664065x_{77} = -97.3893724664065
x78=47.1238904278493x_{78} = 47.1238904278493
x79=3.1415932585699x_{79} = 3.1415932585699
x80=62.831852735923x_{80} = 62.831852735923
x81=−91.1061859802604x_{81} = -91.1061859802604
x82=53.4070741478622x_{82} = 53.4070741478622
x83=−69.1150387500801x_{83} = -69.1150387500801
x84=87.9645943360512x_{84} = 87.9645943360512
x85=−81.6814090384469x_{85} = -81.6814090384469
x86=75.3982227418079x_{86} = 75.3982227418079
x87=−40.8407039100223x_{87} = -40.8407039100223
x88=−62.8318524940769x_{88} = -62.8318524940769
x89=25.1327418431203x_{89} = 25.1327418431203
x90=97.3893713350675x_{90} = 97.3893713350675
x91=65.9734457530642x_{91} = 65.9734457530642
x92=9.4247769576896x_{92} = 9.4247769576896
x93=97.389372654126x_{93} = 97.389372654126
x94=−100.53096457631x_{94} = -100.53096457631
x95=43.9822971695019x_{95} = 43.9822971695019
x96=31.4159255531763x_{96} = 31.4159255531763
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(cos(x)^2).
log⁡(cos⁡2(0))\log{\left(\cos^{2}{\left(0 \right)} \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
−2sin⁡(x)cos⁡(x)=0- \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
The values of the extrema at the points:
(0, 0)

(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:
The function has no minima
Maxima of the function at points:
x2=0x_{2} = 0
x2=πx_{2} = \pi
Decreasing at intervals
(−∞,0]\left(-\infty, 0\right]
Increasing at intervals
[π,∞)\left[\pi, \infty\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
−2(sin⁡2(x)cos⁡2(x)+1)=0- 2 \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞log⁡(cos⁡2(x))=⟨−∞,0⟩\lim_{x \to -\infty} \log{\left(\cos^{2}{\left(x \right)} \right)} = \left\langle -\infty, 0\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,0⟩y = \left\langle -\infty, 0\right\rangle
lim⁡x→∞log⁡(cos⁡2(x))=⟨−∞,0⟩\lim_{x \to \infty} \log{\left(\cos^{2}{\left(x \right)} \right)} = \left\langle -\infty, 0\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,0⟩y = \left\langle -\infty, 0\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(cos(x)^2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(cos⁡2(x))x)=lim⁡x→−∞(log⁡(cos⁡2(x))x)\lim_{x \to -\infty}\left(\frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(log⁡(cos⁡2(x))x)y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{x}\right)
lim⁡x→∞(log⁡(cos⁡2(x))x)=lim⁡x→∞(log⁡(cos⁡2(x))x)\lim_{x \to \infty}\left(\frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(log⁡(cos⁡2(x))x)y = x \lim_{x \to \infty}\left(\frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{x}\right)
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:
log⁡(cos⁡2(x))=log⁡(cos⁡2(x))\log{\left(\cos^{2}{\left(x \right)} \right)} = \log{\left(\cos^{2}{\left(x \right)} \right)}
- Yes
log⁡(cos⁡2(x))=−log⁡(cos⁡2(x))\log{\left(\cos^{2}{\left(x \right)} \right)} = - \log{\left(\cos^{2}{\left(x \right)} \right)}
- No
so, the function
is
even
The graph
Graphing y = log(cos(x)^2)
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