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cos(x)^2
Plotting a function graph/ cos(x)^2

Graphing y = cos(x)^2

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          2   
f(x) = cos (x)
f(x)=cos⁡2(x)f{\left(x \right)} = \cos^{2}{\left(x \right)} 
f = cos(x)^2
The graph of the function
0-40-30-20-10102030405060708002
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⁡2(x)=0\cos^{2}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Numerical solution
x1=89.5353908552844x_{1} = 89.5353908552844
x2=−45.5530935883361x_{2} = -45.5530935883361
x3=−32.9867231091652x_{3} = -32.9867231091652
x4=48.6946859238715x_{4} = 48.6946859238715
x5=83.2522055730903x_{5} = 83.2522055730903
x6=51.8362788999928x_{6} = 51.8362788999928
x7=−39.2699083866483x_{7} = -39.2699083866483
x8=23.5619449395428x_{8} = 23.5619449395428
x9=61.2610569989704x_{9} = 61.2610569989704
x10=−42.4115006098842x_{10} = -42.4115006098842
x11=−17.2787590276524x_{11} = -17.2787590276524
x12=83.2522052340866x_{12} = 83.2522052340866
x13=−32.9867227513827x_{13} = -32.9867227513827
x14=−70.685834448838x_{14} = -70.685834448838
x15=10.9955743696636x_{15} = 10.9955743696636
x16=10.9955740392793x_{16} = 10.9955740392793
x17=95.8185760590309x_{17} = 95.8185760590309
x18=−58.1194639993376x_{18} = -58.1194639993376
x19=−76.9690202568697x_{19} = -76.9690202568697
x20=−23.5619450090417x_{20} = -23.5619450090417
x21=−26.7035372990183x_{21} = -26.7035372990183
x22=−51.8362786897497x_{22} = -51.8362786897497
x23=−39.2699081528781x_{23} = -39.2699081528781
x24=26.7035373461441x_{24} = 26.7035373461441
x25=29.845130320338x_{25} = 29.845130320338
x26=80.1106126771746x_{26} = 80.1106126771746
x27=32.9867226137576x_{27} = 32.9867226137576
x28=42.4115007291722x_{28} = 42.4115007291722
x29=−1.57079642969308x_{29} = -1.57079642969308
x30=−4.7123889912442x_{30} = -4.7123889912442
x31=39.2699081179815x_{31} = 39.2699081179815
x32=−98.960168684456x_{32} = -98.960168684456
x33=−7.85398149857354x_{33} = -7.85398149857354
x34=−83.2522055415057x_{34} = -83.2522055415057
x35=−98.9601684414698x_{35} = -98.9601684414698
x36=86.393797888273x_{36} = 86.393797888273
x37=70.6858345016621x_{37} = 70.6858345016621
x38=−54.9778716831146x_{38} = -54.9778716831146
x39=73.8274274795554x_{39} = 73.8274274795554
x40=−73.8274272800405x_{40} = -73.8274272800405
x41=−89.5353907467661x_{41} = -89.5353907467661
x42=−20.4203520321877x_{42} = -20.4203520321877
x43=−92.6769830239371x_{43} = -92.6769830239371
x44=4.71238876848081x_{44} = 4.71238876848081
x45=−80.1106125795659x_{45} = -80.1106125795659
x46=17.2787595624179x_{46} = 17.2787595624179
x47=76.9690200400775x_{47} = 76.9690200400775
x48=−4.71238872430683x_{48} = -4.71238872430683
x49=36.1283156002139x_{49} = 36.1283156002139
x50=−70.6858346386357x_{50} = -70.6858346386357
x51=32.986722928111x_{51} = 32.986722928111
x52=80.1106131434937x_{52} = 80.1106131434937
x53=−86.393797765473x_{53} = -86.393797765473
x54=−36.1283154192437x_{54} = -36.1283154192437
x55=39.2699084246933x_{55} = 39.2699084246933
x56=−76.9690198771149x_{56} = -76.9690198771149
x57=−95.8185758681287x_{57} = -95.8185758681287
x58=20.4203521497111x_{58} = 20.4203521497111
x59=67.5442422779275x_{59} = 67.5442422779275
x60=54.9778711883962x_{60} = 54.9778711883962
x61=58.1194644379895x_{61} = 58.1194644379895
x62=98.9601683381274x_{62} = 98.9601683381274
x63=−26.7035375427973x_{63} = -26.7035375427973
x64=76.9690197631883x_{64} = 76.9690197631883
x65=23.5619451230057x_{65} = 23.5619451230057
x66=−54.9778713137198x_{66} = -54.9778713137198
x67=−14.1371668392726x_{67} = -14.1371668392726
x68=−48.6946860920117x_{68} = -48.6946860920117
x69=54.9778714849733x_{69} = 54.9778714849733
x70=64.4026493086922x_{70} = 64.4026493086922
x71=−61.2610569641117x_{71} = -61.2610569641117
x72=−64.4026491876462x_{72} = -64.4026491876462
x73=92.6769830795146x_{73} = 92.6769830795146
x74=−17.2787598091171x_{74} = -17.2787598091171
x75=541.924732890135x_{75} = 541.924732890135
x76=−10.9955745350309x_{76} = -10.9955745350309
x77=−67.5442421675773x_{77} = -67.5442421675773
x78=−48.6946858738636x_{78} = -48.6946858738636
x79=76.9690207492347x_{79} = 76.9690207492347
x80=45.553093700501x_{80} = 45.553093700501
x81=7.85398174058521x_{81} = 7.85398174058521
x82=98.9601685932308x_{82} = 98.9601685932308
x83=61.2610566752601x_{83} = 61.2610566752601
x84=14.1371671048484x_{84} = 14.1371671048484
x85=−61.2610562242523x_{85} = -61.2610562242523
x86=−10.9955741902138x_{86} = -10.9955741902138
x87=17.2787598502655x_{87} = 17.2787598502655
x88=1.5707965454425x_{88} = 1.5707965454425
x89=−98.96016883042x_{89} = -98.96016883042
x90=−29.8451300963672x_{90} = -29.8451300963672
x91=−92.6769831823972x_{91} = -92.6769831823972
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)^2.
cos⁡2(0)\cos^{2}{\left(0 \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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- 2 \sin{\left(x \right)} \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}
x3=πx_{3} = \pi
x4=3π2x_{4} = \frac{3 \pi}{2}
The values of the extrema at the points:
(0, 1)

 pi    
(--, 0)
 2     

(pi, 1)

 3*pi    
(----, 0)
  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=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Maxima of the function at points:
x2=0x_{2} = 0
x2=πx_{2} = \pi
Decreasing at intervals
[3π2,∞)\left[\frac{3 \pi}{2}, \infty\right)
Increasing at intervals
(−∞,π2]∪[π,3π2]\left(-\infty, \frac{\pi}{2}\right] \cup \left[\pi, \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
2(sin⁡2(x)−cos⁡2(x))=02 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\pi}{4}
x2=π4x_{2} = \frac{\pi}{4}

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
(−∞,−π4]∪[π4,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[\frac{\pi}{4}, \infty\right)
Convex at the intervals
[−π4,π4]\left[- \frac{\pi}{4}, \frac{\pi}{4}\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⁡2(x)=⟨0,1⟩\lim_{x \to -\infty} \cos^{2}{\left(x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
lim⁡x→∞cos⁡2(x)=⟨0,1⟩\lim_{x \to \infty} \cos^{2}{\left(x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡2(x)x)=0\lim_{x \to -\infty}\left(\frac{\cos^{2}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡2(x)x)=0\lim_{x \to \infty}\left(\frac{\cos^{2}{\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:
cos⁡2(x)=cos⁡2(x)\cos^{2}{\left(x \right)} = \cos^{2}{\left(x \right)}
- Yes
cos⁡2(x)=−cos⁡2(x)\cos^{2}{\left(x \right)} = - \cos^{2}{\left(x \right)}
- No
so, the function
is
even
The graph
Graphing y = cos(x)^2
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