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  • How to use it?

  • Graphing y =:
  • -x^2+5x-4
  • -x^2-2x
  • -x^2+2x-3
  • (x^2-2)/x

  • Identical expressions

  • cos^ two (x)-sin(x)
  • co sinus of e of squared (x) minus sinus of (x)
  • co sinus of e of to the power of two (x) minus sinus of (x)
  • cos2(x)-sin(x)
  • cos2x-sinx
  • cos²(x)-sin(x)
  • cos to the power of 2(x)-sin(x)
  • cos^2x-sinx

  • Similar expressions

  • cos^2(x)+sin(x)
  • cos^2(x)-sinx
Plotting a function graph/ cos^2(x)-sin(x)

Graphing y = cos^2(x)-sin(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          2            
f(x) = cos (x) - sin(x)
f(x)=−sin⁡(x)+cos⁡2(x)f{\left(x \right)} = - \sin{\left(x \right)} + \cos^{2}{\left(x \right)} 
f = -sin(x) + cos(x)^2
The graph of the function
02468-8-6-4-2-10102.5-2.5
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)+cos⁡2(x)=0- \sin{\left(x \right)} + \cos^{2}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=2atan⁡(12+52+21+52)x_{1} = 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)}
x2=2atan⁡(−21+52+12+52)x_{2} = 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}
Numerical solution
x1=−269.51072877623x_{1} = -269.51072877623
x2=−91.7724263865965x_{2} = -91.7724263865965
x3=88.6308337330067x_{3} = 88.6308337330067
x4=−55.8824283321238x_{4} = -55.8824283321238
x5=25.7989806612109x_{5} = 25.7989806612109
x6=−5.61694587468707x_{6} = -5.61694587468707
x7=−60.3564998506986x_{7} = -60.3564998506986
x8=94.9140190401863x_{8} = 94.9140190401863
x9=52.740835678534x_{9} = 52.740835678534
x10=19.5157953540313x_{10} = 19.5157953540313
x11=8.75853852827686x_{11} = 8.75853852827686
x12=−66.6396851578782x_{12} = -66.6396851578782
x13=−41.5069439291598x_{13} = -41.5069439291598
x14=21.324909142636x_{14} = 21.324909142636
x15=90.4399475216115x_{15} = 90.4399475216115
x16=−35.2237586219802x_{16} = -35.2237586219802
x17=15.0417238354565x_{17} = 15.0417238354565
x18=77.8735769072523x_{18} = 77.8735769072523
x19=−49.5992430249442x_{19} = -49.5992430249442
x20=−37.032872410585x_{20} = -37.032872410585
x21=38.36535127557x_{21} = 38.36535127557
x22=−3.80783208608231x_{22} = -3.80783208608231
x23=−16.3742027004415x_{23} = -16.3742027004415
x24=−18.1833164890462x_{24} = -18.1833164890462
x25=0.666239432492515x_{25} = 0.666239432492515
x26=27.6080944498156x_{26} = 27.6080944498156
x27=96.7231328287911x_{27} = 96.7231328287911
x28=−72.9228704650578x_{28} = -72.9228704650578
x29=−54.073314543519x_{29} = -54.073314543519
x30=−22.6573880076211x_{30} = -22.6573880076211
x31=−47.7901292363394x_{31} = -47.7901292363394
x32=76.0644631186476x_{32} = 76.0644631186476
x33=44.6485365827496x_{33} = 44.6485365827496
x34=6.9494247396721x_{34} = 6.9494247396721
x35=82.3476484258271x_{35} = 82.3476484258271
x36=−99.8647254823809x_{36} = -99.8647254823809
x37=−93.5815401752013x_{37} = -93.5815401752013
x38=−68.4487989464829x_{38} = -68.4487989464829
x39=65.3072062928931x_{39} = 65.3072062928931
x40=−11.9001311818667x_{40} = -11.9001311818667
x41=−10.0910173932619x_{41} = -10.0910173932619
x42=33.8912797569952x_{42} = 33.8912797569952
x43=50.9317218899292x_{43} = 50.9317218899292
x44=−24.4665017962258x_{44} = -24.4665017962258
x45=−98.0556116937761x_{45} = -98.0556116937761
x46=46.4576503713544x_{46} = 46.4576503713544
x47=−79.2060557722373x_{47} = -79.2060557722373
x48=−43.3160577177646x_{48} = -43.3160577177646
x49=−81.0151695608421x_{49} = -81.0151695608421
x50=63.4980925042884x_{50} = 63.4980925042884
x51=−85.4892410794169x_{51} = -85.4892410794169
x52=2.47535322109728x_{52} = 2.47535322109728
x53=−30.7496871034054x_{53} = -30.7496871034054
x54=32.0821659683904x_{54} = 32.0821659683904
x55=71.5903916000727x_{55} = 71.5903916000727
x56=40.1744650641748x_{56} = 40.1744650641748
x57=84.1567622144319x_{57} = 84.1567622144319
x58=103.006318135971x_{58} = 103.006318135971
x59=−62.1656136393034x_{59} = -62.1656136393034
x60=69.781277811468x_{60} = 69.781277811468
x61=−87.2983548680217x_{61} = -87.2983548680217
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 - sin(x).
−sin⁡(0)+cos⁡2(0)- \sin{\left(0 \right)} + \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)−cos⁡(x)=0- 2 \sin{\left(x \right)} \cos{\left(x \right)} - \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−5π6x_{1} = - \frac{5 \pi}{6}
x2=−π2x_{2} = - \frac{\pi}{2}
x3=−π6x_{3} = - \frac{\pi}{6}
x4=π2x_{4} = \frac{\pi}{2}
The values of the extrema at the points:
 -5*pi      
(-----, 5/4)
   6        

 -pi     
(----, 1)
  2      

 -pi       
(----, 5/4)
  6        

 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=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
Maxima of the function at points:
x2=−5π6x_{2} = - \frac{5 \pi}{6}
x2=−π6x_{2} = - \frac{\pi}{6}
Decreasing at intervals
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
(−∞,−π2]∪[−π6,π2]\left(-\infty, - \frac{\pi}{2}\right] \cup \left[- \frac{\pi}{6}, \frac{\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
2sin⁡2(x)+sin⁡(x)−2cos⁡2(x)=02 \sin^{2}{\left(x \right)} + \sin{\left(x \right)} - 2 \cos^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−2atan⁡(−14+29−334+334)x_{1} = - 2 \operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} + \frac{\sqrt{33}}{4} \right)}
x2=2atan⁡(14+233+94+334)x_{2} = 2 \operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{2} \sqrt{\sqrt{33} + 9}}{4} + \frac{\sqrt{33}}{4} \right)}
x3=2atan⁡(−334+14+29−334)x_{3} = 2 \operatorname{atan}{\left(- \frac{\sqrt{33}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} \right)}
x4=2atan⁡(−233+94+14+334)x_{4} = 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{\sqrt{33} + 9}}{4} + \frac{1}{4} + \frac{\sqrt{33}}{4} \right)}

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
[−2atan⁡(−14+29−334+334),2atan⁡(−334+14+29−334)]∪[2atan⁡(−233+94+14+334),∞)\left[- 2 \operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} + \frac{\sqrt{33}}{4} \right)}, 2 \operatorname{atan}{\left(- \frac{\sqrt{33}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} \right)}\right] \cup \left[2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{\sqrt{33} + 9}}{4} + \frac{1}{4} + \frac{\sqrt{33}}{4} \right)}, \infty\right)
Convex at the intervals
(−∞,−2atan⁡(−14+29−334+334)]\left(-\infty, - 2 \operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} + \frac{\sqrt{33}}{4} \right)}\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)+cos⁡2(x))=⟨−1,2⟩\lim_{x \to -\infty}\left(- \sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) = \left\langle -1, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,2⟩y = \left\langle -1, 2\right\rangle
lim⁡x→∞(−sin⁡(x)+cos⁡2(x))=⟨−1,2⟩\lim_{x \to \infty}\left(- \sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) = \left\langle -1, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,2⟩y = \left\langle -1, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)^2 - sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−sin⁡(x)+cos⁡2(x)x)=0\lim_{x \to -\infty}\left(\frac{- \sin{\left(x \right)} + \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→∞(−sin⁡(x)+cos⁡2(x)x)=0\lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} + \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:
−sin⁡(x)+cos⁡2(x)=sin⁡(x)+cos⁡2(x)- \sin{\left(x \right)} + \cos^{2}{\left(x \right)} = \sin{\left(x \right)} + \cos^{2}{\left(x \right)}
- No
−sin⁡(x)+cos⁡2(x)=−sin⁡(x)−cos⁡2(x)- \sin{\left(x \right)} + \cos^{2}{\left(x \right)} = - \sin{\left(x \right)} - \cos^{2}{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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