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  • How to use it?

  • Graphing y =:
  • x^3+4x-7
  • x^3-x^2-x-1
  • -x^3-6x
  • x-3/5-x
  • Derivative of:
  • sin(x)/cos(x)^2 sin(x)/cos(x)^2
  • Limit of the function:
  • sin(x)/cos(x)^2 sin(x)/cos(x)^2
  • Integral of d{x}:
  • sin(x)/cos(x)^2 sin(x)/cos(x)^2

  • Identical expressions

  • sin(x)/cos(x)^ two
  • sinus of (x) divide by co sinus of e of (x) squared
  • sinus of (x) divide by co sinus of e of (x) to the power of two
  • sin(x)/cos(x)2
  • sinx/cosx2
  • sin(x)/cos(x)²
  • sin(x)/cos(x) to the power of 2
  • sinx/cosx^2
  • sin(x) divide by cos(x)^2

  • Similar expressions

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

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        sin(x)
f(x) = -------
          2   
       cos (x)
f(x)=sin⁡(x)cos⁡2(x)f{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} 
f = sin(x)/cos(x)^2
The graph of the function
02468-8-6-4-2-1010-1000010000
The domain of the function
The points at which the function is not precisely defined:
x1=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469
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\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=−18.8495559215388x_{1} = -18.8495559215388
x2=−53.4070751110265x_{2} = -53.4070751110265
x3=−37.6991118430775x_{3} = -37.6991118430775
x4=−59.6902604182061x_{4} = -59.6902604182061
x5=−15.707963267949x_{5} = -15.707963267949
x6=−56.5486677646163x_{6} = -56.5486677646163
x7=12.5663706143592x_{7} = 12.5663706143592
x8=3.14159265358979x_{8} = 3.14159265358979
x9=−31.4159265358979x_{9} = -31.4159265358979
x10=84.8230016469244x_{10} = 84.8230016469244
x11=−81.6814089933346x_{11} = -81.6814089933346
x12=94.2477796076938x_{12} = 94.2477796076938
x13=21.9911485751286x_{13} = 21.9911485751286
x14=0x_{14} = 0
x15=−87.9645943005142x_{15} = -87.9645943005142
x16=81.6814089933346x_{16} = 81.6814089933346
x17=40.8407044966673x_{17} = 40.8407044966673
x18=−75.398223686155x_{18} = -75.398223686155
x19=−78.5398163397448x_{19} = -78.5398163397448
x20=62.8318530717959x_{20} = 62.8318530717959
x21=100.530964914873x_{21} = 100.530964914873
x22=−21.9911485751286x_{22} = -21.9911485751286
x23=47.1238898038469x_{23} = 47.1238898038469
x24=91.106186954104x_{24} = 91.106186954104
x25=75.398223686155x_{25} = 75.398223686155
x26=28.2743338823081x_{26} = 28.2743338823081
x27=34.5575191894877x_{27} = 34.5575191894877
x28=6.28318530717959x_{28} = 6.28318530717959
x29=78.5398163397448x_{29} = 78.5398163397448
x30=72.2566310325652x_{30} = 72.2566310325652
x31=−6.28318530717959x_{31} = -6.28318530717959
x32=15.707963267949x_{32} = 15.707963267949
x33=31.4159265358979x_{33} = 31.4159265358979
x34=−47.1238898038469x_{34} = -47.1238898038469
x35=25.1327412287183x_{35} = 25.1327412287183
x36=18.8495559215388x_{36} = 18.8495559215388
x37=−94.2477796076938x_{37} = -94.2477796076938
x38=−3.14159265358979x_{38} = -3.14159265358979
x39=−40.8407044966673x_{39} = -40.8407044966673
x40=56.5486677646163x_{40} = 56.5486677646163
x41=−25.1327412287183x_{41} = -25.1327412287183
x42=53.4070751110265x_{42} = 53.4070751110265
x43=−28.2743338823081x_{43} = -28.2743338823081
x44=−9.42477796076938x_{44} = -9.42477796076938
x45=87.9645943005142x_{45} = 87.9645943005142
x46=−50.2654824574367x_{46} = -50.2654824574367
x47=−100.530964914873x_{47} = -100.530964914873
x48=−43.9822971502571x_{48} = -43.9822971502571
x49=50.2654824574367x_{49} = 50.2654824574367
x50=−97.3893722612836x_{50} = -97.3893722612836
x51=69.1150383789755x_{51} = 69.1150383789755
x52=59.6902604182061x_{52} = 59.6902604182061
x53=97.3893722612836x_{53} = 97.3893722612836
x54=−62.8318530717959x_{54} = -62.8318530717959
x55=−72.2566310325652x_{55} = -72.2566310325652
x56=−91.106186954104x_{56} = -91.106186954104
x57=−12.5663706143592x_{57} = -12.5663706143592
x58=−69.1150383789755x_{58} = -69.1150383789755
x59=37.6991118430775x_{59} = 37.6991118430775
x60=9.42477796076938x_{60} = 9.42477796076938
x61=65.9734457253857x_{61} = 65.9734457253857
x62=−65.9734457253857x_{62} = -65.9734457253857
x63=−84.8230016469244x_{63} = -84.8230016469244
x64=−34.5575191894877x_{64} = -34.5575191894877
x65=43.9822971502571x_{65} = 43.9822971502571
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)/cos(x)^2.
sin⁡(0)cos⁡2(0)\frac{\sin{\left(0 \right)}}{\cos^{2}{\left(0 \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⁡2(x)cos⁡3(x)+cos⁡(x)cos⁡2(x)=0\frac{2 \sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\cos^{2}{\left(x \right)}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
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
(6sin⁡2(x)cos⁡2(x)+5)sin⁡(x)cos⁡2(x)=0\frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469

lim⁡x→1.5707963267949−((6sin⁡2(x)cos⁡2(x)+5)sin⁡(x)cos⁡2(x))=4.2680335647564⋅1065\lim_{x \to 1.5707963267949^-}\left(\frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) = 4.2680335647564 \cdot 10^{65}
lim⁡x→1.5707963267949+((6sin⁡2(x)cos⁡2(x)+5)sin⁡(x)cos⁡2(x))=4.2680335647564⋅1065\lim_{x \to 1.5707963267949^+}\left(\frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) = 4.2680335647564 \cdot 10^{65}
- limits are equal, then skip the corresponding point
lim⁡x→4.71238898038469−((6sin⁡2(x)cos⁡2(x)+5)sin⁡(x)cos⁡2(x))=−5.26917724044⋅1063\lim_{x \to 4.71238898038469^-}\left(\frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) = -5.26917724044 \cdot 10^{63}
lim⁡x→4.71238898038469+((6sin⁡2(x)cos⁡2(x)+5)sin⁡(x)cos⁡2(x))=−5.26917724044⋅1063\lim_{x \to 4.71238898038469^+}\left(\frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) = -5.26917724044 \cdot 10^{63}
- limits are equal, then skip the corresponding point

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
[0,∞)\left[0, \infty\right)
Convex at the intervals
(−∞,0]\left(-\infty, 0\right]
Vertical asymptotes
Have:
x1=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(sin⁡(x)cos⁡2(x))y = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(sin⁡(x)cos⁡2(x))y = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)/cos(x)^2, divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(sin⁡(x)xcos⁡2(x))y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \cos^{2}{\left(x \right)}}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(sin⁡(x)xcos⁡2(x))y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \cos^{2}{\left(x \right)}}\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:
sin⁡(x)cos⁡2(x)=−sin⁡(x)cos⁡2(x)\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} = - \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}
- No
sin⁡(x)cos⁡2(x)=sin⁡(x)cos⁡2(x)\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} = \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}
- No
so, the function
not is
neither even, nor odd
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