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cos(x)/sin(x)
Plotting a function graph/ cos(x)/sin(x)

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

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       cos(x)
f(x) = ------
       sin(x)
f(x)=cos⁡(x)sin⁡(x)f{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} 
f = cos(x)/sin(x)
The graph of the function
0-30-20-10102030405060708090-2500025000
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
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)sin⁡(x)=0\frac{\cos{\left(x \right)}}{\sin{\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=54.9778714378214x_{1} = 54.9778714378214
x2=95.8185759344887x_{2} = 95.8185759344887
x3=−39.2699081698724x_{3} = -39.2699081698724
x4=−29.845130209103x_{4} = -29.845130209103
x5=−54.9778714378214x_{5} = -54.9778714378214
x6=−73.8274273593601x_{6} = -73.8274273593601
x7=−7.85398163397448x_{7} = -7.85398163397448
x8=92.6769832808989x_{8} = 92.6769832808989
x9=80.1106126665397x_{9} = 80.1106126665397
x10=86.3937979737193x_{10} = 86.3937979737193
x11=−76.9690200129499x_{11} = -76.9690200129499
x12=−45.553093477052x_{12} = -45.553093477052
x13=61.261056745001x_{13} = 61.261056745001
x14=−48.6946861306418x_{14} = -48.6946861306418
x15=32.9867228626928x_{15} = 32.9867228626928
x16=20.4203522483337x_{16} = 20.4203522483337
x17=−17.2787595947439x_{17} = -17.2787595947439
x18=23.5619449019235x_{18} = 23.5619449019235
x19=−86.3937979737193x_{19} = -86.3937979737193
x20=−23.5619449019235x_{20} = -23.5619449019235
x21=−67.5442420521806x_{21} = -67.5442420521806
x22=−89.5353906273091x_{22} = -89.5353906273091
x23=−32.9867228626928x_{23} = -32.9867228626928
x24=64.4026493985908x_{24} = 64.4026493985908
x25=4.71238898038469x_{25} = 4.71238898038469
x26=−10.9955742875643x_{26} = -10.9955742875643
x27=−20.4203522483337x_{27} = -20.4203522483337
x28=−80.1106126665397x_{28} = -80.1106126665397
x29=−64.4026493985908x_{29} = -64.4026493985908
x30=−14.1371669411541x_{30} = -14.1371669411541
x31=−26.7035375555132x_{31} = -26.7035375555132
x32=10.9955742875643x_{32} = 10.9955742875643
x33=58.1194640914112x_{33} = 58.1194640914112
x34=−83.2522053201295x_{34} = -83.2522053201295
x35=26.7035375555132x_{35} = 26.7035375555132
x36=−70.6858347057703x_{36} = -70.6858347057703
x37=48.6946861306418x_{37} = 48.6946861306418
x38=−42.4115008234622x_{38} = -42.4115008234622
x39=70.6858347057703x_{39} = 70.6858347057703
x40=−92.6769832808989x_{40} = -92.6769832808989
x41=7.85398163397448x_{41} = 7.85398163397448
x42=−51.8362787842316x_{42} = -51.8362787842316
x43=98.9601685880785x_{43} = 98.9601685880785
x44=42.4115008234622x_{44} = 42.4115008234622
x45=51.8362787842316x_{45} = 51.8362787842316
x46=−58.1194640914112x_{46} = -58.1194640914112
x47=−61.261056745001x_{47} = -61.261056745001
x48=39.2699081698724x_{48} = 39.2699081698724
x49=45.553093477052x_{49} = 45.553093477052
x50=29.845130209103x_{50} = 29.845130209103
x51=−4.71238898038469x_{51} = -4.71238898038469
x52=17.2787595947439x_{52} = 17.2787595947439
x53=89.5353906273091x_{53} = 89.5353906273091
x54=1.5707963267949x_{54} = 1.5707963267949
x55=83.2522053201295x_{55} = 83.2522053201295
x56=−36.1283155162826x_{56} = -36.1283155162826
x57=−95.8185759344887x_{57} = -95.8185759344887
x58=36.1283155162826x_{58} = 36.1283155162826
x59=67.5442420521806x_{59} = 67.5442420521806
x60=73.8274273593601x_{60} = 73.8274273593601
x61=76.9690200129499x_{61} = 76.9690200129499
x62=−1.5707963267949x_{62} = -1.5707963267949
x63=−98.9601685880785x_{63} = -98.9601685880785
x64=14.1371669411541x_{64} = 14.1371669411541
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)/sin(x).
cos⁡(0)sin⁡(0)\frac{\cos{\left(0 \right)}}{\sin{\left(0 \right)}}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
−1−cos⁡2(x)sin⁡2(x)=0-1 - \frac{\cos^{2}{\left(x \right)}}{\sin^{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
(2+2cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡(x)=0\frac{\left(2 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
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=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979

lim⁡x→0−((2+2cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡(x))=−∞\lim_{x \to 0^-}\left(\frac{\left(2 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = -\infty
Let's take the limit
lim⁡x→0+((2+2cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡(x))=∞\lim_{x \to 0^+}\left(\frac{\left(2 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = \infty
Let's take the limit
- the limits are not equal, so
x1=0x_{1} = 0
- is an inflection point
lim⁡x→3.14159265358979−((2+2cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡(x))=−1.08892367577758⋅1048\lim_{x \to 3.14159265358979^-}\left(\frac{\left(2 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = -1.08892367577758 \cdot 10^{48}
Let's take the limit
lim⁡x→3.14159265358979+((2+2cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡(x))=−1.08892367577758⋅1048\lim_{x \to 3.14159265358979^+}\left(\frac{\left(2 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = -1.08892367577758 \cdot 10^{48}
Let's take the limit
- 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
(−∞,π2]\left(-\infty, \frac{\pi}{2}\right]
Convex at the intervals
[3π2,∞)\left[\frac{3 \pi}{2}, \infty\right)
Vertical asymptotes
Have:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
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)sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(cos⁡(x)sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)/sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x)xsin⁡(x))=lim⁡x→−∞(cos⁡(x)xsin⁡(x))\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x \sin{\left(x \right)}}\right) = \lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x \sin{\left(x \right)}}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(cos⁡(x)xsin⁡(x))y = x \lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x \sin{\left(x \right)}}\right)
lim⁡x→∞(cos⁡(x)xsin⁡(x))=lim⁡x→∞(cos⁡(x)xsin⁡(x))\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x \sin{\left(x \right)}}\right) = \lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x \sin{\left(x \right)}}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cos⁡(x)xsin⁡(x))y = x \lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x \sin{\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:
cos⁡(x)sin⁡(x)=−cos⁡(x)sin⁡(x)\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}
- No
cos⁡(x)sin⁡(x)=cos⁡(x)sin⁡(x)\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = cos(x)/sin(x)
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