Mister Exam

Other calculators

sinx*ctgx
Plotting a function graph/ sinx*ctgx

Graphing y = sinx*ctgx

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = sin(x)*cot(x)
f(x)=sin⁡(x)cot⁡(x)f{\left(x \right)} = \sin{\left(x \right)} \cot{\left(x \right)} 
f = sin(x)*cot(x)
The graph of the function
0-30-20-101020304050607080902-2
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)cot⁡(x)=0\sin{\left(x \right)} \cot{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}
x3=πx_{3} = \pi
Numerical solution
x1=36.1283155162826x_{1} = 36.1283155162826
x2=−36.1283155162826x_{2} = -36.1283155162826
x3=98.9601685880785x_{3} = 98.9601685880785
x4=17.2787595947439x_{4} = 17.2787595947439
x5=−48.6946861306418x_{5} = -48.6946861306418
x6=−10.9955742875643x_{6} = -10.9955742875643
x7=−387.986692718339x_{7} = -387.986692718339
x8=48.6946861306418x_{8} = 48.6946861306418
x9=−92.6769832808989x_{9} = -92.6769832808989
x10=45.553093477052x_{10} = 45.553093477052
x11=−17.2787595947439x_{11} = -17.2787595947439
x12=61.261056745001x_{12} = 61.261056745001
x13=23.5619449019235x_{13} = 23.5619449019235
x14=76.9690200129499x_{14} = 76.9690200129499
x15=−86.3937979737193x_{15} = -86.3937979737193
x16=−1.5707963267949x_{16} = -1.5707963267949
x17=−4.71238898038469x_{17} = -4.71238898038469
x18=73.8274273593601x_{18} = 73.8274273593601
x19=51.8362787842316x_{19} = 51.8362787842316
x20=−70.6858347057703x_{20} = -70.6858347057703
x21=−58.1194640914112x_{21} = -58.1194640914112
x22=−54.9778714378214x_{22} = -54.9778714378214
x23=−98.9601685880785x_{23} = -98.9601685880785
x24=−61.261056745001x_{24} = -61.261056745001
x25=70.6858347057703x_{25} = 70.6858347057703
x26=58.1194640914112x_{26} = 58.1194640914112
x27=−23.5619449019235x_{27} = -23.5619449019235
x28=−80.1106126665397x_{28} = -80.1106126665397
x29=83.2522053201295x_{29} = 83.2522053201295
x30=1.5707963267949x_{30} = 1.5707963267949
x31=67.5442420521806x_{31} = 67.5442420521806
x32=−64.4026493985908x_{32} = -64.4026493985908
x33=−51.8362787842316x_{33} = -51.8362787842316
x34=−406.836248639878x_{34} = -406.836248639878
x35=7.85398163397448x_{35} = 7.85398163397448
x36=−26.7035375555132x_{36} = -26.7035375555132
x37=−45.553093477052x_{37} = -45.553093477052
x38=86.3937979737193x_{38} = 86.3937979737193
x39=10.9955742875643x_{39} = 10.9955742875643
x40=−3623.82712591583x_{40} = -3623.82712591583
x41=−73.8274273593601x_{41} = -73.8274273593601
x42=−7.85398163397448x_{42} = -7.85398163397448
x43=92.6769832808989x_{43} = 92.6769832808989
x44=80.1106126665397x_{44} = 80.1106126665397
x45=4.71238898038469x_{45} = 4.71238898038469
x46=32.9867228626928x_{46} = 32.9867228626928
x47=20.4203522483337x_{47} = 20.4203522483337
x48=−95.8185759344887x_{48} = -95.8185759344887
x49=−83.2522053201295x_{49} = -83.2522053201295
x50=−89.5353906273091x_{50} = -89.5353906273091
x51=64.4026493985908x_{51} = 64.4026493985908
x52=−39.2699081698724x_{52} = -39.2699081698724
x53=−20.4203522483337x_{53} = -20.4203522483337
x54=−67.5442420521806x_{54} = -67.5442420521806
x55=42.4115008234622x_{55} = 42.4115008234622
x56=26.7035375555132x_{56} = 26.7035375555132
x57=−14.1371669411541x_{57} = -14.1371669411541
x58=−76.9690200129499x_{58} = -76.9690200129499
x59=89.5353906273091x_{59} = 89.5353906273091
x60=95.8185759344887x_{60} = 95.8185759344887
x61=14.1371669411541x_{61} = 14.1371669411541
x62=−42.4115008234622x_{62} = -42.4115008234622
x63=−2266.65909956504x_{63} = -2266.65909956504
x64=54.9778714378214x_{64} = 54.9778714378214
x65=−32.9867228626928x_{65} = -32.9867228626928
x66=−29.845130209103x_{66} = -29.845130209103
x67=39.2699081698724x_{67} = 39.2699081698724
x68=29.845130209103x_{68} = 29.845130209103
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)*cot(x).
sin⁡(0)cot⁡(0)\sin{\left(0 \right)} \cot{\left(0 \right)}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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
(−cot⁡2(x)−1)sin⁡(x)+cos⁡(x)cot⁡(x)=0\left(- \cot^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)} + \cos{\left(x \right)} \cot{\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(cot⁡2(x)+1)sin⁡(x)cot⁡(x)−2(cot⁡2(x)+1)cos⁡(x)−sin⁡(x)cot⁡(x)=02 \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cot{\left(x \right)} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \cot{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}

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]∪[π2,∞)\left(-\infty, - \frac{\pi}{2}\right] \cup \left[\frac{\pi}{2}, \infty\right)
Convex at the intervals
[−π2,π2]\left[- \frac{\pi}{2}, \frac{\pi}{2}\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)cot⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} \cot{\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→∞(sin⁡(x)cot⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} \cot{\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 sin(x)*cot(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)cot⁡(x)x)=lim⁡x→−∞(sin⁡(x)cot⁡(x)x)\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cot{\left(x \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cot{\left(x \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(sin⁡(x)cot⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cot{\left(x \right)}}{x}\right)
lim⁡x→∞(sin⁡(x)cot⁡(x)x)=lim⁡x→∞(sin⁡(x)cot⁡(x)x)\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cot{\left(x \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cot{\left(x \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(sin⁡(x)cot⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cot{\left(x \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:
sin⁡(x)cot⁡(x)=sin⁡(x)cot⁡(x)\sin{\left(x \right)} \cot{\left(x \right)} = \sin{\left(x \right)} \cot{\left(x \right)}
- No
sin⁡(x)cot⁡(x)=−sin⁡(x)cot⁡(x)\sin{\left(x \right)} \cot{\left(x \right)} = - \sin{\left(x \right)} \cot{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sinx*ctgx
    © Mister Exam – Calculator