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  • Identical expressions

  • cot(x/ two)*sin(x)
  • cotangent of (x divide by 2) multiply by sinus of (x)
  • cotangent of (x divide by two) multiply by sinus of (x)
  • cot(x/2)sin(x)
  • cotx/2sinx
  • cot(x divide by 2)*sin(x)

  • Similar expressions

  • cot(x/2)*sinx
Plotting a function graph/ cot(x/2)*sin(x)

Graphing y = cot(x/2)*sin(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /x\       
f(x) = cot|-|*sin(x)
          \2/
f(x)=sin⁡(x)cot⁡(x2)f{\left(x \right)} = \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)} 
f = sin(x)*cot(x/2)
The graph of the function
02468-8-6-4-2-101004
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⁡(x2)=0\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=91.1061873718352x_{1} = 91.1061873718352
x2=−91.106187265474x_{2} = -91.106187265474
x3=−21.9911490521325x_{3} = -21.9911490521325
x4=65.9734452390837x_{4} = 65.9734452390837
x5=−9.42477744529557x_{5} = -9.42477744529557
x6=−15.7079635641079x_{6} = -15.7079635641079
x7=−34.5575188899093x_{7} = -34.5575188899093
x8=−53.4070745786761x_{8} = -53.4070745786761
x9=−65.9734449870253x_{9} = -65.9734449870253
x10=9.42477748794163x_{10} = 9.42477748794163
x11=34.5575197055812x_{11} = 34.5575197055812
x12=53.4070766553897x_{12} = 53.4070766553897
x13=21.9911489072506x_{13} = 21.9911489072506
x14=9.42477826738203x_{14} = 9.42477826738203
x15=65.9734460390947x_{15} = 65.9734460390947
x16=−21.9911485864417x_{16} = -21.9911485864417
x17=34.5575195449229x_{17} = 34.5575195449229
x18=72.2566310277176x_{18} = 72.2566310277176
x19=−1127.83176318906x_{19} = -1127.83176318906
x20=−84.8230012511693x_{20} = -84.8230012511693
x21=−72.2566308657983x_{21} = -72.2566308657983
x22=−47.1238893275319x_{22} = -47.1238893275319
x23=78.5398152766482x_{23} = 78.5398152766482
x24=3.14159306054457x_{24} = 3.14159306054457
x25=−9.4247781365785x_{25} = -9.4247781365785
x26=−40.8407049290801x_{26} = -40.8407049290801
x27=−15.7079632965989x_{27} = -15.7079632965989
x28=−53.4070745963886x_{28} = -53.4070745963886
x29=34.5575190219169x_{29} = 34.5575190219169
x30=28.2743343711514x_{30} = 28.2743343711514
x31=40.8407045848602x_{31} = 40.8407045848602
x32=65.9734457529812x_{32} = 65.9734457529812
x33=−84.8230020565447x_{33} = -84.8230020565447
x34=78.5398161804942x_{34} = 78.5398161804942
x35=72.2566315166773x_{35} = 72.2566315166773
x36=−59.6902606928653x_{36} = -59.6902606928653
x37=3.1415922548952x_{37} = 3.1415922548952
x38=84.8230021335997x_{38} = 84.8230021335997
x39=59.6902606104322x_{39} = 59.6902606104322
x40=78.5398166181283x_{40} = 78.5398166181283
x41=−91.1061864815274x_{41} = -91.1061864815274
x42=15.7079627593774x_{42} = 15.7079627593774
x43=−47.1238901083229x_{43} = -47.1238901083229
x44=15.7079629803241x_{44} = 15.7079629803241
x45=40.8407045792514x_{45} = 40.8407045792514
x46=47.1238902162437x_{46} = 47.1238902162437
x47=97.389372581711x_{47} = 97.389372581711
x48=−28.2743337069329x_{48} = -28.2743337069329
x49=−65.9734457649277x_{49} = -65.9734457649277
x50=28.2743335663982x_{50} = 28.2743335663982
x51=−97.3893724533348x_{51} = -97.3893724533348
x52=−97.3893716284562x_{52} = -97.3893716284562
x53=−15.707962774825x_{53} = -15.707962774825
x54=−59.6902604578012x_{54} = -59.6902604578012
x55=−72.2566311847166x_{55} = -72.2566311847166
x56=−3.14159217367683x_{56} = -3.14159217367683
x57=78.5398168562347x_{57} = 78.5398168562347
x58=15.7079634518075x_{58} = 15.7079634518075
x59=−28.2743340989896x_{59} = -28.2743340989896
x60=−34.5575196658297x_{60} = -34.5575196658297
x61=40.8407049800347x_{61} = 40.8407049800347
x62=21.9911480932338x_{62} = 21.9911480932338
x63=84.8230013636028x_{63} = 84.8230013636028
x64=−40.8407049008781x_{64} = -40.8407049008781
x65=59.6902599104079x_{65} = 59.6902599104079
x66=−97.3893717476911x_{66} = -97.3893717476911
x67=53.4070746418597x_{67} = 53.4070746418597
x68=−40.8407040952604x_{68} = -40.8407040952604
x69=−78.5398160472843x_{69} = -78.5398160472843
x70=97.3893717959212x_{70} = 97.3893717959212
x71=28.2743338651796x_{71} = 28.2743338651796
x72=91.1061865667532x_{72} = 91.1061865667532
x73=−65.9734453607004x_{73} = -65.9734453607004
x74=−28.2743343914215x_{74} = -28.2743343914215
x75=−3.14159295109225x_{75} = -3.14159295109225
x76=40.8407042062167x_{76} = 40.8407042062167
x77=−53.407075294995x_{77} = -53.407075294995
x78=−21.991148226056x_{78} = -21.991148226056
x79=59.6902600526626x_{79} = 59.6902600526626
x80=−72.2566315419804x_{80} = -72.2566315419804
x81=21.9911485852059x_{81} = 21.9911485852059
x82=72.2566306985x_{82} = 72.2566306985
x83=−78.5398168194507x_{83} = -78.5398168194507
x84=−65.9734461969855x_{84} = -65.9734461969855
x85=78.5398149750205x_{85} = 78.5398149750205
x86=47.123889410773x_{86} = 47.123889410773
x87=53.407075424589x_{87} = 53.407075424589
x88=−59.6902599212271x_{88} = -59.6902599212271
x89=−9.42477752082051x_{89} = -9.42477752082051
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cot(x/2)*sin(x).
sin⁡(0)cot⁡(02)\sin{\left(0 \right)} \cot{\left(\frac{0}{2} \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(x2)2−12)sin⁡(x)+cos⁡(x)cot⁡(x2)=0\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}\right) \sin{\left(x \right)} + \cos{\left(x \right)} \cot{\left(\frac{x}{2} \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
(cot⁡2(x2)+1)sin⁡(x)cot⁡(x2)2−(cot⁡2(x2)+1)cos⁡(x)−sin⁡(x)cot⁡(x2)=0\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}}{2} - \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \cot{\left(\frac{x}{2} \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
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(sin⁡(x)cot⁡(x2))y = \lim_{x \to -\infty}\left(\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(sin⁡(x)cot⁡(x2))y = \lim_{x \to \infty}\left(\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(x/2)*sin(x), 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)cot⁡(x2)x)y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(sin⁡(x)cot⁡(x2)x)y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cot{\left(\frac{x}{2} \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⁡(x2)=sin⁡(x)cot⁡(x2)\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}
- No
sin⁡(x)cot⁡(x2)=−sin⁡(x)cot⁡(x2)\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = - \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}
- No
so, the function
not is
neither even, nor odd
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