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  • Graphing y =:
  • -x^2+4x
  • x^2-10x+27
  • x-8
  • x^3/(x+1)^2

  • Identical expressions

  • (sinx+tgx)/(two *x*ctgx)
  • ( sinus of x plus tgx) divide by (2 multiply by x multiply by ctgx)
  • ( sinus of x plus tgx) divide by (two multiply by x multiply by ctgx)
  • (sinx+tgx)/(2xctgx)
  • sinx+tgx/2xctgx
  • (sinx+tgx) divide by (2*x*ctgx)

  • Similar expressions

  • (sinx-tgx)/(2*x*ctgx)
Plotting a function graph/ (sinx+tgx)/(2*x*ctgx)

Graphing y = (sinx+tgx)/(2*x*ctgx)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       sin(x) + tan(x)
f(x) = ---------------
          2*x*cot(x)
f(x)=sin⁡(x)+tan⁡(x)2xcot⁡(x)f{\left(x \right)} = \frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x \cot{\left(x \right)}} 
f = (sin(x) + tan(x))/(((2*x)*cot(x)))
The graph of the function
02468-8-6-4-2-1010-50005000
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
x2=1.5707963267949x_{2} = 1.5707963267949
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)+tan⁡(x)2xcot⁡(x)=0\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x \cot{\left(x \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−πx_{1} = - \pi
x2=πx_{2} = \pi
x3=2πx_{3} = 2 \pi
Numerical solution
x1=28.2742637865287x_{1} = 28.2742637865287
x2=−34.5557272431001x_{2} = -34.5557272431001
x3=−97.3903823516316x_{3} = -97.3903823516316
x4=−84.8200753771042x_{4} = -84.8200753771042
x5=81.6814092182089x_{5} = 81.6814092182089
x6=47.1209763081767x_{6} = 47.1209763081767
x7=−3.14326707194023x_{7} = -3.14326707194023
x8=50.2654824463177x_{8} = 50.2654824463177
x9=−18.8495568079986x_{9} = -18.8495568079986
x10=−91.1080860717611x_{10} = -91.1080860717611
x11=−12.5663731336943x_{11} = -12.5663731336943
x12=9.42658199496347x_{12} = 9.42658199496347
x13=−25.1327416534355x_{13} = -25.1327416534355
x14=−81.6814090386147x_{14} = -81.6814090386147
x15=18.849555492068x_{15} = 18.849555492068
x16=43.9822971695215x_{16} = 43.9822971695215
x17=31.4159239820201x_{17} = 31.4159239820201
x18=12.5663700981581x_{18} = 12.5663700981581
x19=−25.1327382847962x_{19} = -25.1327382847962
x20=25.1327403244751x_{20} = 25.1327403244751
x21=12.5663704177689x_{21} = 12.5663704177689
x22=−87.9645943582203x_{22} = -87.9645943582203
x23=−9.42567821453527x_{23} = -9.42567821453527
x24=−31.41592673575x_{24} = -31.41592673575
x25=−40.8437719191275x_{25} = -40.8437719191275
x26=59.6912473597501x_{26} = 59.6912473597501
x27=−43.9822971744317x_{27} = -43.9822971744317
x28=91.1033742827819x_{28} = 91.1033742827819
x29=−15.7080841076436x_{29} = -15.7080841076436
x30=−47.1257220786859x_{30} = -47.1257220786859
x31=−37.6991118774179x_{31} = -37.6991118774179
x32=97.3913416584044x_{32} = 97.3913416584044
x33=−75.3982155408245x_{33} = -75.3982155408245
x34=−72.2558000971762x_{34} = -72.2558000971762
x35=62.8318526701909x_{35} = 62.8318526701909
x36=75.3982213788012x_{36} = 75.3982213788012
x37=−59.6904319845233x_{37} = -59.6904319845233
x38=37.6991120547296x_{38} = 37.6991120547296
x39=−56.5486673390634x_{39} = -56.5486673390634
x40=72.2566119315779x_{40} = 72.2566119315779
x41=−62.8318540332717x_{41} = -62.8318540332717
x42=91.1095039335036x_{42} = 91.1095039335036
x43=−62.8318521658186x_{43} = -62.8318521658186
x44=100.530964745217x_{44} = 100.530964745217
x45=−94.247779428311x_{45} = -94.247779428311
x46=94.247779609352x_{46} = 94.247779609352
x47=56.5486675835781x_{47} = 56.5486675835781
x48=69.1150394191749x_{48} = 69.1150394191749
x49=69.1150375370464x_{49} = 69.1150375370464
x50=6.28318528394097x_{50} = 6.28318528394097
x51=−56.5486706799278x_{51} = -56.5486706799278
x52=75.3982241638922x_{52} = 75.3982241638922
x53=34.5566546899828x_{53} = 34.5566546899828
x54=31.4159269876038x_{54} = 31.4159269876038
x55=25.132742193978x_{55} = 25.132742193978
x56=−18.849554936845x_{56} = -18.849554936845
x57=34.5611202441731x_{57} = 34.5611202441731
x58=−65.9735393758622x_{58} = -65.9735393758622
x59=−12.5663701515109x_{59} = -12.5663701515109
x60=−75.3982238987855x_{60} = -75.3982238987855
x61=−78.5380926204531x_{61} = -78.5380926204531
x62=−50.2654822658225x_{62} = -50.2654822658225
x63=−6.28318509237865x_{63} = -6.28318509237865
x64=87.9645943361683x_{64} = 87.9645943361683
x65=62.8318564649026x_{65} = 62.8318564649026
x66=65.9735389843344x_{66} = 65.9735389843344
x67=78.5390028929786x_{67} = 78.5390028929786
x68=−69.1150388295625x_{68} = -69.1150388295625
x69=−21.9911796982577x_{69} = -21.9911796982577
x70=53.4089756107496x_{70} = 53.4089756107496
x71=−53.4080340967238x_{71} = -53.4080340967238
x72=−69.1150357551256x_{72} = -69.1150357551256
x73=21.9911796938987x_{73} = 21.9911796938987
x74=84.8213388084006x_{74} = 84.8213388084006
x75=81.68140841851x_{75} = 81.68140841851
x76=−84.8261831073356x_{76} = -84.8261831073356
x77=−100.530964509946x_{77} = -100.530964509946
x78=47.1270880348955x_{78} = 47.1270880348955
x79=18.849558891133x_{79} = 18.849558891133
x80=−28.2734476034419x_{80} = -28.2734476034419
x81=15.7088922765864x_{81} = 15.7088922765864
x82=−40.8376695273069x_{82} = -40.8376695273069
x83=40.8389774408875x_{83} = 40.8389774408875
x84=−31.4159219217159x_{84} = -31.4159219217159
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (sin(x) + tan(x))/(((2*x)*cot(x))).
sin⁡(0)+tan⁡(0)0⋅2cot⁡(0)\frac{\sin{\left(0 \right)} + \tan{\left(0 \right)}}{0 \cdot 2 \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
12xcot⁡(x)(cos⁡(x)+tan⁡2(x)+1)+(−2x(−cot⁡2(x)−1)−2cot⁡(x))(sin⁡(x)+tan⁡(x))4x2cot⁡2(x)=0\frac{1}{2 x \cot{\left(x \right)}} \left(\cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right) + \frac{\left(- 2 x \left(- \cot^{2}{\left(x \right)} - 1\right) - 2 \cot{\left(x \right)}\right) \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)}{4 x^{2} \cot^{2}{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=−97.3894633954265x_{1} = -97.3894633954265
x2=81.6814089933346x_{2} = 81.6814089933346
x3=−43.9822971502571x_{3} = -43.9822971502571
x4=−100.530964914873x_{4} = -100.530964914873
x5=53.4072667518135x_{5} = 53.4072667518135
x6=−21.99115164051x_{6} = -21.99115164051
x7=−47.1240716887076x_{7} = -47.1240716887076
x8=−12.5663706143592x_{8} = -12.5663706143592
x9=−31.4159265358979x_{9} = -31.4159265358979
x10=59.6903504740077x_{10} = 59.6903504740077
x11=56.5486677646163x_{11} = 56.5486677646163
x12=84.8228398927363x_{12} = 84.8228398927363
x13=−34.5573403476478x_{13} = -34.5573403476478
x14=−56.5486677646163x_{14} = -56.5486677646163
x15=12.5663706143592x_{15} = 12.5663706143592
x16=−59.6902758155797x_{16} = -59.6902758155797
x17=−91.1063775106345x_{17} = -91.1063775106345
x18=43.9822971502571x_{18} = 43.9822971502571
x19=100.530964914873x_{19} = 100.530964914873
x20=21.991151641644x_{20} = 21.991151641644
x21=−84.8226545167245x_{21} = -84.8226545167245
x22=72.2566292955242x_{22} = 72.2566292955242
x23=−78.5396461911396x_{23} = -78.5396461911396
x24=−3.14175093489092x_{24} = -3.14175093489092
x25=−53.4071612202693x_{25} = -53.4071612202693
x26=6.28318530717959x_{26} = 6.28318530717959
x27=91.1066053663186x_{27} = 91.1066053663186
x28=−84.8233930037709x_{28} = -84.8233930037709
x29=−87.9645943005142x_{29} = -87.9645943005142
x30=69.1150383789755x_{30} = 69.1150383789755
x31=87.9645943005142x_{31} = 87.9645943005142
x32=18.8495559215388x_{32} = 18.8495559215388
x33=−40.8403379519905x_{33} = -40.8403379519905
x34=−9.42485781925051x_{34} = -9.42485781925051
x35=−40.8410747688973x_{35} = -40.8410747688973
x36=−28.2742529777451x_{36} = -28.2742529777451
x37=37.6991118430775x_{37} = 37.6991118430775
x38=25.1327412287183x_{38} = 25.1327412287183
x39=50.2654824574367x_{39} = 50.2654824574367
x40=65.97345482794x_{40} = 65.97345482794
x41=−6.28318530717959x_{41} = -6.28318530717959
x42=91.1058606234583x_{42} = 91.1058606234583
x43=−62.8318530717959x_{43} = -62.8318530717959
x44=75.398223686155x_{44} = 75.398223686155
x45=−25.1327412287183x_{45} = -25.1327412287183
x46=15.7080473943606x_{46} = 15.7080473943606
x47=47.1235460529235x_{47} = 47.1235460529235
x48=−69.1150383789755x_{48} = -69.1150383789755
x49=−15.7079741591662x_{49} = -15.7079741591662
x50=34.5574417217695x_{50} = 34.5574417217695
x51=94.2477796076938x_{51} = 94.2477796076938
x52=−18.8495559215388x_{52} = -18.8495559215388
x53=−50.2654824574367x_{53} = -50.2654824574367
x54=9.42495587951317x_{54} = 9.42495587951317
x55=78.5397437779814x_{55} = 78.5397437779814
x56=−37.6991118430775x_{56} = -37.6991118430775
x57=28.274327526925x_{57} = 28.274327526925
x58=−81.6814089933346x_{58} = -81.6814089933346
x59=62.8318530717959x_{59} = 62.8318530717959
x60=31.4159265358979x_{60} = 31.4159265358979
x61=97.389573015133x_{61} = 97.389573015133
x62=−65.9734547229338x_{62} = -65.9734547229338
x63=−75.398223686155x_{63} = -75.398223686155
x64=40.8405347827156x_{64} = 40.8405347827156
x65=−94.2477796076938x_{65} = -94.2477796076938
x66=−72.2565554808085x_{66} = -72.2565554808085
The values of the extrema at the points:
(-97.38946339542655, -1.77073058979625e-19)

(81.68140899333463, 1.88255223925939e-31)

(-43.982297150257104, -6.68343712010972e-32)

(-100.53096491487338, -1.52764277031079e-31)

(53.40726675181354, 6.3138091250563e-18)

(-21.991151640509973, -1.00376583622348e-24)

(-47.12407168870756, -5.806085316048e-18)

(-12.566370614359172, -1.90955346288849e-32)

(-31.41592653589793, -4.77388365722123e-32)

(59.690350474007744, 2.75475304481601e-19)

(56.548667764616276, 8.59299058299821e-32)

(84.82283989273634, 2.01766630804503e-18)

(-34.55734034764782, -7.40075829164777e-18)

(-56.548667764616276, -8.59299058299821e-32)

(12.566370614359172, 1.90955346288849e-32)

(-59.69027581557969, -2.35408833715755e-22)

(-91.10637751063449, -3.61815017326727e-18)

(43.982297150257104, 6.68343712010972e-32)

(100.53096491487338, 1.52764277031079e-31)

(21.991151641643985, 1.00525924381744e-24)

(-84.82265451672453, -4.27954673616227e-17)

(72.25662929552416, 3.15009033747855e-26)

(-78.53964619113955, -2.66786990270092e-18)

(-3.141750934890923, -4.99443894134616e-17)

(-53.40716122026933, -2.57359145580604e-19)

(6.283185307179586, 9.54776731444245e-33)

(91.10660536631859, 8.41022481934631e-17)

(-84.82339300377086, -6.91378946352496e-17)

(-87.96459430051421, -1.33668742402194e-31)

(69.11503837897546, 2.81541104661861e-31)

(87.96459430051421, 1.33668742402194e-31)

(18.84955592153876, 2.86433019433274e-32)

(-40.8403379519905, -1.10499008809096e-16)

(-9.424857819250505, -1.07882095276346e-18)

(-40.84107476889726, -1.15060815318006e-16)

(-28.274252977745054, -3.78827011247992e-19)

(37.69911184307752, 5.72866038866547e-32)

(25.132741228718345, 3.81910692577698e-32)

(50.26548245743669, 7.63821385155396e-32)

(65.97345482794003, 2.60149754277779e-23)

(-6.283185307179586, -9.54776731444245e-33)

(91.10586062345827, 3.11189662741211e-17)

(-62.83185307179586, -9.54776731444245e-32)

(75.39822368615503, 1.14573207773309e-31)

(-25.132741228718345, -3.81910692577698e-32)

(15.708047394360637, 7.97163235164562e-19)

(47.12354605292351, 7.40760077486107e-17)

(-69.11503837897546, -2.81541104661861e-31)

(-15.707974159166167, -2.23936882398876e-22)

(34.557441721769464, 2.60544241909504e-19)

(94.2477796076938, 1.24937720620631e-31)

(-18.84955592153876, -2.86433019433274e-32)

(-50.26548245743669, -7.63821385155396e-32)

(9.424955879513174, 2.65795427218555e-17)

(78.53974377798144, 8.82433229843939e-20)

(-37.69911184307752, -5.72866038866547e-32)

(28.274327526924953, 1.44249900139147e-23)

(-81.68140899333463, -1.88255223925939e-31)

(62.83185307179586, 9.54776731444245e-32)

(31.41592653589793, 4.77388365722123e-32)

(97.38957301513302, 4.169491411353e-18)

(-65.97345472293381, -2.48351882236391e-23)

(-75.39822368615503, -1.14573207773309e-31)

(40.840534782715636, 5.07830496843494e-18)

(-94.2477796076938, -1.24937720620631e-31)

(-72.25655548080847, -1.1273039336493e-19)


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=81.6814089933346x_{1} = 81.6814089933346
x2=56.5486677646163x_{2} = 56.5486677646163
x3=12.5663706143592x_{3} = 12.5663706143592
x4=43.9822971502571x_{4} = 43.9822971502571
x5=100.530964914873x_{5} = 100.530964914873
x6=6.28318530717959x_{6} = 6.28318530717959
x7=69.1150383789755x_{7} = 69.1150383789755
x8=87.9645943005142x_{8} = 87.9645943005142
x9=18.8495559215388x_{9} = 18.8495559215388
x10=37.6991118430775x_{10} = 37.6991118430775
x11=25.1327412287183x_{11} = 25.1327412287183
x12=50.2654824574367x_{12} = 50.2654824574367
x13=75.398223686155x_{13} = 75.398223686155
x14=94.2477796076938x_{14} = 94.2477796076938
x15=62.8318530717959x_{15} = 62.8318530717959
x16=31.4159265358979x_{16} = 31.4159265358979
Maxima of the function at points:
x16=−43.9822971502571x_{16} = -43.9822971502571
x16=−100.530964914873x_{16} = -100.530964914873
x16=−12.5663706143592x_{16} = -12.5663706143592
x16=−31.4159265358979x_{16} = -31.4159265358979
x16=−56.5486677646163x_{16} = -56.5486677646163
x16=−87.9645943005142x_{16} = -87.9645943005142
x16=−6.28318530717959x_{16} = -6.28318530717959
x16=−62.8318530717959x_{16} = -62.8318530717959
x16=−25.1327412287183x_{16} = -25.1327412287183
x16=−69.1150383789755x_{16} = -69.1150383789755
x16=−18.8495559215388x_{16} = -18.8495559215388
x16=−50.2654824574367x_{16} = -50.2654824574367
x16=−37.6991118430775x_{16} = -37.6991118430775
x16=−81.6814089933346x_{16} = -81.6814089933346
x16=−75.398223686155x_{16} = -75.398223686155
x16=−94.2477796076938x_{16} = -94.2477796076938
Decreasing at intervals
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Increasing at intervals
[−6.28318530717959,6.28318530717959]\left[-6.28318530717959, 6.28318530717959\right]
Vertical asymptotes
Have:
x1=0x_{1} = 0
x2=1.5707963267949x_{2} = 1.5707963267949
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)+tan⁡(x)2xcot⁡(x))y = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x \cot{\left(x \right)}}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(sin⁡(x)+tan⁡(x)2xcot⁡(x))y = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x \cot{\left(x \right)}}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (sin(x) + tan(x))/(((2*x)*cot(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→−∞(12xcot⁡(x)(sin⁡(x)+tan⁡(x))x)y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{2 x \cot{\left(x \right)}} \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(12xcot⁡(x)(sin⁡(x)+tan⁡(x))x)y = x \lim_{x \to \infty}\left(\frac{\frac{1}{2 x \cot{\left(x \right)}} \left(\sin{\left(x \right)} + \tan{\left(x \right)}\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)+tan⁡(x)2xcot⁡(x)=−sin⁡(x)−tan⁡(x)2xcot⁡(x)\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x \cot{\left(x \right)}} = \frac{- \sin{\left(x \right)} - \tan{\left(x \right)}}{2 x \cot{\left(x \right)}}
- No
sin⁡(x)+tan⁡(x)2xcot⁡(x)=−−sin⁡(x)−tan⁡(x)2xcot⁡(x)\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x \cot{\left(x \right)}} = - \frac{- \sin{\left(x \right)} - \tan{\left(x \right)}}{2 x \cot{\left(x \right)}}
- No
so, the function
not is
neither even, nor odd
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